3.1640 \(\int \sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=187 \[ -\frac {12 b^5 (d+e x)^{13/2} (b d-a e)}{13 e^7}+\frac {30 b^4 (d+e x)^{11/2} (b d-a e)^2}{11 e^7}-\frac {40 b^3 (d+e x)^{9/2} (b d-a e)^3}{9 e^7}+\frac {30 b^2 (d+e x)^{7/2} (b d-a e)^4}{7 e^7}-\frac {12 b (d+e x)^{5/2} (b d-a e)^5}{5 e^7}+\frac {2 (d+e x)^{3/2} (b d-a e)^6}{3 e^7}+\frac {2 b^6 (d+e x)^{15/2}}{15 e^7} \]

[Out]

2/3*(-a*e+b*d)^6*(e*x+d)^(3/2)/e^7-12/5*b*(-a*e+b*d)^5*(e*x+d)^(5/2)/e^7+30/7*b^2*(-a*e+b*d)^4*(e*x+d)^(7/2)/e
^7-40/9*b^3*(-a*e+b*d)^3*(e*x+d)^(9/2)/e^7+30/11*b^4*(-a*e+b*d)^2*(e*x+d)^(11/2)/e^7-12/13*b^5*(-a*e+b*d)*(e*x
+d)^(13/2)/e^7+2/15*b^6*(e*x+d)^(15/2)/e^7

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Rubi [A]  time = 0.06, antiderivative size = 187, 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} \[ -\frac {12 b^5 (d+e x)^{13/2} (b d-a e)}{13 e^7}+\frac {30 b^4 (d+e x)^{11/2} (b d-a e)^2}{11 e^7}-\frac {40 b^3 (d+e x)^{9/2} (b d-a e)^3}{9 e^7}+\frac {30 b^2 (d+e x)^{7/2} (b d-a e)^4}{7 e^7}-\frac {12 b (d+e x)^{5/2} (b d-a e)^5}{5 e^7}+\frac {2 (d+e x)^{3/2} (b d-a e)^6}{3 e^7}+\frac {2 b^6 (d+e x)^{15/2}}{15 e^7} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(3/2))/(3*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^(5/2))/(5*e^7) + (30*b^2*(b*d - a*e)
^4*(d + e*x)^(7/2))/(7*e^7) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^7) + (30*b^4*(b*d - a*e)^2*(d + e*x)
^(11/2))/(11*e^7) - (12*b^5*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^7) + (2*b^6*(d + e*x)^(15/2))/(15*e^7)

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 \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 \sqrt {d+e x} \, dx\\ &=\int \left (\frac {(-b d+a e)^6 \sqrt {d+e x}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{3/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{7/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{9/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{11/2}}{e^6}+\frac {b^6 (d+e x)^{13/2}}{e^6}\right ) \, dx\\ &=\frac {2 (b d-a e)^6 (d+e x)^{3/2}}{3 e^7}-\frac {12 b (b d-a e)^5 (d+e x)^{5/2}}{5 e^7}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{7/2}}{7 e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{9/2}}{9 e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{11/2}}{11 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{13/2}}{13 e^7}+\frac {2 b^6 (d+e x)^{15/2}}{15 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 145, normalized size = 0.78 \[ \frac {2 (d+e x)^{3/2} \left (-20790 b^5 (d+e x)^5 (b d-a e)+61425 b^4 (d+e x)^4 (b d-a e)^2-100100 b^3 (d+e x)^3 (b d-a e)^3+96525 b^2 (d+e x)^2 (b d-a e)^4-54054 b (d+e x) (b d-a e)^5+15015 (b d-a e)^6+3003 b^6 (d+e x)^6\right )}{45045 e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*(d + e*x)^(3/2)*(15015*(b*d - a*e)^6 - 54054*b*(b*d - a*e)^5*(d + e*x) + 96525*b^2*(b*d - a*e)^4*(d + e*x)^
2 - 100100*b^3*(b*d - a*e)^3*(d + e*x)^3 + 61425*b^4*(b*d - a*e)^2*(d + e*x)^4 - 20790*b^5*(b*d - a*e)*(d + e*
x)^5 + 3003*b^6*(d + e*x)^6))/(45045*e^7)

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fricas [B]  time = 1.03, size = 447, normalized size = 2.39 \[ \frac {2 \, {\left (3003 \, b^{6} e^{7} x^{7} + 1024 \, b^{6} d^{7} - 7680 \, a b^{5} d^{6} e + 24960 \, a^{2} b^{4} d^{5} e^{2} - 45760 \, a^{3} b^{3} d^{4} e^{3} + 51480 \, a^{4} b^{2} d^{3} e^{4} - 36036 \, a^{5} b d^{2} e^{5} + 15015 \, a^{6} d e^{6} + 231 \, {\left (b^{6} d e^{6} + 90 \, a b^{5} e^{7}\right )} x^{6} - 63 \, {\left (4 \, b^{6} d^{2} e^{5} - 30 \, a b^{5} d e^{6} - 975 \, a^{2} b^{4} e^{7}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{3} e^{4} - 60 \, a b^{5} d^{2} e^{5} + 195 \, a^{2} b^{4} d e^{6} + 2860 \, a^{3} b^{3} e^{7}\right )} x^{4} - 5 \, {\left (64 \, b^{6} d^{4} e^{3} - 480 \, a b^{5} d^{3} e^{4} + 1560 \, a^{2} b^{4} d^{2} e^{5} - 2860 \, a^{3} b^{3} d e^{6} - 19305 \, a^{4} b^{2} e^{7}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{5} e^{2} - 960 \, a b^{5} d^{4} e^{3} + 3120 \, a^{2} b^{4} d^{3} e^{4} - 5720 \, a^{3} b^{3} d^{2} e^{5} + 6435 \, a^{4} b^{2} d e^{6} + 18018 \, a^{5} b e^{7}\right )} x^{2} - {\left (512 \, b^{6} d^{6} e - 3840 \, a b^{5} d^{5} e^{2} + 12480 \, a^{2} b^{4} d^{4} e^{3} - 22880 \, a^{3} b^{3} d^{3} e^{4} + 25740 \, a^{4} b^{2} d^{2} e^{5} - 18018 \, a^{5} b d e^{6} - 15015 \, a^{6} e^{7}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*b^6*e^7*x^7 + 1024*b^6*d^7 - 7680*a*b^5*d^6*e + 24960*a^2*b^4*d^5*e^2 - 45760*a^3*b^3*d^4*e^3 +
51480*a^4*b^2*d^3*e^4 - 36036*a^5*b*d^2*e^5 + 15015*a^6*d*e^6 + 231*(b^6*d*e^6 + 90*a*b^5*e^7)*x^6 - 63*(4*b^6
*d^2*e^5 - 30*a*b^5*d*e^6 - 975*a^2*b^4*e^7)*x^5 + 35*(8*b^6*d^3*e^4 - 60*a*b^5*d^2*e^5 + 195*a^2*b^4*d*e^6 +
2860*a^3*b^3*e^7)*x^4 - 5*(64*b^6*d^4*e^3 - 480*a*b^5*d^3*e^4 + 1560*a^2*b^4*d^2*e^5 - 2860*a^3*b^3*d*e^6 - 19
305*a^4*b^2*e^7)*x^3 + 3*(128*b^6*d^5*e^2 - 960*a*b^5*d^4*e^3 + 3120*a^2*b^4*d^3*e^4 - 5720*a^3*b^3*d^2*e^5 +
6435*a^4*b^2*d*e^6 + 18018*a^5*b*e^7)*x^2 - (512*b^6*d^6*e - 3840*a*b^5*d^5*e^2 + 12480*a^2*b^4*d^4*e^3 - 2288
0*a^3*b^3*d^3*e^4 + 25740*a^4*b^2*d^2*e^5 - 18018*a^5*b*d*e^6 - 15015*a^6*e^7)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.21, size = 886, normalized size = 4.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(90090*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*d*e^(-1) + 45045*(3*(x*e + d)^(5/2) - 10*(x*e + d)^
(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4*b^2*d*e^(-2) + 25740*(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^3*d*e^(-3) + 2145*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 37
8*(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^4*d*e^(-4) + 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)*a*b^5*d*e^(-5) + 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)*b^6*d*e^(-6) + 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^5*b*e^(-1) +
19305*(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*b^2*e^(-2
) + 2860*(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^3*e^(-3) + 975*(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^4*e^(-4) + 90*(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^5*e^(-5) + 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^6*e^(-6) + 45045*sqrt(x*e + d)*a^
6*d + 15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^6)*e^(-1)

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maple [B]  time = 0.05, size = 377, normalized size = 2.02 \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 b^{6} e^{6} x^{6}+20790 a \,b^{5} e^{6} x^{5}-2772 b^{6} d \,e^{5} x^{5}+61425 a^{2} b^{4} e^{6} x^{4}-18900 a \,b^{5} d \,e^{5} x^{4}+2520 b^{6} d^{2} e^{4} x^{4}+100100 a^{3} b^{3} e^{6} x^{3}-54600 a^{2} b^{4} d \,e^{5} x^{3}+16800 a \,b^{5} d^{2} e^{4} x^{3}-2240 b^{6} d^{3} e^{3} x^{3}+96525 a^{4} b^{2} e^{6} x^{2}-85800 a^{3} b^{3} d \,e^{5} x^{2}+46800 a^{2} b^{4} d^{2} e^{4} x^{2}-14400 a \,b^{5} d^{3} e^{3} x^{2}+1920 b^{6} d^{4} e^{2} x^{2}+54054 a^{5} b \,e^{6} x -77220 a^{4} b^{2} d \,e^{5} x +68640 a^{3} b^{3} d^{2} e^{4} x -37440 a^{2} b^{4} d^{3} e^{3} x +11520 a \,b^{5} d^{4} e^{2} x -1536 b^{6} d^{5} e x +15015 a^{6} e^{6}-36036 a^{5} b d \,e^{5}+51480 a^{4} b^{2} d^{2} e^{4}-45760 a^{3} b^{3} d^{3} e^{3}+24960 a^{2} b^{4} d^{4} e^{2}-7680 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{45045 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x)

[Out]

2/45045*(e*x+d)^(3/2)*(3003*b^6*e^6*x^6+20790*a*b^5*e^6*x^5-2772*b^6*d*e^5*x^5+61425*a^2*b^4*e^6*x^4-18900*a*b
^5*d*e^5*x^4+2520*b^6*d^2*e^4*x^4+100100*a^3*b^3*e^6*x^3-54600*a^2*b^4*d*e^5*x^3+16800*a*b^5*d^2*e^4*x^3-2240*
b^6*d^3*e^3*x^3+96525*a^4*b^2*e^6*x^2-85800*a^3*b^3*d*e^5*x^2+46800*a^2*b^4*d^2*e^4*x^2-14400*a*b^5*d^3*e^3*x^
2+1920*b^6*d^4*e^2*x^2+54054*a^5*b*e^6*x-77220*a^4*b^2*d*e^5*x+68640*a^3*b^3*d^2*e^4*x-37440*a^2*b^4*d^3*e^3*x
+11520*a*b^5*d^4*e^2*x-1536*b^6*d^5*e*x+15015*a^6*e^6-36036*a^5*b*d*e^5+51480*a^4*b^2*d^2*e^4-45760*a^3*b^3*d^
3*e^3+24960*a^2*b^4*d^4*e^2-7680*a*b^5*d^5*e+1024*b^6*d^6)/e^7

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maxima [B]  time = 1.06, size = 350, normalized size = 1.87 \[ \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} b^{6} - 20790 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 61425 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 100100 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 96525 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 54054 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*b^6 - 20790*(b^6*d - a*b^5*e)*(e*x + d)^(13/2) + 61425*(b^6*d^2 - 2*a*b^5*d*e +
a^2*b^4*e^2)*(e*x + d)^(11/2) - 100100*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*(e*x + d)^(9
/2) + 96525*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*(e*x + d)^(7/2) - 54
054*(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*(e*x + d
)^(5/2) + 15015*(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^
5*b*d*e^5 + a^6*e^6)*(e*x + d)^(3/2))/e^7

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mupad [B]  time = 0.55, size = 162, normalized size = 0.87 \[ \frac {2\,b^6\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}+\frac {30\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {30\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {12\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

(2*b^6*(d + e*x)^(15/2))/(15*e^7) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(13/2))/(13*e^7) + (2*(a*e - b*d)^6*(d
+ e*x)^(3/2))/(3*e^7) + (30*b^2*(a*e - b*d)^4*(d + e*x)^(7/2))/(7*e^7) + (40*b^3*(a*e - b*d)^3*(d + e*x)^(9/2)
)/(9*e^7) + (30*b^4*(a*e - b*d)^2*(d + e*x)^(11/2))/(11*e^7) + (12*b*(a*e - b*d)^5*(d + e*x)^(5/2))/(5*e^7)

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sympy [B]  time = 7.77, size = 422, normalized size = 2.26 \[ \frac {2 \left (\frac {b^{6} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \left (6 a b^{5} e - 6 b^{6} d\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (6 a^{5} b e^{5} - 30 a^{4} b^{2} d e^{4} + 60 a^{3} b^{3} d^{2} e^{3} - 60 a^{2} b^{4} d^{3} e^{2} + 30 a b^{5} d^{4} e - 6 b^{6} d^{5}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}\right )}{3 e^{6}}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3*(e*x+d)**(1/2),x)

[Out]

2*(b**6*(d + e*x)**(15/2)/(15*e**6) + (d + e*x)**(13/2)*(6*a*b**5*e - 6*b**6*d)/(13*e**6) + (d + e*x)**(11/2)*
(15*a**2*b**4*e**2 - 30*a*b**5*d*e + 15*b**6*d**2)/(11*e**6) + (d + e*x)**(9/2)*(20*a**3*b**3*e**3 - 60*a**2*b
**4*d*e**2 + 60*a*b**5*d**2*e - 20*b**6*d**3)/(9*e**6) + (d + e*x)**(7/2)*(15*a**4*b**2*e**4 - 60*a**3*b**3*d*
e**3 + 90*a**2*b**4*d**2*e**2 - 60*a*b**5*d**3*e + 15*b**6*d**4)/(7*e**6) + (d + e*x)**(5/2)*(6*a**5*b*e**5 -
30*a**4*b**2*d*e**4 + 60*a**3*b**3*d**2*e**3 - 60*a**2*b**4*d**3*e**2 + 30*a*b**5*d**4*e - 6*b**6*d**5)/(5*e**
6) + (d + e*x)**(3/2)*(a**6*e**6 - 6*a**5*b*d*e**5 + 15*a**4*b**2*d**2*e**4 - 20*a**3*b**3*d**3*e**3 + 15*a**2
*b**4*d**4*e**2 - 6*a*b**5*d**5*e + b**6*d**6)/(3*e**6))/e

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