3.21.85 \(\int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=358 \[ \frac {(b d-a e)^5 (5 a B e-12 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{512 b^{7/2} e^{9/2}}-\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^4 (5 a B e-12 A b e+7 b B d)}{512 b^3 e^4}+\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^3 (5 a B e-12 A b e+7 b B d)}{768 b^3 e^3}-\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e)^2 (5 a B e-12 A b e+7 b B d)}{960 b^3 e^2}-\frac {(a+b x)^{7/2} \sqrt {d+e x} (b d-a e) (5 a B e-12 A b e+7 b B d)}{160 b^3 e}-\frac {(a+b x)^{7/2} (d+e x)^{3/2} (5 a B e-12 A b e+7 b B d)}{60 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e} \]

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Rubi [A]  time = 0.30, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^4 (5 a B e-12 A b e+7 b B d)}{512 b^3 e^4}+\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^3 (5 a B e-12 A b e+7 b B d)}{768 b^3 e^3}-\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e)^2 (5 a B e-12 A b e+7 b B d)}{960 b^3 e^2}+\frac {(b d-a e)^5 (5 a B e-12 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{512 b^{7/2} e^{9/2}}-\frac {(a+b x)^{7/2} \sqrt {d+e x} (b d-a e) (5 a B e-12 A b e+7 b B d)}{160 b^3 e}-\frac {(a+b x)^{7/2} (d+e x)^{3/2} (5 a B e-12 A b e+7 b B d)}{60 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/2)*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

-((b*d - a*e)^4*(7*b*B*d - 12*A*b*e + 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(512*b^3*e^4) + ((b*d - a*e)^3*(7*
b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(768*b^3*e^3) - ((b*d - a*e)^2*(7*b*B*d - 12*A*b*e
+ 5*a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(960*b^3*e^2) - ((b*d - a*e)*(7*b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x
)^(7/2)*Sqrt[d + e*x])/(160*b^3*e) - ((7*b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(60*b^2*
e) + (B*(a + b*x)^(7/2)*(d + e*x)^(5/2))/(6*b*e) + ((b*d - a*e)^5*(7*b*B*d - 12*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt
[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(512*b^(7/2)*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)^{3/2} \, dx &=\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac {\left (6 A b e-B \left (\frac {7 b d}{2}+\frac {5 a e}{2}\right )\right ) \int (a+b x)^{5/2} (d+e x)^{3/2} \, dx}{6 b e}\\ &=-\frac {(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac {((b d-a e) (7 b B d-12 A b e+5 a B e)) \int (a+b x)^{5/2} \sqrt {d+e x} \, dx}{40 b^2 e}\\ &=-\frac {(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{160 b^3 e}-\frac {(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac {\left ((b d-a e)^2 (7 b B d-12 A b e+5 a B e)\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {d+e x}} \, dx}{320 b^3 e}\\ &=-\frac {(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{960 b^3 e^2}-\frac {(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{160 b^3 e}-\frac {(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac {\left ((b d-a e)^3 (7 b B d-12 A b e+5 a B e)\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{384 b^3 e^2}\\ &=\frac {(b d-a e)^3 (7 b B d-12 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^3 e^3}-\frac {(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{960 b^3 e^2}-\frac {(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{160 b^3 e}-\frac {(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac {\left ((b d-a e)^4 (7 b B d-12 A b e+5 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{512 b^3 e^3}\\ &=-\frac {(b d-a e)^4 (7 b B d-12 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{512 b^3 e^4}+\frac {(b d-a e)^3 (7 b B d-12 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^3 e^3}-\frac {(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{960 b^3 e^2}-\frac {(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{160 b^3 e}-\frac {(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac {\left ((b d-a e)^5 (7 b B d-12 A b e+5 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{1024 b^3 e^4}\\ &=-\frac {(b d-a e)^4 (7 b B d-12 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{512 b^3 e^4}+\frac {(b d-a e)^3 (7 b B d-12 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^3 e^3}-\frac {(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{960 b^3 e^2}-\frac {(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{160 b^3 e}-\frac {(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac {\left ((b d-a e)^5 (7 b B d-12 A b e+5 a B e)\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 )}{512 b^4 e^4}\\ &=-\frac {(b d-a e)^4 (7 b B d-12 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{512 b^3 e^4}+\frac {(b d-a e)^3 (7 b B d-12 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^3 e^3}-\frac {(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{960 b^3 e^2}-\frac {(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{160 b^3 e}-\frac {(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac {\left ((b d-a e)^5 (7 b B d-12 A b e+5 a B e)\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 )}{512 b^4 e^4}\\ &=-\frac {(b d-a e)^4 (7 b B d-12 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{512 b^3 e^4}+\frac {(b d-a e)^3 (7 b B d-12 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^3 e^3}-\frac {(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{960 b^3 e^2}-\frac {(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{160 b^3 e}-\frac {(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac {(b d-a e)^5 (7 b B d-12 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{512 b^{7/2} e^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 4.61, size = 360, normalized size = 1.01 \begin {gather*} \frac {\sqrt {b d-a e} \left (\frac {b (d+e x)}{b d-a e}\right )^{3/2} \left (-\frac {5 a B e}{2}+6 A b e-\frac {7}{2} b B d\right ) \left (-10 e^{3/2} (a+b x)^2 (b d-a e)^{9/2} \sqrt {\frac {b (d+e x)}{b d-a e}}+8 e^{5/2} (a+b x)^3 (b d-a e)^{7/2} \sqrt {\frac {b (d+e x)}{b d-a e}}+16 e^{7/2} (a+b x)^4 (b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}} (-3 a e+11 b d+8 b e x)+15 \sqrt {e} (a+b x) (b d-a e)^{11/2} \sqrt {\frac {b (d+e x)}{b d-a e}}-15 \sqrt {a+b x} (b d-a e)^6 \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )\right )+640 b^4 B e^{7/2} (a+b x)^4 (d+e x)^4}{3840 b^5 e^{9/2} \sqrt {a+b x} (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/2)*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

(640*b^4*B*e^(7/2)*(a + b*x)^4*(d + e*x)^4 + Sqrt[b*d - a*e]*((-7*b*B*d)/2 + 6*A*b*e - (5*a*B*e)/2)*((b*(d + e
*x))/(b*d - a*e))^(3/2)*(15*Sqrt[e]*(b*d - a*e)^(11/2)*(a + b*x)*Sqrt[(b*(d + e*x))/(b*d - a*e)] - 10*e^(3/2)*
(b*d - a*e)^(9/2)*(a + b*x)^2*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 8*e^(5/2)*(b*d - a*e)^(7/2)*(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)]*(11*b*d -
3*a*e + 8*b*e*x) - 15*(b*d - a*e)^6*Sqrt[a + b*x]*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]]))/(3840*b^
5*e^(9/2)*Sqrt[a + b*x]*(d + e*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.91, size = 489, normalized size = 1.37 \begin {gather*} \frac {(b d-a e)^5 (5 a B e-12 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{512 b^{7/2} e^{9/2}}+\frac {\sqrt {a+b x} (b d-a e)^5 \left (-\frac {1020 A b^5 e^2 (a+b x)}{d+e x}+\frac {2376 A b^4 e^3 (a+b x)^2}{(d+e x)^2}-\frac {696 A b^3 e^4 (a+b x)^3}{(d+e x)^3}-\frac {1020 A b^2 e^5 (a+b x)^4}{(d+e x)^4}+\frac {180 A b e^6 (a+b x)^5}{(d+e x)^5}+\frac {595 b^5 B d e (a+b x)}{d+e x}-75 a b^5 B e+\frac {425 a b^4 B e^2 (a+b x)}{d+e x}-\frac {1386 b^4 B d e^2 (a+b x)^2}{(d+e x)^2}-\frac {990 a b^3 B e^3 (a+b x)^2}{(d+e x)^2}+\frac {1686 b^3 B d e^3 (a+b x)^3}{(d+e x)^3}-\frac {990 a b^2 B e^4 (a+b x)^3}{(d+e x)^3}+\frac {595 b^2 B d e^4 (a+b x)^4}{(d+e x)^4}-\frac {75 a B e^6 (a+b x)^5}{(d+e x)^5}+\frac {425 a b B e^5 (a+b x)^4}{(d+e x)^4}-\frac {105 b B d e^5 (a+b x)^5}{(d+e x)^5}+180 A b^6 e-105 b^6 B d\right )}{7680 b^3 e^4 \sqrt {d+e x} \left (b-\frac {e (a+b x)}{d+e x}\right )^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(5/2)*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

((b*d - a*e)^5*Sqrt[a + b*x]*(-105*b^6*B*d + 180*A*b^6*e - 75*a*b^5*B*e - (105*b*B*d*e^5*(a + b*x)^5)/(d + e*x
)^5 + (180*A*b*e^6*(a + b*x)^5)/(d + e*x)^5 - (75*a*B*e^6*(a + b*x)^5)/(d + e*x)^5 + (595*b^2*B*d*e^4*(a + b*x
)^4)/(d + e*x)^4 - (1020*A*b^2*e^5*(a + b*x)^4)/(d + e*x)^4 + (425*a*b*B*e^5*(a + b*x)^4)/(d + e*x)^4 + (1686*
b^3*B*d*e^3*(a + b*x)^3)/(d + e*x)^3 - (696*A*b^3*e^4*(a + b*x)^3)/(d + e*x)^3 - (990*a*b^2*B*e^4*(a + b*x)^3)
/(d + e*x)^3 - (1386*b^4*B*d*e^2*(a + b*x)^2)/(d + e*x)^2 + (2376*A*b^4*e^3*(a + b*x)^2)/(d + e*x)^2 - (990*a*
b^3*B*e^3*(a + b*x)^2)/(d + e*x)^2 + (595*b^5*B*d*e*(a + b*x))/(d + e*x) - (1020*A*b^5*e^2*(a + b*x))/(d + e*x
) + (425*a*b^4*B*e^2*(a + b*x))/(d + e*x)))/(7680*b^3*e^4*Sqrt[d + e*x]*(b - (e*(a + b*x))/(d + e*x))^6) + ((b
*d - a*e)^5*(7*b*B*d - 12*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(512*b^(7
/2)*e^(9/2))

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fricas [B]  time = 1.75, size = 1388, normalized size = 3.88

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)^(3/2),x, algorithm="fricas")

[Out]

[1/30720*(15*(7*B*b^6*d^6 - 6*(5*B*a*b^5 + 2*A*b^6)*d^5*e + 15*(3*B*a^2*b^4 + 4*A*a*b^5)*d^4*e^2 - 20*(B*a^3*b
^3 + 6*A*a^2*b^4)*d^3*e^3 - 15*(B*a^4*b^2 - 8*A*a^3*b^3)*d^2*e^4 + 6*(3*B*a^5*b - 10*A*a^4*b^2)*d*e^5 - (5*B*a
^6 - 12*A*a^5*b)*e^6)*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)*sq
rt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) + 4*(1280*B*b^6*e^6*x^5 - 105*B*b^6*d^5*e + 5*(
83*B*a*b^5 + 36*A*b^6)*d^4*e^2 - 42*(13*B*a^2*b^4 + 20*A*a*b^5)*d^3*e^3 + 6*(25*B*a^3*b^3 + 256*A*a^2*b^4)*d^2
*e^4 - 35*(7*B*a^4*b^2 - 24*A*a^3*b^3)*d*e^5 + 15*(5*B*a^5*b - 12*A*a^4*b^2)*e^6 + 128*(13*B*b^6*d*e^5 + (25*B
*a*b^5 + 12*A*b^6)*e^6)*x^4 + 16*(3*B*b^6*d^2*e^4 + 2*(139*B*a*b^5 + 66*A*b^6)*d*e^5 + 9*(15*B*a^2*b^4 + 28*A*
a*b^5)*e^6)*x^3 - 8*(7*B*b^6*d^3*e^3 - 3*(9*B*a*b^5 + 4*A*b^6)*d^2*e^4 - 3*(141*B*a^2*b^4 + 256*A*a*b^5)*d*e^5
- (5*B*a^3*b^3 + 372*A*a^2*b^4)*e^6)*x^2 + 2*(35*B*b^6*d^4*e^2 - 4*(34*B*a*b^5 + 15*A*b^6)*d^3*e^3 + 6*(29*B*
a^2*b^4 + 46*A*a*b^5)*d^2*e^4 + 4*(20*B*a^3*b^3 + 699*A*a^2*b^4)*d*e^5 - 5*(5*B*a^4*b^2 - 12*A*a^3*b^3)*e^6)*x
)*sqrt(b*x + a)*sqrt(e*x + d))/(b^4*e^5), -1/15360*(15*(7*B*b^6*d^6 - 6*(5*B*a*b^5 + 2*A*b^6)*d^5*e + 15*(3*B*
a^2*b^4 + 4*A*a*b^5)*d^4*e^2 - 20*(B*a^3*b^3 + 6*A*a^2*b^4)*d^3*e^3 - 15*(B*a^4*b^2 - 8*A*a^3*b^3)*d^2*e^4 + 6
*(3*B*a^5*b - 10*A*a^4*b^2)*d*e^5 - (5*B*a^6 - 12*A*a^5*b)*e^6)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sq
rt(-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*(1280*B*b^6*e^6*x^5
- 105*B*b^6*d^5*e + 5*(83*B*a*b^5 + 36*A*b^6)*d^4*e^2 - 42*(13*B*a^2*b^4 + 20*A*a*b^5)*d^3*e^3 + 6*(25*B*a^3*b
^3 + 256*A*a^2*b^4)*d^2*e^4 - 35*(7*B*a^4*b^2 - 24*A*a^3*b^3)*d*e^5 + 15*(5*B*a^5*b - 12*A*a^4*b^2)*e^6 + 128*
(13*B*b^6*d*e^5 + (25*B*a*b^5 + 12*A*b^6)*e^6)*x^4 + 16*(3*B*b^6*d^2*e^4 + 2*(139*B*a*b^5 + 66*A*b^6)*d*e^5 +
9*(15*B*a^2*b^4 + 28*A*a*b^5)*e^6)*x^3 - 8*(7*B*b^6*d^3*e^3 - 3*(9*B*a*b^5 + 4*A*b^6)*d^2*e^4 - 3*(141*B*a^2*b
^4 + 256*A*a*b^5)*d*e^5 - (5*B*a^3*b^3 + 372*A*a^2*b^4)*e^6)*x^2 + 2*(35*B*b^6*d^4*e^2 - 4*(34*B*a*b^5 + 15*A*
b^6)*d^3*e^3 + 6*(29*B*a^2*b^4 + 46*A*a*b^5)*d^2*e^4 + 4*(20*B*a^3*b^3 + 699*A*a^2*b^4)*d*e^5 - 5*(5*B*a^4*b^2
- 12*A*a^3*b^3)*e^6)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^4*e^5)]

________________________________________________________________________________________

giac [B]  time = 5.27, size = 3935, normalized size = 10.99

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)^(3/2),x, algorithm="giac")

[Out]

1/7680*(960*(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*abs(b) + 120*(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(-sqr
t(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*abs(b) - 7680*((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*abs(b)/b^2 + 960*(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*abs(b)/b + 40*(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*abs(b) + 4*(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*abs(b) + 120*(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^1
4) + 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*abs(b)*e + 12*(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^2
1*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*abs(b)*e + 320*(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(ab
s(-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*abs(b)*e/b^2 + 960*(sq
rt(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*abs(b)*e/b + 120*(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*abs(b)*e/b + 4*(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*abs(b)*e + (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 + 1
26*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*b*abs(b)*e + 1920*((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)*s
qrt(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*abs(b)/b^3 + 5760*((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*abs(b)/b^2 + 1920*((
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*abs(b)*e/b^3)/b

________________________________________________________________________________________

maple [B]  time = 0.02, size = 2198, normalized size = 6.14

result 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)^(3/2),x)

[Out]

1/15360*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(1800*A*a^3*b^3*d^2*e^4*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))+180*A*a^5*b*e^6*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))-75*B*a^6*e^6*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))+105*B*b^6*d^6*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))-1
80*A*b^6*d^5*e*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))+150*(b*e*x^
2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a^5*e^5-210*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*b^5*d^5+96*(b
*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*b^5*d^2*e^3*x^3+5952*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A
*a^2*b^3*e^5*x^2+192*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*b^5*d^2*e^3*x^2+80*(b*e*x^2+a*e*x+b*d*x+a*d
)^(1/2)*(b*e)^(1/2)*B*a^3*b^2*e^5*x^2+240*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a^3*b^2*e^5*x-240*(b*e
*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*b^5*d^3*e^2*x-490*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a^4*
b*d*e^4+300*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a^3*b^2*d^2*e^3-1092*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)
*(b*e)^(1/2)*B*a^2*b^3*d^3*e^2+830*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a*b^4*d^4*e-900*A*a^4*b^2*d*e
^5*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))+270*B*a^5*b*d*e^5*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))-225*B*a^4*b^2*d^2*e^4*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))-300*B*a^3*b^3*d^3*e^3*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))+675*B*a^2*b^4*d^4*e^2*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))-1800*A*a^2*b^4*d^3*e^3*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))+900*A*a*b^5*d^4*e^2*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))-450*B*a*b^5*d^5*e*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))-360*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a^4*b*e^5+
360*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*b^5*d^4*e+2560*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B
*b^5*e^5*x^5+3072*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*b^5*e^5*x^4+140*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2
)*(b*e)^(1/2)*B*b^5*d^4*e*x+1680*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a^3*b^2*d*e^4-1680*(b*e*x^2+a*e
*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a*b^4*d^3*e^2+8064*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a*b^4*e^5*x
^3+4224*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*b^5*d*e^4*x^3+4320*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)
^(1/2)*B*a^2*b^3*e^5*x^3+6768*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a^2*b^3*d*e^4*x^2+432*(b*e*x^2+a*e
*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a*b^4*d^2*e^3*x^2+8896*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a*b^4*d
*e^4*x^3+12288*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a*b^4*d*e^4*x^2+696*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/
2)*(b*e)^(1/2)*B*a^2*b^3*d^2*e^3*x-544*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a*b^4*d^3*e^2*x+1104*(b*e
*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a*b^4*d^2*e^3*x+320*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a^
3*b^2*d*e^4*x+11184*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a^2*b^3*d*e^4*x-112*(b*e*x^2+a*e*x+b*d*x+a*d
)^(1/2)*(b*e)^(1/2)*B*b^5*d^3*e^2*x^2+3072*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a^2*b^3*d^2*e^3+6400*
(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a*b^4*e^5*x^4+3328*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B
*b^5*d*e^4*x^4-100*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a^4*b*e^5*x)/b^3/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 \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)^(3/2),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 \begin {gather*} \int \left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2} \,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)^(3/2),x)

[Out]

int((A + B*x)*(a + b*x)^(5/2)*(d + e*x)^(3/2), x)

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sympy [F(-1)]  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)**(3/2),x)

[Out]

Timed out

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