3.2223 \(\int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx\)

Optimal. Leaf size=412 \[ -\frac {5 (b d-a e)^6 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{1024 b^{9/2} e^{9/2}}+\frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^5 (2 A b e-B (a e+b d))}{1024 b^4 e^4}-\frac {5 (a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^4 (2 A b e-B (a e+b d))}{1536 b^4 e^3}+\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{384 b^4 e^2}+\frac {(a+b x)^{7/2} \sqrt {d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^4 e}+\frac {(a+b x)^{7/2} (d+e x)^{3/2} (b d-a e) (2 A b e-B (a e+b d))}{24 b^3 e}+\frac {(a+b x)^{7/2} (d+e x)^{5/2} (2 A b e-B (a e+b d))}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e} \]

[Out]

1/24*(-a*e+b*d)*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(7/2)*(e*x+d)^(3/2)/b^3/e+1/12*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(7/
2)*(e*x+d)^(5/2)/b^2/e+1/7*B*(b*x+a)^(7/2)*(e*x+d)^(7/2)/b/e-5/1024*(-a*e+b*d)^6*(2*A*b*e-B*(a*e+b*d))*arctanh
(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(9/2)/e^(9/2)-5/1536*(-a*e+b*d)^4*(2*A*b*e-B*(a*e+b*d))*(b*x+a
)^(3/2)*(e*x+d)^(1/2)/b^4/e^3+1/384*(-a*e+b*d)^3*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(5/2)*(e*x+d)^(1/2)/b^4/e^2+1/6
4*(-a*e+b*d)^2*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(7/2)*(e*x+d)^(1/2)/b^4/e+5/1024*(-a*e+b*d)^5*(2*A*b*e-B*(a*e+b*d
))*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^4/e^4

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Rubi [A]  time = 0.38, antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {80, 50, 63, 217, 206} \[ \frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^5 (2 A b e-B (a e+b d))}{1024 b^4 e^4}-\frac {5 (a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^4 (2 A b e-B (a e+b d))}{1536 b^4 e^3}+\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{384 b^4 e^2}-\frac {5 (b d-a e)^6 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{1024 b^{9/2} e^{9/2}}+\frac {(a+b x)^{7/2} \sqrt {d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^4 e}+\frac {(a+b x)^{7/2} (d+e x)^{3/2} (b d-a e) (2 A b e-B (a e+b d))}{24 b^3 e}+\frac {(a+b x)^{7/2} (d+e x)^{5/2} (2 A b e-B (a e+b d))}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/2)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(5*(b*d - a*e)^5*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(1024*b^4*e^4) - (5*(b*d - a*e)^4*(2*A
*b*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x])/(1536*b^4*e^3) + ((b*d - a*e)^3*(2*A*b*e - B*(b*d + a*e))
*(a + b*x)^(5/2)*Sqrt[d + e*x])/(384*b^4*e^2) + ((b*d - a*e)^2*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(7/2)*Sqrt[
d + e*x])/(64*b^4*e) + ((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(24*b^3*e) + ((
2*A*b*e - B*(b*d + a*e))*(a + b*x)^(7/2)*(d + e*x)^(5/2))/(12*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(7/2))/(7*
b*e) - (5*(b*d - a*e)^6*(2*A*b*e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(1
024*b^(9/2)*e^(9/2))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx &=\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}+\frac {\left (7 A b e-B \left (\frac {7 b d}{2}+\frac {7 a e}{2}\right )\right ) \int (a+b x)^{5/2} (d+e x)^{5/2} \, dx}{7 b e}\\ &=\frac {(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}+\frac {\left (5 (b d-a e) \left (7 A b e-B \left (\frac {7 b d}{2}+\frac {7 a e}{2}\right )\right )\right ) \int (a+b x)^{5/2} (d+e x)^{3/2} \, dx}{84 b^2 e}\\ &=\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}+\frac {\left ((b d-a e)^2 \left (7 A b e-B \left (\frac {7 b d}{2}+\frac {7 a e}{2}\right )\right )\right ) \int (a+b x)^{5/2} \sqrt {d+e x} \, dx}{56 b^3 e}\\ &=\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt {d+e x}}{64 b^4 e}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}+\frac {\left ((b d-a e)^3 \left (7 A b e-B \left (\frac {7 b d}{2}+\frac {7 a e}{2}\right )\right )\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {d+e x}} \, dx}{448 b^4 e}\\ &=\frac {(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{384 b^4 e^2}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt {d+e x}}{64 b^4 e}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac {\left (5 (b d-a e)^4 \left (7 A b e-B \left (\frac {7 b d}{2}+\frac {7 a e}{2}\right )\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{2688 b^4 e^2}\\ &=-\frac {5 (b d-a e)^4 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{1536 b^4 e^3}+\frac {(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{384 b^4 e^2}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt {d+e x}}{64 b^4 e}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}+\frac {\left (5 (b d-a e)^5 \left (7 A b e-B \left (\frac {7 b d}{2}+\frac {7 a e}{2}\right )\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{3584 b^4 e^3}\\ &=\frac {5 (b d-a e)^5 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{1024 b^4 e^4}-\frac {5 (b d-a e)^4 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{1536 b^4 e^3}+\frac {(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{384 b^4 e^2}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt {d+e x}}{64 b^4 e}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac {\left (5 (b d-a e)^6 \left (7 A b e-B \left (\frac {7 b d}{2}+\frac {7 a e}{2}\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{7168 b^4 e^4}\\ &=\frac {5 (b d-a e)^5 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{1024 b^4 e^4}-\frac {5 (b d-a e)^4 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{1536 b^4 e^3}+\frac {(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{384 b^4 e^2}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt {d+e x}}{64 b^4 e}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac {\left (5 (b d-a e)^6 \left (7 A b e-B \left (\frac {7 b d}{2}+\frac {7 a e}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{3584 b^5 e^4}\\ &=\frac {5 (b d-a e)^5 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{1024 b^4 e^4}-\frac {5 (b d-a e)^4 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{1536 b^4 e^3}+\frac {(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{384 b^4 e^2}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt {d+e x}}{64 b^4 e}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac {\left (5 (b d-a e)^6 \left (7 A b e-B \left (\frac {7 b d}{2}+\frac {7 a e}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{3584 b^5 e^4}\\ &=\frac {5 (b d-a e)^5 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{1024 b^4 e^4}-\frac {5 (b d-a e)^4 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{1536 b^4 e^3}+\frac {(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{384 b^4 e^2}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt {d+e x}}{64 b^4 e}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac {5 (b d-a e)^6 (2 A b e-B (b d+a e)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{1024 b^{9/2} e^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 5.73, size = 388, normalized size = 0.94 \[ \frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac {\sqrt {b d-a e} \left (\frac {b (d+e x)}{b d-a e}\right )^{3/2} (a B e-2 A b e+b B d) \left (16 e^{7/2} (a+b x)^4 (b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}} \left (3 a^2 e^2-2 a b e (7 d+4 e x)+b^2 \left (27 d^2+40 d e x+16 e^2 x^2\right )\right )-10 e^{3/2} (a+b x)^2 (b d-a e)^{11/2} \sqrt {\frac {b (d+e x)}{b d-a e}}+8 e^{5/2} (a+b x)^3 (b d-a e)^{9/2} \sqrt {\frac {b (d+e x)}{b d-a e}}+15 \sqrt {e} (a+b x) (b d-a e)^{13/2} \sqrt {\frac {b (d+e x)}{b d-a e}}-15 \sqrt {a+b x} (b d-a e)^7 \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )\right )}{3072 b^6 e^{9/2} \sqrt {a+b x} (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/2)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(B*(a + b*x)^(7/2)*(d + e*x)^(7/2))/(7*b*e) - (Sqrt[b*d - a*e]*(b*B*d - 2*A*b*e + a*B*e)*((b*(d + e*x))/(b*d -
a*e))^(3/2)*(15*Sqrt[e]*(b*d - a*e)^(13/2)*(a + b*x)*Sqrt[(b*(d + e*x))/(b*d - a*e)] - 10*e^(3/2)*(b*d - a*e)
^(11/2)*(a + b*x)^2*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 8*e^(5/2)*(b*d - a*e)^(9/2)*(a + b*x)^3*Sqrt[(b*(d + e*x
))/(b*d - a*e)] + 16*e^(7/2)*(b*d - a*e)^(3/2)*(a + b*x)^4*Sqrt[(b*(d + e*x))/(b*d - a*e)]*(3*a^2*e^2 - 2*a*b*
e*(7*d + 4*e*x) + b^2*(27*d^2 + 40*d*e*x + 16*e^2*x^2)) - 15*(b*d - a*e)^7*Sqrt[a + b*x]*ArcSinh[(Sqrt[e]*Sqrt
[a + b*x])/Sqrt[b*d - a*e]]))/(3072*b^6*e^(9/2)*Sqrt[a + b*x]*(d + e*x)^(3/2))

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fricas [B]  time = 1.71, size = 1758, normalized size = 4.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)*(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

[1/86016*(105*(B*b^7*d^7 - (5*B*a*b^6 + 2*A*b^7)*d^6*e + 3*(3*B*a^2*b^5 + 4*A*a*b^6)*d^5*e^2 - 5*(B*a^3*b^4 +
6*A*a^2*b^5)*d^4*e^3 - 5*(B*a^4*b^3 - 8*A*a^3*b^4)*d^3*e^4 + 3*(3*B*a^5*b^2 - 10*A*a^4*b^3)*d^2*e^5 - (5*B*a^6
*b - 12*A*a^5*b^2)*d*e^6 + (B*a^7 - 2*A*a^6*b)*e^7)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^
2 + 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) + 4*(3072*B*b^7*e
^7*x^6 - 105*B*b^7*d^6*e + 70*(7*B*a*b^6 + 3*A*b^7)*d^5*e^2 - 7*(113*B*a^2*b^5 + 170*A*a*b^6)*d^4*e^3 + 12*(25
*B*a^3*b^4 + 231*A*a^2*b^5)*d^3*e^4 - 7*(113*B*a^4*b^3 - 396*A*a^3*b^4)*d^2*e^5 + 70*(7*B*a^5*b^2 - 17*A*a^4*b
^3)*d*e^6 - 105*(B*a^6*b - 2*A*a^5*b^2)*e^7 + 256*(29*B*b^7*d*e^6 + (29*B*a*b^6 + 14*A*b^7)*e^7)*x^5 + 128*(37
*B*b^7*d^2*e^5 + 2*(73*B*a*b^6 + 35*A*b^7)*d*e^6 + (37*B*a^2*b^5 + 70*A*a*b^6)*e^7)*x^4 + 16*(3*B*b^7*d^3*e^4
+ (797*B*a*b^6 + 378*A*b^7)*d^2*e^5 + (797*B*a^2*b^5 + 1484*A*a*b^6)*d*e^6 + 3*(B*a^3*b^4 + 126*A*a^2*b^5)*e^7
)*x^3 - 8*(7*B*b^7*d^4*e^3 - 2*(16*B*a*b^6 + 7*A*b^7)*d^3*e^4 - 6*(205*B*a^2*b^5 + 371*A*a*b^6)*d^2*e^5 - 2*(1
6*B*a^3*b^4 + 1113*A*a^2*b^5)*d*e^6 + 7*(B*a^4*b^3 - 2*A*a^3*b^4)*e^7)*x^2 + 2*(35*B*b^7*d^5*e^2 - 7*(23*B*a*b
^6 + 10*A*b^7)*d^4*e^3 + 2*(127*B*a^2*b^5 + 196*A*a*b^6)*d^3*e^4 + 2*(127*B*a^3*b^4 + 4158*A*a^2*b^5)*d^2*e^5
- 7*(23*B*a^4*b^3 - 56*A*a^3*b^4)*d*e^6 + 35*(B*a^5*b^2 - 2*A*a^4*b^3)*e^7)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b
^5*e^5), -1/43008*(105*(B*b^7*d^7 - (5*B*a*b^6 + 2*A*b^7)*d^6*e + 3*(3*B*a^2*b^5 + 4*A*a*b^6)*d^5*e^2 - 5*(B*a
^3*b^4 + 6*A*a^2*b^5)*d^4*e^3 - 5*(B*a^4*b^3 - 8*A*a^3*b^4)*d^3*e^4 + 3*(3*B*a^5*b^2 - 10*A*a^4*b^3)*d^2*e^5 -
(5*B*a^6*b - 12*A*a^5*b^2)*d*e^6 + (B*a^7 - 2*A*a^6*b)*e^7)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(
-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) - 2*(3072*B*b^7*e^7*x^6 - 1
05*B*b^7*d^6*e + 70*(7*B*a*b^6 + 3*A*b^7)*d^5*e^2 - 7*(113*B*a^2*b^5 + 170*A*a*b^6)*d^4*e^3 + 12*(25*B*a^3*b^4
+ 231*A*a^2*b^5)*d^3*e^4 - 7*(113*B*a^4*b^3 - 396*A*a^3*b^4)*d^2*e^5 + 70*(7*B*a^5*b^2 - 17*A*a^4*b^3)*d*e^6
- 105*(B*a^6*b - 2*A*a^5*b^2)*e^7 + 256*(29*B*b^7*d*e^6 + (29*B*a*b^6 + 14*A*b^7)*e^7)*x^5 + 128*(37*B*b^7*d^2
*e^5 + 2*(73*B*a*b^6 + 35*A*b^7)*d*e^6 + (37*B*a^2*b^5 + 70*A*a*b^6)*e^7)*x^4 + 16*(3*B*b^7*d^3*e^4 + (797*B*a
*b^6 + 378*A*b^7)*d^2*e^5 + (797*B*a^2*b^5 + 1484*A*a*b^6)*d*e^6 + 3*(B*a^3*b^4 + 126*A*a^2*b^5)*e^7)*x^3 - 8*
(7*B*b^7*d^4*e^3 - 2*(16*B*a*b^6 + 7*A*b^7)*d^3*e^4 - 6*(205*B*a^2*b^5 + 371*A*a*b^6)*d^2*e^5 - 2*(16*B*a^3*b^
4 + 1113*A*a^2*b^5)*d*e^6 + 7*(B*a^4*b^3 - 2*A*a^3*b^4)*e^7)*x^2 + 2*(35*B*b^7*d^5*e^2 - 7*(23*B*a*b^6 + 10*A*
b^7)*d^4*e^3 + 2*(127*B*a^2*b^5 + 196*A*a*b^6)*d^3*e^4 + 2*(127*B*a^3*b^4 + 4158*A*a^2*b^5)*d^2*e^5 - 7*(23*B*
a^4*b^3 - 56*A*a^3*b^4)*d*e^6 + 35*(B*a^5*b^2 - 2*A*a^4*b^3)*e^7)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e^5)]

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giac [B]  time = 8.22, size = 6900, normalized size = 16.75 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)*(e*x+d)^(5/2),x, algorithm="giac")

[Out]

1/107520*(13440*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3
- 13*a*b^5*e^4)*e^(-4)/b^7) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^
2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*e^(-5/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*
b*e - a*b*e)))/b^(3/2))*A*a*d^2*abs(b) + 1680*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(
6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)*e^(-6)/b^14) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11
*e^6)*e^(-6)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)*e^(-6)/b^14)*
sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*e^(-7/2)*log(a
bs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*B*a*d^2*abs(b) - 107520*((b
^2*d - a*b*e)*e^(-1/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b)
- sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a))*A*a^3*d^2*abs(b)/b^2 + 13440*(sqrt(b^2*d + (b*x + a)*b*e
- a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)*e^(-4)/b^7) - 3*(b^7*d^2*e^2
+ 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*e^(-5/2
)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*B*a^2*d^2*abs(b)/b +
560*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11
*e^6)*e^(-6)/b^14) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)*e^(-6)/b^14) + 3*(5*b^14*d^3*e^3 +
9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)*e^(-6)/b^14)*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^
3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(
b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*A*b*d^2*abs(b) + 56*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(b*x
+ a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*d*e^7 - 41*a*b^19*e^8)*e^(-8)/b^23) - (7*b^21*d^2*e^6 + 26*a*b^20*d
*e^7 - 513*a^2*b^19*e^8)*e^(-8)/b^23) + 5*(7*b^22*d^3*e^5 + 19*a*b^21*d^2*e^6 + 37*a^2*b^20*d*e^7 - 447*a^3*b^
19*e^8)*e^(-8)/b^23)*(b*x + a) - 15*(7*b^23*d^4*e^4 + 12*a*b^22*d^3*e^5 + 18*a^2*b^21*d^2*e^6 + 28*a^3*b^20*d*
e^7 - 193*a^4*b^19*e^8)*e^(-8)/b^23)*sqrt(b*x + a) - 15*(7*b^5*d^5 + 5*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 + 10*a^
3*b^2*d^2*e^3 + 35*a^4*b*d*e^4 - 63*a^5*e^5)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b
*x + a)*b*e - a*b*e)))/b^(7/2))*B*b*d^2*abs(b) + 3360*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*
x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)*e^(-6)/b^14) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*
a^2*b^11*e^6)*e^(-6)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)*e^(-6
)/b^14)*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*e^(-7/
2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*A*a*d*abs(b)*e + 33
6*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*d*e^7 - 41*a*b^19
*e^8)*e^(-8)/b^23) - (7*b^21*d^2*e^6 + 26*a*b^20*d*e^7 - 513*a^2*b^19*e^8)*e^(-8)/b^23) + 5*(7*b^22*d^3*e^5 +
19*a*b^21*d^2*e^6 + 37*a^2*b^20*d*e^7 - 447*a^3*b^19*e^8)*e^(-8)/b^23)*(b*x + a) - 15*(7*b^23*d^4*e^4 + 12*a*b
^22*d^3*e^5 + 18*a^2*b^21*d^2*e^6 + 28*a^3*b^20*d*e^7 - 193*a^4*b^19*e^8)*e^(-8)/b^23)*sqrt(b*x + a) - 15*(7*b
^5*d^5 + 5*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 35*a^4*b*d*e^4 - 63*a^5*e^5)*e^(-9/2)*log(ab
s(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2))*B*a*d*abs(b)*e + 8960*(sqrt(
b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)*e^(-4)
/b^7) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^
2 - 5*a^3*e^3)*e^(-5/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2)
)*B*a^3*d*abs(b)*e/b^2 + 26880*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^
2 + (b^6*d*e^3 - 13*a*b^5*e^4)*e^(-4)/b^7) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/b^7) - 3*
(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*e^(-5/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2
*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*A*a^2*d*abs(b)*e/b + 3360*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x
+ a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)*e^(-6)/b^14) - (5*b^13*d^2*e^4 + 14*a*b^12*d
*e^5 - 163*a^2*b^11*e^6)*e^(-6)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11
*e^6)*e^(-6)/b^14)*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*
e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*B*a^2*d*
abs(b)*e/b + 112*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*d*
e^7 - 41*a*b^19*e^8)*e^(-8)/b^23) - (7*b^21*d^2*e^6 + 26*a*b^20*d*e^7 - 513*a^2*b^19*e^8)*e^(-8)/b^23) + 5*(7*
b^22*d^3*e^5 + 19*a*b^21*d^2*e^6 + 37*a^2*b^20*d*e^7 - 447*a^3*b^19*e^8)*e^(-8)/b^23)*(b*x + a) - 15*(7*b^23*d
^4*e^4 + 12*a*b^22*d^3*e^5 + 18*a^2*b^21*d^2*e^6 + 28*a^3*b^20*d*e^7 - 193*a^4*b^19*e^8)*e^(-8)/b^23)*sqrt(b*x
+ a) - 15*(7*b^5*d^5 + 5*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 35*a^4*b*d*e^4 - 63*a^5*e^5)*
e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2))*A*b*d*abs(b)*
e + 28*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^5 + (b^30*d*e^9 -
61*a*b^29*e^10)*e^(-10)/b^34) - 3*(3*b^31*d^2*e^8 + 14*a*b^30*d*e^9 - 417*a^2*b^29*e^10)*e^(-10)/b^34) + (21*b
^32*d^3*e^7 + 77*a*b^31*d^2*e^8 + 183*a^2*b^30*d*e^9 - 3481*a^3*b^29*e^10)*e^(-10)/b^34)*(b*x + a) - 5*(21*b^3
3*d^4*e^6 + 56*a*b^32*d^3*e^7 + 106*a^2*b^31*d^2*e^8 + 176*a^3*b^30*d*e^9 - 2279*a^4*b^29*e^10)*e^(-10)/b^34)*
(b*x + a) + 15*(21*b^34*d^5*e^5 + 35*a*b^33*d^4*e^6 + 50*a^2*b^32*d^3*e^7 + 70*a^3*b^31*d^2*e^8 + 105*a^4*b^30
*d*e^9 - 793*a^5*b^29*e^10)*e^(-10)/b^34)*sqrt(b*x + a) + 15*(21*b^6*d^6 + 14*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2
+ 20*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 - 231*a^6*e^6)*e^(-11/2)*log(abs(-sqrt(b*x + a)*s
qrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(9/2))*B*b*d*abs(b)*e + 26880*((b^3*d^2 + 2*a*b^2*d*e
- 3*a^2*b*e^2)*e^(-3/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b
) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + (b*d*e - 5*a*e^2)*e^(-2) + 2*a)*sqrt(b*x + a))*B*a^3*d^2*abs(
b)/b^3 + 80640*((b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*e^(-3/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b
^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + (b*d*e - 5*a*e^2)*e^(-2
) + 2*a)*sqrt(b*x + a))*A*a^2*d^2*abs(b)/b^2 + 168*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(b*x + a)*(6*(b*
x + a)*(8*(b*x + a)/b^4 + (b^20*d*e^7 - 41*a*b^19*e^8)*e^(-8)/b^23) - (7*b^21*d^2*e^6 + 26*a*b^20*d*e^7 - 513*
a^2*b^19*e^8)*e^(-8)/b^23) + 5*(7*b^22*d^3*e^5 + 19*a*b^21*d^2*e^6 + 37*a^2*b^20*d*e^7 - 447*a^3*b^19*e^8)*e^(
-8)/b^23)*(b*x + a) - 15*(7*b^23*d^4*e^4 + 12*a*b^22*d^3*e^5 + 18*a^2*b^21*d^2*e^6 + 28*a^3*b^20*d*e^7 - 193*a
^4*b^19*e^8)*e^(-8)/b^23)*sqrt(b*x + a) - 15*(7*b^5*d^5 + 5*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e
^3 + 35*a^4*b*d*e^4 - 63*a^5*e^5)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e
- a*b*e)))/b^(7/2))*A*a*abs(b)*e^2 + 42*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*
(10*(b*x + a)/b^5 + (b^30*d*e^9 - 61*a*b^29*e^10)*e^(-10)/b^34) - 3*(3*b^31*d^2*e^8 + 14*a*b^30*d*e^9 - 417*a^
2*b^29*e^10)*e^(-10)/b^34) + (21*b^32*d^3*e^7 + 77*a*b^31*d^2*e^8 + 183*a^2*b^30*d*e^9 - 3481*a^3*b^29*e^10)*e
^(-10)/b^34)*(b*x + a) - 5*(21*b^33*d^4*e^6 + 56*a*b^32*d^3*e^7 + 106*a^2*b^31*d^2*e^8 + 176*a^3*b^30*d*e^9 -
2279*a^4*b^29*e^10)*e^(-10)/b^34)*(b*x + a) + 15*(21*b^34*d^5*e^5 + 35*a*b^33*d^4*e^6 + 50*a^2*b^32*d^3*e^7 +
70*a^3*b^31*d^2*e^8 + 105*a^4*b^30*d*e^9 - 793*a^5*b^29*e^10)*e^(-10)/b^34)*sqrt(b*x + a) + 15*(21*b^6*d^6 + 1
4*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 20*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 - 231*a^6*e^6)*
e^(-11/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(9/2))*B*a*abs(b)*e
^2 + 4480*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a
*b^5*e^4)*e^(-4)/b^7) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^2*d^2*
e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*e^(-5/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e -
a*b*e)))/b^(3/2))*A*a^3*abs(b)*e^2/b^2 + 560*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6
*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)*e^(-6)/b^14) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*
e^6)*e^(-6)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)*e^(-6)/b^14)*s
qrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*e^(-7/2)*log(ab
s(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*B*a^3*abs(b)*e^2/b^2 + 1680*
(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)
*e^(-6)/b^14) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)*e^(-6)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b
^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)*e^(-6)/b^14)*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e +
6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d
+ (b*x + a)*b*e - a*b*e)))/b^(5/2))*A*a^2*abs(b)*e^2/b + 168*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(b*x
+ a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*d*e^7 - 41*a*b^19*e^8)*e^(-8)/b^23) - (7*b^21*d^2*e^6 + 26*a*b^20*d
*e^7 - 513*a^2*b^19*e^8)*e^(-8)/b^23) + 5*(7*b^22*d^3*e^5 + 19*a*b^21*d^2*e^6 + 37*a^2*b^20*d*e^7 - 447*a^3*b^
19*e^8)*e^(-8)/b^23)*(b*x + a) - 15*(7*b^23*d^4*e^4 + 12*a*b^22*d^3*e^5 + 18*a^2*b^21*d^2*e^6 + 28*a^3*b^20*d*
e^7 - 193*a^4*b^19*e^8)*e^(-8)/b^23)*sqrt(b*x + a) - 15*(7*b^5*d^5 + 5*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 + 10*a^
3*b^2*d^2*e^3 + 35*a^4*b*d*e^4 - 63*a^5*e^5)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b
*x + a)*b*e - a*b*e)))/b^(7/2))*B*a^2*abs(b)*e^2/b + 14*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(2*(b*x + a
)*(8*(b*x + a)*(10*(b*x + a)/b^5 + (b^30*d*e^9 - 61*a*b^29*e^10)*e^(-10)/b^34) - 3*(3*b^31*d^2*e^8 + 14*a*b^30
*d*e^9 - 417*a^2*b^29*e^10)*e^(-10)/b^34) + (21*b^32*d^3*e^7 + 77*a*b^31*d^2*e^8 + 183*a^2*b^30*d*e^9 - 3481*a
^3*b^29*e^10)*e^(-10)/b^34)*(b*x + a) - 5*(21*b^33*d^4*e^6 + 56*a*b^32*d^3*e^7 + 106*a^2*b^31*d^2*e^8 + 176*a^
3*b^30*d*e^9 - 2279*a^4*b^29*e^10)*e^(-10)/b^34)*(b*x + a) + 15*(21*b^34*d^5*e^5 + 35*a*b^33*d^4*e^6 + 50*a^2*
b^32*d^3*e^7 + 70*a^3*b^31*d^2*e^8 + 105*a^4*b^30*d*e^9 - 793*a^5*b^29*e^10)*e^(-10)/b^34)*sqrt(b*x + a) + 15*
(21*b^6*d^6 + 14*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 20*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5
- 231*a^6*e^6)*e^(-11/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(9/2
))*A*b*abs(b)*e^2 + (sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(2*(8*(b*x + a)*(10*(b*x + a)*(12*(b*x + a)/b^6
+ (b^42*d*e^11 - 85*a*b^41*e^12)*e^(-12)/b^47) - (11*b^43*d^2*e^10 + 62*a*b^42*d*e^11 - 2593*a^2*b^41*e^12)*e
^(-12)/b^47) + 3*(33*b^44*d^3*e^9 + 153*a*b^43*d^2*e^10 + 435*a^2*b^42*d*e^11 - 11821*a^3*b^41*e^12)*e^(-12)/b
^47)*(b*x + a) - 7*(33*b^45*d^4*e^8 + 120*a*b^44*d^3*e^9 + 282*a^2*b^43*d^2*e^10 + 544*a^3*b^42*d*e^11 - 10579
*a^4*b^41*e^12)*e^(-12)/b^47)*(b*x + a) + 35*(33*b^46*d^5*e^7 + 87*a*b^45*d^4*e^8 + 162*a^2*b^44*d^3*e^9 + 262
*a^3*b^43*d^2*e^10 + 397*a^4*b^42*d*e^11 - 5549*a^5*b^41*e^12)*e^(-12)/b^47)*(b*x + a) - 105*(33*b^47*d^6*e^6
+ 54*a*b^46*d^5*e^7 + 75*a^2*b^45*d^4*e^8 + 100*a^3*b^44*d^3*e^9 + 135*a^4*b^43*d^2*e^10 + 198*a^5*b^42*d*e^11
- 1619*a^6*b^41*e^12)*e^(-12)/b^47)*sqrt(b*x + a) - 105*(33*b^7*d^7 + 21*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 + 2
5*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 + 63*a^5*b^2*d^2*e^5 + 231*a^6*b*d*e^6 - 429*a^7*e^7)*e^(-13/2)*log(abs
(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(11/2))*B*b*abs(b)*e^2 + 53760*((b^3
*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*e^(-3/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e
- a*b*e)))/sqrt(b) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + (b*d*e - 5*a*e^2)*e^(-2) + 2*a)*sqrt(b*x +
a))*A*a^3*d*abs(b)*e/b^3)/b

________________________________________________________________________________________

maple [B]  time = 0.03, size = 2851, normalized size = 6.92 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)*(e*x+d)^(5/2),x)

[Out]

-1/43008*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(525*B*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1
/2))/(b*e)^(1/2))*a*b^6*d^6*e-105*B*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*
e)^(1/2))*a^7*e^7-105*B*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*b^
7*d^7+210*A*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^6*b*e^7+210*
A*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*b^7*d^6*e+210*B*(b*e)^(1
/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^6*e^6+210*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*b^6*d^6-1016*B*(
b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^3*b^3*d^2*e^4-1016*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2
)*x*a^2*b^4*d^3*e^3+644*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a*b^5*d^4*e^2-33264*A*(b*e)^(1/2)*(b*e
*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*b^4*d^2*e^4-1568*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a*b^5*d^3*e
^3+644*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^4*b^2*d*e^5-1568*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a
*d)^(1/2)*x*a^3*b^3*d*e^5-512*B*x^2*a*b^5*d^3*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-19680*B*x^2*a^2*
b^4*d^2*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-37376*B*x^4*a*b^5*d*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2
)*(b*e)^(1/2)-47488*A*x^3*a*b^5*d*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-25504*B*x^3*a^2*b^4*d*e^5*(b
*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-25504*B*x^3*a*b^5*d^2*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2
)-35616*A*x^2*a^2*b^4*d*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-35616*A*x^2*a*b^5*d^2*e^4*(b*e*x^2+a*e
*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-512*B*x^2*a^3*b^3*d*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-5544*A*(b*
e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*b^4*d^3*e^3+2380*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*
b^5*d^4*e^2-14848*B*x^5*a*b^5*e^6*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-14848*B*x^5*b^6*d*e^5*(b*e*x^2+a
*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-17920*A*x^4*a*b^5*e^6*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-17920*A*x^
4*b^6*d*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-9472*B*x^4*a^2*b^4*e^6*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)
*(b*e)^(1/2)-9472*B*x^4*b^6*d^2*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-12096*A*x^3*a^2*b^4*e^6*(b*e*x
^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-12096*A*x^3*b^6*d^2*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-96*B
*x^3*a^3*b^3*e^6*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-96*B*x^3*b^6*d^3*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1
/2)*(b*e)^(1/2)-224*A*x^2*a^3*b^3*e^6*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-224*A*x^2*b^6*d^3*e^3*(b*e*x
^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+112*B*x^2*a^4*b^2*e^6*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+112*B*
x^2*b^6*d^4*e^2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-980*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*
a^5*b*d*e^5+1582*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^4*b^2*d^2*e^4-600*B*(b*e)^(1/2)*(b*e*x^2+a*e*
x+b*d*x+a*d)^(1/2)*a^3*b^3*d^3*e^3+280*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^4*b^2*e^6+1582*B*(b*e
)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*b^4*d^4*e^2-980*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*b^
5*d^5*e+525*B*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^4*b^3*d^3*
e^4+525*B*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^3*b^4*d^4*e^3-
945*B*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^5*b^2*d^2*e^5-945*
B*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^2*b^5*d^5*e^2-4200*A*l
n(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^3*b^4*d^3*e^4+3150*A*ln(1
/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^2*b^5*d^4*e^3-1260*A*ln(1/2*
(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a*b^6*d^5*e^2+525*B*ln(1/2*(2*b*e
*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^6*b*d*e^6-1260*A*ln(1/2*(2*b*e*x+a*e+
b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^5*b^2*d*e^6+3150*A*ln(1/2*(2*b*e*x+a*e+b*d+2
*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^4*b^3*d^2*e^5-420*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*
d*x+a*d)^(1/2)*a^5*b*e^6-420*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*b^6*d^5*e-6144*B*x^6*b^6*e^6*(b*e*x
^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-7168*A*x^5*b^6*e^6*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+280*A*(b*
e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*b^6*d^4*e^2-140*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^5
*b*e^6-140*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*b^6*d^5*e+2380*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a
*d)^(1/2)*a^4*b^2*d*e^5-5544*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^3*b^3*d^2*e^4)/b^4/e^4/(b*e*x^2+a
*e*x+b*d*x+a*d)^(1/2)/(b*e)^(1/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)*(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^(5/2)*(d + e*x)^(5/2),x)

[Out]

int((A + B*x)*(a + b*x)^(5/2)*(d + e*x)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(B*x+A)*(e*x+d)**(5/2),x)

[Out]

Timed out

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