3.23.23 \(\int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx\) [2223]

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

[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 = 0.24, 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 = {81, 52, 65, 223, 212} \begin {gather*} -\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} \end {gather*}

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 52

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 65

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 81

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 212

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

Rule 223

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 ) \text {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 ) \text {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 = 1.38, size = 514, normalized size = 1.25 \begin {gather*} \frac {(b d-a e)^6 \left (-\frac {\sqrt {b} \sqrt {e} \sqrt {a+b x} \sqrt {d+e x} \left (105 b B d e^6 (a+b x)^6-210 A b e^7 (a+b x)^6+105 a B e^7 (a+b x)^6-700 b^2 B d e^5 (a+b x)^5 (d+e x)+1400 A b^2 e^6 (a+b x)^5 (d+e x)-700 a b B e^6 (a+b x)^5 (d+e x)+1981 b^3 B d e^4 (a+b x)^4 (d+e x)^2-3962 A b^3 e^5 (a+b x)^4 (d+e x)^2+1981 a b^2 B e^5 (a+b x)^4 (d+e x)^2+3072 b^4 B d e^3 (a+b x)^3 (d+e x)^3-3072 a b^3 B e^4 (a+b x)^3 (d+e x)^3-1981 b^5 B d e^2 (a+b x)^2 (d+e x)^4+3962 A b^5 e^3 (a+b x)^2 (d+e x)^4-1981 a b^4 B e^3 (a+b x)^2 (d+e x)^4+700 b^6 B d e (a+b x) (d+e x)^5-1400 A b^6 e^2 (a+b x) (d+e x)^5+700 a b^5 B e^2 (a+b x) (d+e x)^5-105 b^7 B d (d+e x)^6+210 A b^7 e (d+e x)^6-105 a b^6 B e (d+e x)^6\right )}{(-b d+a e)^7}+105 (b B d-2 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )\right )}{21504 b^{9/2} e^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)^6*(-((Sqrt[b]*Sqrt[e]*Sqrt[a + b*x]*Sqrt[d + e*x]*(105*b*B*d*e^6*(a + b*x)^6 - 210*A*b*e^7*(a + b
*x)^6 + 105*a*B*e^7*(a + b*x)^6 - 700*b^2*B*d*e^5*(a + b*x)^5*(d + e*x) + 1400*A*b^2*e^6*(a + b*x)^5*(d + e*x)
- 700*a*b*B*e^6*(a + b*x)^5*(d + e*x) + 1981*b^3*B*d*e^4*(a + b*x)^4*(d + e*x)^2 - 3962*A*b^3*e^5*(a + b*x)^4
*(d + e*x)^2 + 1981*a*b^2*B*e^5*(a + b*x)^4*(d + e*x)^2 + 3072*b^4*B*d*e^3*(a + b*x)^3*(d + e*x)^3 - 3072*a*b^
3*B*e^4*(a + b*x)^3*(d + e*x)^3 - 1981*b^5*B*d*e^2*(a + b*x)^2*(d + e*x)^4 + 3962*A*b^5*e^3*(a + b*x)^2*(d + e
*x)^4 - 1981*a*b^4*B*e^3*(a + b*x)^2*(d + e*x)^4 + 700*b^6*B*d*e*(a + b*x)*(d + e*x)^5 - 1400*A*b^6*e^2*(a + b
*x)*(d + e*x)^5 + 700*a*b^5*B*e^2*(a + b*x)*(d + e*x)^5 - 105*b^7*B*d*(d + e*x)^6 + 210*A*b^7*e*(d + e*x)^6 -
105*a*b^6*B*e*(d + e*x)^6))/(-(b*d) + a*e)^7) + 105*(b*B*d - 2*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/
(Sqrt[e]*Sqrt[a + b*x])]))/(21504*b^(9/2)*e^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2395\) vs. \(2(356)=712\).
time = 0.10, size = 2396, normalized size = 5.82

method	result	size
default	\(\text {Expression too large to display}\)	\(2396\)

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,method=_RETURNVERBOSE)

[Out]

-1/43008*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-6144*B*b^6*e^6*x^6*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-7168*A*b^6*e^6*x
^5*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-420*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^5*b*e^6-420*A*(b*e)^(1/2)*(
(b*x+a)*(e*x+d))^(1/2)*b^6*d^5*e-1260*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(
1/2))*a^5*b^2*d*e^6+3150*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*b^3
*d^2*e^5-4200*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^4*d^3*e^4-94
5*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^5*b^2*d^2*e^5+525*B*ln(1/2*(
2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*b^3*d^3*e^4+525*B*ln(1/2*(2*b*e*x+2*((
b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^4*d^4*e^3-945*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+
d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^5*d^5*e^2+525*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b
*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^6*d^6*e+3150*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b
*d)/(b*e)^(1/2))*a^2*b^5*d^4*e^3-1260*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(
1/2))*a*b^6*d^5*e^2+525*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^6*b*d*
e^6-105*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^7*e^7-105*B*ln(1/2*(2*
b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^7*d^7-96*B*a^3*b^3*e^6*x^3*(b*e)^(1/2)*((b
*x+a)*(e*x+d))^(1/2)-96*B*b^6*d^3*e^3*x^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-224*A*a^3*b^3*e^6*x^2*(b*e)^(1/2
)*((b*x+a)*(e*x+d))^(1/2)-224*A*b^6*d^3*e^3*x^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+112*B*a^4*b^2*e^6*x^2*(b*e
)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+112*B*b^6*d^4*e^2*x^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-1568*A*(b*e)^(1/2)*(
(b*x+a)*(e*x+d))^(1/2)*a^3*b^3*d*e^5*x-33264*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b^4*d^2*e^4*x-1568*A*(b
*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^5*d^3*e^3*x+644*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^4*b^2*d*e^5*x-10
16*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b^3*d^2*e^4*x-1016*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b^4*
d^3*e^3*x+644*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^5*d^4*e^2*x-14848*B*a*b^5*e^6*x^5*(b*e)^(1/2)*((b*x+a)
*(e*x+d))^(1/2)-14848*B*b^6*d*e^5*x^5*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-17920*A*a*b^5*e^6*x^4*(b*e)^(1/2)*((
b*x+a)*(e*x+d))^(1/2)-17920*A*b^6*d*e^5*x^4*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-9472*B*a^2*b^4*e^6*x^4*(b*e)^(
1/2)*((b*x+a)*(e*x+d))^(1/2)-9472*B*b^6*d^2*e^4*x^4*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-12096*A*a^2*b^4*e^6*x^
3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-12096*A*b^6*d^2*e^4*x^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-37376*B*a*b^
5*d*e^5*x^4*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-47488*A*a*b^5*d*e^5*x^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-25
504*B*a^2*b^4*d*e^5*x^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-25504*B*a*b^5*d^2*e^4*x^3*(b*e)^(1/2)*((b*x+a)*(e*
x+d))^(1/2)-35616*A*a^2*b^4*d*e^5*x^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-35616*A*a*b^5*d^2*e^4*x^2*(b*e)^(1/2
)*((b*x+a)*(e*x+d))^(1/2)-512*B*a^3*b^3*d*e^5*x^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-19680*B*a^2*b^4*d^2*e^4*
x^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-512*B*a*b^5*d^3*e^3*x^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+210*A*ln(1
/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^6*b*e^7+210*A*ln(1/2*(2*b*e*x+2*((b*
x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^7*d^6*e+210*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^6*
e^6+210*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^6*d^6+280*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^4*b^2*e^6*x+
2380*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^5*d^4*e^2-600*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b^3*d^3
*e^3+1582*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b^4*d^4*e^2-980*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^
5*d^5*e+280*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^6*d^4*e^2*x-140*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^5*
b*e^6*x-140*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^6*d^5*e*x+2380*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^4*b
^2*d*e^5-980*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^5*b*d*e^5+1582*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^4*
b^2*d^2*e^4-5544*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b^3*d^2*e^4-5544*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1
/2)*a^2*b^4*d^3*e^3)/b^4/e^4/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 838 vs. \(2 (375) = 750\).
time = 1.48, size = 1686, normalized size = 4.09 \begin {gather*} \text {Too large to display} \end {gather*}

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^(1/2)*log(b^2*d^2 + 4*(b*d + (2*b*x + a)*e)*sqrt
(b*x + a)*sqrt(x*e + d)*sqrt(b)*e^(1/2) + (8*b^2*x^2 + 8*a*b*x + a^2)*e^2 + 2*(4*b^2*d*x + 3*a*b*d)*e) - 4*(10
5*B*b^7*d^6*e - (3072*B*b^7*x^6 - 105*B*a^6*b + 210*A*a^5*b^2 + 256*(29*B*a*b^6 + 14*A*b^7)*x^5 + 128*(37*B*a^
2*b^5 + 70*A*a*b^6)*x^4 + 48*(B*a^3*b^4 + 126*A*a^2*b^5)*x^3 - 56*(B*a^4*b^3 - 2*A*a^3*b^4)*x^2 + 70*(B*a^5*b^
2 - 2*A*a^4*b^3)*x)*e^7 - 2*(3712*B*b^7*d*x^5 + 128*(73*B*a*b^6 + 35*A*b^7)*d*x^4 + 8*(797*B*a^2*b^5 + 1484*A*
a*b^6)*d*x^3 + 8*(16*B*a^3*b^4 + 1113*A*a^2*b^5)*d*x^2 - 7*(23*B*a^4*b^3 - 56*A*a^3*b^4)*d*x + 35*(7*B*a^5*b^2
- 17*A*a^4*b^3)*d)*e^6 - (4736*B*b^7*d^2*x^4 + 16*(797*B*a*b^6 + 378*A*b^7)*d^2*x^3 + 48*(205*B*a^2*b^5 + 371
*A*a*b^6)*d^2*x^2 + 4*(127*B*a^3*b^4 + 4158*A*a^2*b^5)*d^2*x - 7*(113*B*a^4*b^3 - 396*A*a^3*b^4)*d^2)*e^5 - 4*
(12*B*b^7*d^3*x^3 + 4*(16*B*a*b^6 + 7*A*b^7)*d^3*x^2 + (127*B*a^2*b^5 + 196*A*a*b^6)*d^3*x + 3*(25*B*a^3*b^4 +
231*A*a^2*b^5)*d^3)*e^4 + 7*(8*B*b^7*d^4*x^2 + 2*(23*B*a*b^6 + 10*A*b^7)*d^4*x + (113*B*a^2*b^5 + 170*A*a*b^6
)*d^4)*e^3 - 70*(B*b^7*d^5*x + (7*B*a*b^6 + 3*A*b^7)*d^5)*e^2)*sqrt(b*x + a)*sqrt(x*e + d))*e^(-5)/b^5, -1/430
08*(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 - 1
2*A*a^5*b^2)*d*e^6 + (B*a^7 - 2*A*a^6*b)*e^7)*sqrt(-b*e)*arctan(1/2*(b*d + (2*b*x + a)*e)*sqrt(b*x + a)*sqrt(-
b*e)*sqrt(x*e + d)/((b^2*x^2 + a*b*x)*e^2 + (b^2*d*x + a*b*d)*e)) + 2*(105*B*b^7*d^6*e - (3072*B*b^7*x^6 - 105
*B*a^6*b + 210*A*a^5*b^2 + 256*(29*B*a*b^6 + 14*A*b^7)*x^5 + 128*(37*B*a^2*b^5 + 70*A*a*b^6)*x^4 + 48*(B*a^3*b
^4 + 126*A*a^2*b^5)*x^3 - 56*(B*a^4*b^3 - 2*A*a^3*b^4)*x^2 + 70*(B*a^5*b^2 - 2*A*a^4*b^3)*x)*e^7 - 2*(3712*B*b
^7*d*x^5 + 128*(73*B*a*b^6 + 35*A*b^7)*d*x^4 + 8*(797*B*a^2*b^5 + 1484*A*a*b^6)*d*x^3 + 8*(16*B*a^3*b^4 + 1113
*A*a^2*b^5)*d*x^2 - 7*(23*B*a^4*b^3 - 56*A*a^3*b^4)*d*x + 35*(7*B*a^5*b^2 - 17*A*a^4*b^3)*d)*e^6 - (4736*B*b^7
*d^2*x^4 + 16*(797*B*a*b^6 + 378*A*b^7)*d^2*x^3 + 48*(205*B*a^2*b^5 + 371*A*a*b^6)*d^2*x^2 + 4*(127*B*a^3*b^4
+ 4158*A*a^2*b^5)*d^2*x - 7*(113*B*a^4*b^3 - 396*A*a^3*b^4)*d^2)*e^5 - 4*(12*B*b^7*d^3*x^3 + 4*(16*B*a*b^6 + 7
*A*b^7)*d^3*x^2 + (127*B*a^2*b^5 + 196*A*a*b^6)*d^3*x + 3*(25*B*a^3*b^4 + 231*A*a^2*b^5)*d^3)*e^4 + 7*(8*B*b^7
*d^4*x^2 + 2*(23*B*a*b^6 + 10*A*b^7)*d^4*x + (113*B*a^2*b^5 + 170*A*a*b^6)*d^4)*e^3 - 70*(B*b^7*d^5*x + (7*B*a
*b^6 + 3*A*b^7)*d^5)*e^2)*sqrt(b*x + a)*sqrt(x*e + d))*e^(-5)/b^5]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 6900 vs. \(2 (375) = 750\).
time = 2.43, size = 6900, normalized size = 16.75 \begin {gather*} \text {Too large to display} \end {gather*}

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 - ...

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

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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