3.1735 \(\int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx\)
Optimal. Leaf size=173 \[ -\frac {2 b^2 (d+e x)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5}+\frac {6 b (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5}-\frac {2 (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5}+\frac {2 (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5}+\frac {2 b^3 B (d+e x)^{15/2}}{15 e^5} \]
[Out]
2/7*(-a*e+b*d)^3*(-A*e+B*d)*(e*x+d)^(7/2)/e^5-2/9*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b*d)*(e*x+d)^(9/2)/e^5+6/11
*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(11/2)/e^5-2/13*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^(13/2)/e^5+2
/15*b^3*B*(e*x+d)^(15/2)/e^5
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Rubi [A] time = 0.08, antiderivative size = 173, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used =
{77} \[ -\frac {2 b^2 (d+e x)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5}+\frac {6 b (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5}-\frac {2 (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5}+\frac {2 (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5}+\frac {2 b^3 B (d+e x)^{15/2}}{15 e^5} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^3*(A + B*x)*(d + e*x)^(5/2),x]
[Out]
(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
^(9/2))/(9*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^5) - (2*b^2*(4*b*B*d - A*
b*e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^5) + (2*b^3*B*(d + e*x)^(15/2))/(15*e^5)
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps
\begin {align*} \int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx &=\int \left (\frac {(-b d+a e)^3 (-B d+A e) (d+e x)^{5/2}}{e^4}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{7/2}}{e^4}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^4}+\frac {b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{11/2}}{e^4}+\frac {b^3 B (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{7/2}}{7 e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{9/2}}{9 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{11/2}}{11 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{13/2}}{13 e^5}+\frac {2 b^3 B (d+e x)^{15/2}}{15 e^5}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 145, normalized size = 0.84 \[ \frac {2 (d+e x)^{7/2} \left (-3465 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+12285 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-5005 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+6435 (b d-a e)^3 (B d-A e)+3003 b^3 B (d+e x)^4\right )}{45045 e^5} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^(5/2),x]
[Out]
(2*(d + e*x)^(7/2)*(6435*(b*d - a*e)^3*(B*d - A*e) - 5005*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
+ 12285*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 3465*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^3
+ 3003*b^3*B*(d + e*x)^4))/(45045*e^5)
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fricas [B] time = 0.66, size = 539, normalized size = 3.12 \[ \frac {2 \, {\left (3003 \, B b^{3} e^{7} x^{7} + 128 \, B b^{3} d^{7} + 6435 \, A a^{3} d^{3} e^{4} - 240 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e + 1560 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{2} - 1430 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{3} + 231 \, {\left (31 \, B b^{3} d e^{6} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{7}\right )} x^{6} + 63 \, {\left (71 \, B b^{3} d^{2} e^{5} + 135 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{6} + 195 \, {\left (B a^{2} b + A a b^{2}\right )} e^{7}\right )} x^{5} + 35 \, {\left (B b^{3} d^{3} e^{4} + 159 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{5} + 897 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{6} + 143 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{4} e^{3} - 1287 \, A a^{3} e^{7} - 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{4} - 4407 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{5} - 2717 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{5} e^{2} + 6435 \, A a^{3} d e^{6} - 30 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{3} + 195 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{4} + 3575 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{6} e - 19305 \, A a^{3} d^{2} e^{5} - 120 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{2} + 780 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{3} - 715 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x, algorithm="fricas")
[Out]
2/45045*(3003*B*b^3*e^7*x^7 + 128*B*b^3*d^7 + 6435*A*a^3*d^3*e^4 - 240*(3*B*a*b^2 + A*b^3)*d^6*e + 1560*(B*a^2
*b + A*a*b^2)*d^5*e^2 - 1430*(B*a^3 + 3*A*a^2*b)*d^4*e^3 + 231*(31*B*b^3*d*e^6 + 15*(3*B*a*b^2 + A*b^3)*e^7)*x
^6 + 63*(71*B*b^3*d^2*e^5 + 135*(3*B*a*b^2 + A*b^3)*d*e^6 + 195*(B*a^2*b + A*a*b^2)*e^7)*x^5 + 35*(B*b^3*d^3*e
^4 + 159*(3*B*a*b^2 + A*b^3)*d^2*e^5 + 897*(B*a^2*b + A*a*b^2)*d*e^6 + 143*(B*a^3 + 3*A*a^2*b)*e^7)*x^4 - 5*(8
*B*b^3*d^4*e^3 - 1287*A*a^3*e^7 - 15*(3*B*a*b^2 + A*b^3)*d^3*e^4 - 4407*(B*a^2*b + A*a*b^2)*d^2*e^5 - 2717*(B*
a^3 + 3*A*a^2*b)*d*e^6)*x^3 + 3*(16*B*b^3*d^5*e^2 + 6435*A*a^3*d*e^6 - 30*(3*B*a*b^2 + A*b^3)*d^4*e^3 + 195*(B
*a^2*b + A*a*b^2)*d^3*e^4 + 3575*(B*a^3 + 3*A*a^2*b)*d^2*e^5)*x^2 - (64*B*b^3*d^6*e - 19305*A*a^3*d^2*e^5 - 12
0*(3*B*a*b^2 + A*b^3)*d^5*e^2 + 780*(B*a^2*b + A*a*b^2)*d^4*e^3 - 715*(B*a^3 + 3*A*a^2*b)*d^3*e^4)*x)*sqrt(e*x
+ d)/e^5
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giac [B] time = 1.49, size = 2062, normalized size = 11.92 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x, algorithm="giac")
[Out]
2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*d^3*e^(-1) + 45045*((x*e + d)^(3/2) - 3*sqrt(x*e +
d)*d)*A*a^2*b*d^3*e^(-1) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^2*b*d^3*
e^(-2) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a*b^2*d^3*e^(-2) + 3861*(5*(
x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a*b^2*d^3*e^(-3) + 12
87*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*b^3*d^3*e^(-3)
+ 143*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*s
qrt(x*e + d)*d^4)*B*b^3*d^3*e^(-4) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*
a^3*d^2*e^(-1) + 27027*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*b*d^2*e^(-1) +
11583*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^2*b*d^2*e
^(-2) + 11583*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a*b
^2*d^2*e^(-2) + 1287*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/
2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a*b^2*d^2*e^(-3) + 429*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*
e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*b^3*d^2*e^(-3) + 195*(63*(x*e + d)^(11/2
) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 69
3*sqrt(x*e + d)*d^5)*B*b^3*d^2*e^(-4) + 45045*sqrt(x*e + d)*A*a^3*d^3 + 45045*((x*e + d)^(3/2) - 3*sqrt(x*e +
d)*d)*A*a^3*d^2 + 3861*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d
^3)*B*a^3*d*e^(-1) + 11583*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e +
d)*d^3)*A*a^2*b*d*e^(-1) + 1287*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x
*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a^2*b*d*e^(-2) + 1287*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d
+ 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a*b^2*d*e^(-2) + 585*(63*(x*e
+ d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2
)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a*b^2*d*e^(-3) + 195*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e
+ d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*b^3*d*e^(-3) +
45*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 90
09*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*b^3*d*e^(-4) + 9009*(3*(x*e + d)
^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^3*d + 143*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*
d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a^3*e^(-1) + 429*(35*(x*e + d
)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A
*a^2*b*e^(-1) + 195*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5
/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^2*b*e^(-2) + 195*(63*(x*e + d)^(11/2) - 385*(x
*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e
+ d)*d^5)*A*a*b^2*e^(-2) + 45*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 85
80*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*a*b^2
*e^(-3) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)
*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*b^3*e^(-3) + 7*(429*(x*
e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e
+ d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*b^3*e^(-4)
+ 1287*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^3)*e^(-1
)
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maple [A] time = 0.01, size = 301, normalized size = 1.74 \[ \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 B \,b^{3} x^{4} e^{4}+3465 A \,b^{3} e^{4} x^{3}+10395 B a \,b^{2} e^{4} x^{3}-1848 B \,b^{3} d \,e^{3} x^{3}+12285 A a \,b^{2} e^{4} x^{2}-1890 A \,b^{3} d \,e^{3} x^{2}+12285 B \,a^{2} b \,e^{4} x^{2}-5670 B a \,b^{2} d \,e^{3} x^{2}+1008 B \,b^{3} d^{2} e^{2} x^{2}+15015 A \,a^{2} b \,e^{4} x -5460 A a \,b^{2} d \,e^{3} x +840 A \,b^{3} d^{2} e^{2} x +5005 B \,a^{3} e^{4} x -5460 B \,a^{2} b d \,e^{3} x +2520 B a \,b^{2} d^{2} e^{2} x -448 B \,b^{3} d^{3} e x +6435 a^{3} A \,e^{4}-4290 A \,a^{2} b d \,e^{3}+1560 A a \,b^{2} d^{2} e^{2}-240 A \,b^{3} d^{3} e -1430 B \,a^{3} d \,e^{3}+1560 B \,a^{2} b \,d^{2} e^{2}-720 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right )}{45045 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x)
[Out]
2/45045*(e*x+d)^(7/2)*(3003*B*b^3*e^4*x^4+3465*A*b^3*e^4*x^3+10395*B*a*b^2*e^4*x^3-1848*B*b^3*d*e^3*x^3+12285*
A*a*b^2*e^4*x^2-1890*A*b^3*d*e^3*x^2+12285*B*a^2*b*e^4*x^2-5670*B*a*b^2*d*e^3*x^2+1008*B*b^3*d^2*e^2*x^2+15015
*A*a^2*b*e^4*x-5460*A*a*b^2*d*e^3*x+840*A*b^3*d^2*e^2*x+5005*B*a^3*e^4*x-5460*B*a^2*b*d*e^3*x+2520*B*a*b^2*d^2
*e^2*x-448*B*b^3*d^3*e*x+6435*A*a^3*e^4-4290*A*a^2*b*d*e^3+1560*A*a*b^2*d^2*e^2-240*A*b^3*d^3*e-1430*B*a^3*d*e
^3+1560*B*a^2*b*d^2*e^2-720*B*a*b^2*d^3*e+128*B*b^3*d^4)/e^5
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maxima [A] time = 0.49, size = 265, normalized size = 1.53 \[ \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} B b^{3} - 3465 \, {\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (2 \, B b^{3} d^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e + {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 5005 \, {\left (4 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{45045 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x, algorithm="maxima")
[Out]
2/45045*(3003*(e*x + d)^(15/2)*B*b^3 - 3465*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*(e*x + d)^(13/2) + 12285*(2*B*
b^3*d^2 - (3*B*a*b^2 + A*b^3)*d*e + (B*a^2*b + A*a*b^2)*e^2)*(e*x + d)^(11/2) - 5005*(4*B*b^3*d^3 - 3*(3*B*a*b
^2 + A*b^3)*d^2*e + 6*(B*a^2*b + A*a*b^2)*d*e^2 - (B*a^3 + 3*A*a^2*b)*e^3)*(e*x + d)^(9/2) + 6435*(B*b^3*d^4 +
A*a^3*e^4 - (3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 - (B*a^3 + 3*A*a^2*b)*d*e^3)*(e*x + d)^
(7/2))/e^5
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mupad [B] time = 1.23, size = 154, normalized size = 0.89 \[ \frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{13\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{9\,e^5}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{11\,e^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)*(a + b*x)^3*(d + e*x)^(5/2),x)
[Out]
((d + e*x)^(13/2)*(2*A*b^3*e - 8*B*b^3*d + 6*B*a*b^2*e))/(13*e^5) + (2*(a*e - b*d)^2*(d + e*x)^(9/2)*(3*A*b*e
+ B*a*e - 4*B*b*d))/(9*e^5) + (2*B*b^3*(d + e*x)^(15/2))/(15*e^5) + (2*(A*e - B*d)*(a*e - b*d)^3*(d + e*x)^(7/
2))/(7*e^5) + (6*b*(a*e - b*d)*(d + e*x)^(11/2)*(A*b*e + B*a*e - 2*B*b*d))/(11*e^5)
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sympy [A] time = 52.65, size = 1564, normalized size = 9.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(5/2),x)
[Out]
A*a**3*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*A*a**3*d*(-d*(d + e*x)**(3/
2)/3 + (d + e*x)**(5/2)/5)/e + 2*A*a**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7
)/e + 6*A*a**2*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*A*a**2*b*d*(d**2*(d + e*x)**(3/2)
/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*A*a**2*b*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*
x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**2 + 6*A*a*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2
*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*A*a*b**2*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)*
*(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 6*A*a*b**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d
+ e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 2*A*b**3*
d**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4
+ 4*A*b**3*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)
**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 2*A*b**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*
(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 2*B*
a**3*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*B*a**3*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e
*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*B*a**3*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d
*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**2 + 6*B*a**2*b*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/
2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*B*a**2*b*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d
+ e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 6*B*a**2*b*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5
+ 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 6*B*a*b**2*d**2*(-d**3*(d
+ e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 12*B*a*b**2*
d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 +
(d + e*x)**(11/2)/11)/e**4 + 6*B*a*b**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)*
*(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 2*B*b**3*d**2*
(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d
+ e*x)**(11/2)/11)/e**5 + 4*B*b**3*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(
7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 2*B*b**3*(d**6*(
d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d
**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5
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