3.1726 \(\int (a+b x)^2 (A+B x) (d+e x)^{7/2} \, dx\)
Optimal. Leaf size=128 \[ -\frac {2 b (d+e x)^{13/2} (-2 a B e-A b e+3 b B d)}{13 e^4}+\frac {2 (d+e x)^{11/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{11 e^4}-\frac {2 (d+e x)^{9/2} (b d-a e)^2 (B d-A e)}{9 e^4}+\frac {2 b^2 B (d+e x)^{15/2}}{15 e^4} \]
[Out]
-2/9*(-a*e+b*d)^2*(-A*e+B*d)*(e*x+d)^(9/2)/e^4+2/11*(-a*e+b*d)*(-2*A*b*e-B*a*e+3*B*b*d)*(e*x+d)^(11/2)/e^4-2/1
3*b*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^(13/2)/e^4+2/15*b^2*B*(e*x+d)^(15/2)/e^4
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Rubi [A] time = 0.07, antiderivative size = 128, 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} \[ -\frac {2 b (d+e x)^{13/2} (-2 a B e-A b e+3 b B d)}{13 e^4}+\frac {2 (d+e x)^{11/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{11 e^4}-\frac {2 (d+e x)^{9/2} (b d-a e)^2 (B d-A e)}{9 e^4}+\frac {2 b^2 B (d+e x)^{15/2}}{15 e^4} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^2*(A + B*x)*(d + e*x)^(7/2),x]
[Out]
(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^4) + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^
(11/2))/(11*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(13/2))/(13*e^4) + (2*b^2*B*(d + e*x)^(15/2))/(1
5*e^4)
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)^2 (A+B x) (d+e x)^{7/2} \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^{7/2}}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{9/2}}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^{11/2}}{e^3}+\frac {b^2 B (d+e x)^{13/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{9/2}}{9 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{11/2}}{11 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{13/2}}{13 e^4}+\frac {2 b^2 B (d+e x)^{15/2}}{15 e^4}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 107, normalized size = 0.84 \[ \frac {2 (d+e x)^{9/2} \left (-495 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+585 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-715 (b d-a e)^2 (B d-A e)+429 b^2 B (d+e x)^3\right )}{6435 e^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^(7/2),x]
[Out]
(2*(d + e*x)^(9/2)*(-715*(b*d - a*e)^2*(B*d - A*e) + 585*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x) - 4
95*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^2 + 429*b^2*B*(d + e*x)^3))/(6435*e^4)
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fricas [B] time = 0.68, size = 424, normalized size = 3.31 \[ \frac {2 \, {\left (429 \, B b^{2} e^{7} x^{7} - 16 \, B b^{2} d^{7} + 715 \, A a^{2} d^{4} e^{3} + 40 \, {\left (2 \, B a b + A b^{2}\right )} d^{6} e - 130 \, {\left (B a^{2} + 2 \, A a b\right )} d^{5} e^{2} + 33 \, {\left (46 \, B b^{2} d e^{6} + 15 \, {\left (2 \, B a b + A b^{2}\right )} e^{7}\right )} x^{6} + 9 \, {\left (206 \, B b^{2} d^{2} e^{5} + 200 \, {\left (2 \, B a b + A b^{2}\right )} d e^{6} + 65 \, {\left (B a^{2} + 2 \, A a b\right )} e^{7}\right )} x^{5} + 5 \, {\left (160 \, B b^{2} d^{3} e^{4} + 143 \, A a^{2} e^{7} + 458 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{5} + 442 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{6}\right )} x^{4} + 5 \, {\left (B b^{2} d^{4} e^{3} + 572 \, A a^{2} d e^{6} + 212 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{4} + 598 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{5}\right )} x^{3} - 3 \, {\left (2 \, B b^{2} d^{5} e^{2} - 1430 \, A a^{2} d^{2} e^{5} - 5 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e^{3} - 520 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{4}\right )} x^{2} + {\left (8 \, B b^{2} d^{6} e + 2860 \, A a^{2} d^{3} e^{4} - 20 \, {\left (2 \, B a b + A b^{2}\right )} d^{5} e^{2} + 65 \, {\left (B a^{2} + 2 \, A a b\right )} d^{4} e^{3}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(7/2),x, algorithm="fricas")
[Out]
2/6435*(429*B*b^2*e^7*x^7 - 16*B*b^2*d^7 + 715*A*a^2*d^4*e^3 + 40*(2*B*a*b + A*b^2)*d^6*e - 130*(B*a^2 + 2*A*a
*b)*d^5*e^2 + 33*(46*B*b^2*d*e^6 + 15*(2*B*a*b + A*b^2)*e^7)*x^6 + 9*(206*B*b^2*d^2*e^5 + 200*(2*B*a*b + A*b^2
)*d*e^6 + 65*(B*a^2 + 2*A*a*b)*e^7)*x^5 + 5*(160*B*b^2*d^3*e^4 + 143*A*a^2*e^7 + 458*(2*B*a*b + A*b^2)*d^2*e^5
+ 442*(B*a^2 + 2*A*a*b)*d*e^6)*x^4 + 5*(B*b^2*d^4*e^3 + 572*A*a^2*d*e^6 + 212*(2*B*a*b + A*b^2)*d^3*e^4 + 598
*(B*a^2 + 2*A*a*b)*d^2*e^5)*x^3 - 3*(2*B*b^2*d^5*e^2 - 1430*A*a^2*d^2*e^5 - 5*(2*B*a*b + A*b^2)*d^4*e^3 - 520*
(B*a^2 + 2*A*a*b)*d^3*e^4)*x^2 + (8*B*b^2*d^6*e + 2860*A*a^2*d^3*e^4 - 20*(2*B*a*b + A*b^2)*d^5*e^2 + 65*(B*a^
2 + 2*A*a*b)*d^4*e^3)*x)*sqrt(e*x + d)/e^4
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giac [B] time = 1.49, size = 1913, normalized size = 14.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(7/2),x, algorithm="giac")
[Out]
2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^2*d^4*e^(-1) + 30030*((x*e + d)^(3/2) - 3*sqrt(x*e +
d)*d)*A*a*b*d^4*e^(-1) + 6006*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a*b*d^4*e^(-
2) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b^2*d^4*e^(-2) + 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)*B*b^2*d^4*e^(-3) + 12012*(3*(
x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^2*d^3*e^(-1) + 24024*(3*(x*e + d)^(5/2) - 10
*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a*b*d^3*e^(-1) + 10296*(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*d^3*e^(-2) + 5148*(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^2*d^3*e^(-2) + 572*(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*b^2*d^3*e^(-3) + 450
45*sqrt(x*e + d)*A*a^2*d^4 + 60060*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^2*d^3 + 7722*(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*d^2*e^(-1) + 15444*(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*d^2*e^(-1) + 1716*(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*d^2*e^(-2) + 858*(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^2*d^2*e^(-2) + 390*(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*b^2*d^2*e
^(-3) + 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*d^2 + 572*(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*d*e^(-1) + 1144*(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*d*e^(-1) + 520*(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*d*e^(-2) + 26
0*(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^2*d*e^(-2) + 60*(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)*B*b^2*d*e^(-3) + 5148*(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*d + 65*(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*e^(-1) + 130*(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*e^(-1) + 30*(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)*B*a*b*e^(-2) + 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^2*e^(-2
) + 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^2*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*sqrt(x*e + d)*d^4)*A*a^2)*e^(-1)
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maple [A] time = 0.01, size = 169, normalized size = 1.32 \[ \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (429 b^{2} B \,x^{3} e^{3}+495 A \,b^{2} e^{3} x^{2}+990 B a b \,e^{3} x^{2}-198 B \,b^{2} d \,e^{2} x^{2}+1170 A a b \,e^{3} x -180 A \,b^{2} d \,e^{2} x +585 B \,a^{2} e^{3} x -360 B a b d \,e^{2} x +72 B \,b^{2} d^{2} e x +715 a^{2} A \,e^{3}-260 A a b d \,e^{2}+40 A \,b^{2} d^{2} e -130 B \,a^{2} d \,e^{2}+80 B a b \,d^{2} e -16 B \,b^{2} d^{3}\right )}{6435 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^2*(B*x+A)*(e*x+d)^(7/2),x)
[Out]
2/6435*(e*x+d)^(9/2)*(429*B*b^2*e^3*x^3+495*A*b^2*e^3*x^2+990*B*a*b*e^3*x^2-198*B*b^2*d*e^2*x^2+1170*A*a*b*e^3
*x-180*A*b^2*d*e^2*x+585*B*a^2*e^3*x-360*B*a*b*d*e^2*x+72*B*b^2*d^2*e*x+715*A*a^2*e^3-260*A*a*b*d*e^2+40*A*b^2
*d^2*e-130*B*a^2*d*e^2+80*B*a*b*d^2*e-16*B*b^2*d^3)/e^4
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maxima [A] time = 0.56, size = 159, normalized size = 1.24 \[ \frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} B b^{2} - 495 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 585 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 715 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{6435 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(7/2),x, algorithm="maxima")
[Out]
2/6435*(429*(e*x + d)^(15/2)*B*b^2 - 495*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x + d)^(13/2) + 585*(3*B*b^2*d^2
- 2*(2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*a*b)*e^2)*(e*x + d)^(11/2) - 715*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b +
A*b^2)*d^2*e + (B*a^2 + 2*A*a*b)*d*e^2)*(e*x + d)^(9/2))/e^4
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mupad [B] time = 0.11, size = 115, normalized size = 0.90 \[ \frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{13\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{11\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)*(a + b*x)^2*(d + e*x)^(7/2),x)
[Out]
((d + e*x)^(13/2)*(2*A*b^2*e - 6*B*b^2*d + 4*B*a*b*e))/(13*e^4) + (2*B*b^2*(d + e*x)^(15/2))/(15*e^4) + (2*(a*
e - b*d)*(d + e*x)^(11/2)*(2*A*b*e + B*a*e - 3*B*b*d))/(11*e^4) + (2*(A*e - B*d)*(a*e - b*d)^2*(d + e*x)^(9/2)
)/(9*e^4)
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sympy [A] time = 10.81, size = 1020, normalized size = 7.97 \[ \begin {cases} \frac {2 A a^{2} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 A a^{2} d^{3} x \sqrt {d + e x}}{9} + \frac {4 A a^{2} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 A a^{2} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 A a^{2} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {8 A a b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {4 A a b d^{4} x \sqrt {d + e x}}{99 e} + \frac {32 A a b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {184 A a b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {136 A a b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {4 A a b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 A b^{2} d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 A b^{2} d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 A b^{2} d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 A b^{2} d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 A b^{2} d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 A b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 A b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {4 B a^{2} d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 B a^{2} d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 B a^{2} d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 B a^{2} d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 B a^{2} d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 B a^{2} e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {32 B a b d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {16 B a b d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {4 B a b d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {848 B a b d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {1832 B a b d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {160 B a b d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {4 B a b e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {32 B b^{2} d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {16 B b^{2} d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {4 B b^{2} d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {2 B b^{2} d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {320 B b^{2} d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {412 B b^{2} d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {92 B b^{2} d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {2 B b^{2} e^{3} x^{7} \sqrt {d + e x}}{15} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (A a^{2} x + A a b x^{2} + \frac {A b^{2} x^{3}}{3} + \frac {B a^{2} x^{2}}{2} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**2*(B*x+A)*(e*x+d)**(7/2),x)
[Out]
Piecewise((2*A*a**2*d**4*sqrt(d + e*x)/(9*e) + 8*A*a**2*d**3*x*sqrt(d + e*x)/9 + 4*A*a**2*d**2*e*x**2*sqrt(d +
e*x)/3 + 8*A*a**2*d*e**2*x**3*sqrt(d + e*x)/9 + 2*A*a**2*e**3*x**4*sqrt(d + e*x)/9 - 8*A*a*b*d**5*sqrt(d + e*
x)/(99*e**2) + 4*A*a*b*d**4*x*sqrt(d + e*x)/(99*e) + 32*A*a*b*d**3*x**2*sqrt(d + e*x)/33 + 184*A*a*b*d**2*e*x*
*3*sqrt(d + e*x)/99 + 136*A*a*b*d*e**2*x**4*sqrt(d + e*x)/99 + 4*A*a*b*e**3*x**5*sqrt(d + e*x)/11 + 16*A*b**2*
d**6*sqrt(d + e*x)/(1287*e**3) - 8*A*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*A*b**2*d**4*x**2*sqrt(d + e*x)/
(429*e) + 424*A*b**2*d**3*x**3*sqrt(d + e*x)/1287 + 916*A*b**2*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*A*b**2*d*e*
*2*x**5*sqrt(d + e*x)/143 + 2*A*b**2*e**3*x**6*sqrt(d + e*x)/13 - 4*B*a**2*d**5*sqrt(d + e*x)/(99*e**2) + 2*B*
a**2*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a**2*d**3*x**2*sqrt(d + e*x)/33 + 92*B*a**2*d**2*e*x**3*sqrt(d + e*x)/
99 + 68*B*a**2*d*e**2*x**4*sqrt(d + e*x)/99 + 2*B*a**2*e**3*x**5*sqrt(d + e*x)/11 + 32*B*a*b*d**6*sqrt(d + e*x
)/(1287*e**3) - 16*B*a*b*d**5*x*sqrt(d + e*x)/(1287*e**2) + 4*B*a*b*d**4*x**2*sqrt(d + e*x)/(429*e) + 848*B*a*
b*d**3*x**3*sqrt(d + e*x)/1287 + 1832*B*a*b*d**2*e*x**4*sqrt(d + e*x)/1287 + 160*B*a*b*d*e**2*x**5*sqrt(d + e*
x)/143 + 4*B*a*b*e**3*x**6*sqrt(d + e*x)/13 - 32*B*b**2*d**7*sqrt(d + e*x)/(6435*e**4) + 16*B*b**2*d**6*x*sqrt
(d + e*x)/(6435*e**3) - 4*B*b**2*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 2*B*b**2*d**4*x**3*sqrt(d + e*x)/(1287*
e) + 320*B*b**2*d**3*x**4*sqrt(d + e*x)/1287 + 412*B*b**2*d**2*e*x**5*sqrt(d + e*x)/715 + 92*B*b**2*d*e**2*x**
6*sqrt(d + e*x)/195 + 2*B*b**2*e**3*x**7*sqrt(d + e*x)/15, Ne(e, 0)), (d**(7/2)*(A*a**2*x + A*a*b*x**2 + A*b**
2*x**3/3 + B*a**2*x**2/2 + 2*B*a*b*x**3/3 + B*b**2*x**4/4), True))
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