3.14.31 \(\int (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=129 \[ -\frac {8 b^3 (d+e x)^{15/2} (b d-a e)}{15 e^5}+\frac {12 b^2 (d+e x)^{13/2} (b d-a e)^2}{13 e^5}-\frac {8 b (d+e x)^{11/2} (b d-a e)^3}{11 e^5}+\frac {2 (d+e x)^{9/2} (b d-a e)^4}{9 e^5}+\frac {2 b^4 (d+e x)^{17/2}}{17 e^5} \]

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Rubi [A]  time = 0.07, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 43} \begin {gather*} -\frac {8 b^3 (d+e x)^{15/2} (b d-a e)}{15 e^5}+\frac {12 b^2 (d+e x)^{13/2} (b d-a e)^2}{13 e^5}-\frac {8 b (d+e x)^{11/2} (b d-a e)^3}{11 e^5}+\frac {2 (d+e x)^{9/2} (b d-a e)^4}{9 e^5}+\frac {2 b^4 (d+e x)^{17/2}}{17 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(9/2))/(9*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(11/2))/(11*e^5) + (12*b^2*(b*d - a*e
)^2*(d + e*x)^(13/2))/(13*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(15/2))/(15*e^5) + (2*b^4*(d + e*x)^(17/2))/(17*
e^5)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (d+e x)^{7/2}}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{9/2}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{11/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{13/2}}{e^4}+\frac {b^4 (d+e x)^{15/2}}{e^4}\right ) \, dx\\ &=\frac {2 (b d-a e)^4 (d+e x)^{9/2}}{9 e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{11/2}}{11 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{13/2}}{13 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{15/2}}{15 e^5}+\frac {2 b^4 (d+e x)^{17/2}}{17 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 101, normalized size = 0.78 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (-29172 b^3 (d+e x)^3 (b d-a e)+50490 b^2 (d+e x)^2 (b d-a e)^2-39780 b (d+e x) (b d-a e)^3+12155 (b d-a e)^4+6435 b^4 (d+e x)^4\right )}{109395 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(9/2)*(12155*(b*d - a*e)^4 - 39780*b*(b*d - a*e)^3*(d + e*x) + 50490*b^2*(b*d - a*e)^2*(d + e*x)^
2 - 29172*b^3*(b*d - a*e)*(d + e*x)^3 + 6435*b^4*(d + e*x)^4))/(109395*e^5)

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IntegrateAlgebraic [A]  time = 0.09, size = 213, normalized size = 1.65 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (12155 a^4 e^4+39780 a^3 b e^3 (d+e x)-48620 a^3 b d e^3+72930 a^2 b^2 d^2 e^2+50490 a^2 b^2 e^2 (d+e x)^2-119340 a^2 b^2 d e^2 (d+e x)-48620 a b^3 d^3 e+119340 a b^3 d^2 e (d+e x)+29172 a b^3 e (d+e x)^3-100980 a b^3 d e (d+e x)^2+12155 b^4 d^4-39780 b^4 d^3 (d+e x)+50490 b^4 d^2 (d+e x)^2+6435 b^4 (d+e x)^4-29172 b^4 d (d+e x)^3\right )}{109395 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(9/2)*(12155*b^4*d^4 - 48620*a*b^3*d^3*e + 72930*a^2*b^2*d^2*e^2 - 48620*a^3*b*d*e^3 + 12155*a^4*
e^4 - 39780*b^4*d^3*(d + e*x) + 119340*a*b^3*d^2*e*(d + e*x) - 119340*a^2*b^2*d*e^2*(d + e*x) + 39780*a^3*b*e^
3*(d + e*x) + 50490*b^4*d^2*(d + e*x)^2 - 100980*a*b^3*d*e*(d + e*x)^2 + 50490*a^2*b^2*e^2*(d + e*x)^2 - 29172
*b^4*d*(d + e*x)^3 + 29172*a*b^3*e*(d + e*x)^3 + 6435*b^4*(d + e*x)^4))/(109395*e^5)

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fricas [B]  time = 0.40, size = 444, normalized size = 3.44 \begin {gather*} \frac {2 \, {\left (6435 \, b^{4} e^{8} x^{8} + 128 \, b^{4} d^{8} - 1088 \, a b^{3} d^{7} e + 4080 \, a^{2} b^{2} d^{6} e^{2} - 8840 \, a^{3} b d^{5} e^{3} + 12155 \, a^{4} d^{4} e^{4} + 1716 \, {\left (13 \, b^{4} d e^{7} + 17 \, a b^{3} e^{8}\right )} x^{7} + 66 \, {\left (401 \, b^{4} d^{2} e^{6} + 1564 \, a b^{3} d e^{7} + 765 \, a^{2} b^{2} e^{8}\right )} x^{6} + 36 \, {\left (303 \, b^{4} d^{3} e^{5} + 3502 \, a b^{3} d^{2} e^{6} + 5100 \, a^{2} b^{2} d e^{7} + 1105 \, a^{3} b e^{8}\right )} x^{5} + 5 \, {\left (7 \, b^{4} d^{4} e^{4} + 10880 \, a b^{3} d^{3} e^{5} + 46716 \, a^{2} b^{2} d^{2} e^{6} + 30056 \, a^{3} b d e^{7} + 2431 \, a^{4} e^{8}\right )} x^{4} - 20 \, {\left (2 \, b^{4} d^{5} e^{3} - 17 \, a b^{3} d^{4} e^{4} - 5406 \, a^{2} b^{2} d^{3} e^{5} - 10166 \, a^{3} b d^{2} e^{6} - 2431 \, a^{4} d e^{7}\right )} x^{3} + 6 \, {\left (8 \, b^{4} d^{6} e^{2} - 68 \, a b^{3} d^{5} e^{3} + 255 \, a^{2} b^{2} d^{4} e^{4} + 17680 \, a^{3} b d^{3} e^{5} + 12155 \, a^{4} d^{2} e^{6}\right )} x^{2} - 4 \, {\left (16 \, b^{4} d^{7} e - 136 \, a b^{3} d^{6} e^{2} + 510 \, a^{2} b^{2} d^{5} e^{3} - 1105 \, a^{3} b d^{4} e^{4} - 12155 \, a^{4} d^{3} e^{5}\right )} x\right )} \sqrt {e x + d}}{109395 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

2/109395*(6435*b^4*e^8*x^8 + 128*b^4*d^8 - 1088*a*b^3*d^7*e + 4080*a^2*b^2*d^6*e^2 - 8840*a^3*b*d^5*e^3 + 1215
5*a^4*d^4*e^4 + 1716*(13*b^4*d*e^7 + 17*a*b^3*e^8)*x^7 + 66*(401*b^4*d^2*e^6 + 1564*a*b^3*d*e^7 + 765*a^2*b^2*
e^8)*x^6 + 36*(303*b^4*d^3*e^5 + 3502*a*b^3*d^2*e^6 + 5100*a^2*b^2*d*e^7 + 1105*a^3*b*e^8)*x^5 + 5*(7*b^4*d^4*
e^4 + 10880*a*b^3*d^3*e^5 + 46716*a^2*b^2*d^2*e^6 + 30056*a^3*b*d*e^7 + 2431*a^4*e^8)*x^4 - 20*(2*b^4*d^5*e^3
- 17*a*b^3*d^4*e^4 - 5406*a^2*b^2*d^3*e^5 - 10166*a^3*b*d^2*e^6 - 2431*a^4*d*e^7)*x^3 + 6*(8*b^4*d^6*e^2 - 68*
a*b^3*d^5*e^3 + 255*a^2*b^2*d^4*e^4 + 17680*a^3*b*d^3*e^5 + 12155*a^4*d^2*e^6)*x^2 - 4*(16*b^4*d^7*e - 136*a*b
^3*d^6*e^2 + 510*a^2*b^2*d^5*e^3 - 1105*a^3*b*d^4*e^4 - 12155*a^4*d^3*e^5)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.30, size = 1765, normalized size = 13.68

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2/765765*(1021020*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*b*d^4*e^(-1) + 306306*(3*(x*e + d)^(5/2) - 10*(x*e
+ d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b^2*d^4*e^(-2) + 87516*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 3
5*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*b^3*d^4*e^(-3) + 2431*(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^4*d^4*e^(-4) + 816816*(3*(x
*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*b*d^3*e^(-1) + 525096*(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^2*b^2*d^3*e^(-2) + 38896*(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^3*e^(-3) + 4420*(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^4*d^3*e^(-4) + 765765*sqrt(x*e + d)*a^4*d^4 + 10
21020*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*d^3 + 525096*(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^3*b*d^2*e^(-1) + 87516*(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^2*b^2*d^2*e^(-2) + 26520*(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^2*e^(-3) + 1530*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5
005*(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 + 300
3*sqrt(x*e + d)*d^6)*b^4*d^2*e^(-4) + 306306*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)
*a^4*d^2 + 38896*(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^3*b*d*e^(-1) + 26520*(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^2*b^2*d*e^(-2) +
4080*(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 + 9
009*(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*d*e^(-3) + 476*(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^4*d*e^(-4) + 87
516*(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^4*d + 4420*(6
3*(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^3*b*e^(-1) + 1530*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 50
05*(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^2*b^2*e^(-2) + 476*(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)*a*b^3*e^(-3) + 7*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 23562
0*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5
+ 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*b^4*e^(-4) + 2431*(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^4)*e^(-1)

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maple [A]  time = 0.05, size = 186, normalized size = 1.44 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (6435 b^{4} e^{4} x^{4}+29172 a \,b^{3} e^{4} x^{3}-3432 b^{4} d \,e^{3} x^{3}+50490 a^{2} b^{2} e^{4} x^{2}-13464 a \,b^{3} d \,e^{3} x^{2}+1584 b^{4} d^{2} e^{2} x^{2}+39780 a^{3} b \,e^{4} x -18360 a^{2} b^{2} d \,e^{3} x +4896 a \,b^{3} d^{2} e^{2} x -576 b^{4} d^{3} e x +12155 a^{4} e^{4}-8840 a^{3} b d \,e^{3}+4080 a^{2} b^{2} d^{2} e^{2}-1088 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{109395 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

2/109395*(e*x+d)^(9/2)*(6435*b^4*e^4*x^4+29172*a*b^3*e^4*x^3-3432*b^4*d*e^3*x^3+50490*a^2*b^2*e^4*x^2-13464*a*
b^3*d*e^3*x^2+1584*b^4*d^2*e^2*x^2+39780*a^3*b*e^4*x-18360*a^2*b^2*d*e^3*x+4896*a*b^3*d^2*e^2*x-576*b^4*d^3*e*
x+12155*a^4*e^4-8840*a^3*b*d*e^3+4080*a^2*b^2*d^2*e^2-1088*a*b^3*d^3*e+128*b^4*d^4)/e^5

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maxima [A]  time = 1.06, size = 181, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left (6435 \, {\left (e x + d\right )}^{\frac {17}{2}} b^{4} - 29172 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 50490 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 39780 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 12155 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{109395 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

2/109395*(6435*(e*x + d)^(17/2)*b^4 - 29172*(b^4*d - a*b^3*e)*(e*x + d)^(15/2) + 50490*(b^4*d^2 - 2*a*b^3*d*e
+ a^2*b^2*e^2)*(e*x + d)^(13/2) - 39780*(b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*(e*x + d)^(11/
2) + 12155*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(e*x + d)^(9/2))/e^5

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mupad [B]  time = 0.06, size = 112, normalized size = 0.87 \begin {gather*} \frac {2\,b^4\,{\left (d+e\,x\right )}^{17/2}}{17\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

(2*b^4*(d + e*x)^(17/2))/(17*e^5) - ((8*b^4*d - 8*a*b^3*e)*(d + e*x)^(15/2))/(15*e^5) + (2*(a*e - b*d)^4*(d +
e*x)^(9/2))/(9*e^5) + (12*b^2*(a*e - b*d)^2*(d + e*x)^(13/2))/(13*e^5) + (8*b*(a*e - b*d)^3*(d + e*x)^(11/2))/
(11*e^5)

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sympy [A]  time = 14.07, size = 903, normalized size = 7.00 \begin {gather*} \begin {cases} \frac {2 a^{4} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 a^{4} d^{3} x \sqrt {d + e x}}{9} + \frac {4 a^{4} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 a^{4} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 a^{4} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {16 a^{3} b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {8 a^{3} b d^{4} x \sqrt {d + e x}}{99 e} + \frac {64 a^{3} b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {368 a^{3} b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {272 a^{3} b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {8 a^{3} b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {32 a^{2} b^{2} d^{6} \sqrt {d + e x}}{429 e^{3}} - \frac {16 a^{2} b^{2} d^{5} x \sqrt {d + e x}}{429 e^{2}} + \frac {4 a^{2} b^{2} d^{4} x^{2} \sqrt {d + e x}}{143 e} + \frac {848 a^{2} b^{2} d^{3} x^{3} \sqrt {d + e x}}{429} + \frac {1832 a^{2} b^{2} d^{2} e x^{4} \sqrt {d + e x}}{429} + \frac {480 a^{2} b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {12 a^{2} b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {128 a b^{3} d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {64 a b^{3} d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {16 a b^{3} d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {8 a b^{3} d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {1280 a b^{3} d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {1648 a b^{3} d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {368 a b^{3} d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {8 a b^{3} e^{3} x^{7} \sqrt {d + e x}}{15} + \frac {256 b^{4} d^{8} \sqrt {d + e x}}{109395 e^{5}} - \frac {128 b^{4} d^{7} x \sqrt {d + e x}}{109395 e^{4}} + \frac {32 b^{4} d^{6} x^{2} \sqrt {d + e x}}{36465 e^{3}} - \frac {16 b^{4} d^{5} x^{3} \sqrt {d + e x}}{21879 e^{2}} + \frac {14 b^{4} d^{4} x^{4} \sqrt {d + e x}}{21879 e} + \frac {2424 b^{4} d^{3} x^{5} \sqrt {d + e x}}{12155} + \frac {1604 b^{4} d^{2} e x^{6} \sqrt {d + e x}}{3315} + \frac {104 b^{4} d e^{2} x^{7} \sqrt {d + e x}}{255} + \frac {2 b^{4} e^{3} x^{8} \sqrt {d + e x}}{17} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Piecewise((2*a**4*d**4*sqrt(d + e*x)/(9*e) + 8*a**4*d**3*x*sqrt(d + e*x)/9 + 4*a**4*d**2*e*x**2*sqrt(d + e*x)/
3 + 8*a**4*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**4*e**3*x**4*sqrt(d + e*x)/9 - 16*a**3*b*d**5*sqrt(d + e*x)/(99*e
**2) + 8*a**3*b*d**4*x*sqrt(d + e*x)/(99*e) + 64*a**3*b*d**3*x**2*sqrt(d + e*x)/33 + 368*a**3*b*d**2*e*x**3*sq
rt(d + e*x)/99 + 272*a**3*b*d*e**2*x**4*sqrt(d + e*x)/99 + 8*a**3*b*e**3*x**5*sqrt(d + e*x)/11 + 32*a**2*b**2*
d**6*sqrt(d + e*x)/(429*e**3) - 16*a**2*b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 4*a**2*b**2*d**4*x**2*sqrt(d +
e*x)/(143*e) + 848*a**2*b**2*d**3*x**3*sqrt(d + e*x)/429 + 1832*a**2*b**2*d**2*e*x**4*sqrt(d + e*x)/429 + 480*
a**2*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 12*a**2*b**2*e**3*x**6*sqrt(d + e*x)/13 - 128*a*b**3*d**7*sqrt(d + e
*x)/(6435*e**4) + 64*a*b**3*d**6*x*sqrt(d + e*x)/(6435*e**3) - 16*a*b**3*d**5*x**2*sqrt(d + e*x)/(2145*e**2) +
8*a*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 1280*a*b**3*d**3*x**4*sqrt(d + e*x)/1287 + 1648*a*b**3*d**2*e*x**
5*sqrt(d + e*x)/715 + 368*a*b**3*d*e**2*x**6*sqrt(d + e*x)/195 + 8*a*b**3*e**3*x**7*sqrt(d + e*x)/15 + 256*b**
4*d**8*sqrt(d + e*x)/(109395*e**5) - 128*b**4*d**7*x*sqrt(d + e*x)/(109395*e**4) + 32*b**4*d**6*x**2*sqrt(d +
e*x)/(36465*e**3) - 16*b**4*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 14*b**4*d**4*x**4*sqrt(d + e*x)/(21879*e) +
2424*b**4*d**3*x**5*sqrt(d + e*x)/12155 + 1604*b**4*d**2*e*x**6*sqrt(d + e*x)/3315 + 104*b**4*d*e**2*x**7*sqr
t(d + e*x)/255 + 2*b**4*e**3*x**8*sqrt(d + e*x)/17, Ne(e, 0)), (d**(7/2)*(a**4*x + 2*a**3*b*x**2 + 2*a**2*b**2
*x**3 + a*b**3*x**4 + b**4*x**5/5), True))

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