3.1639 \(\int (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)
Optimal. Leaf size=187 \[ -\frac {4 b^5 (d+e x)^{15/2} (b d-a e)}{5 e^7}+\frac {30 b^4 (d+e x)^{13/2} (b d-a e)^2}{13 e^7}-\frac {40 b^3 (d+e x)^{11/2} (b d-a e)^3}{11 e^7}+\frac {10 b^2 (d+e x)^{9/2} (b d-a e)^4}{3 e^7}-\frac {12 b (d+e x)^{7/2} (b d-a e)^5}{7 e^7}+\frac {2 (d+e x)^{5/2} (b d-a e)^6}{5 e^7}+\frac {2 b^6 (d+e x)^{17/2}}{17 e^7} \]
[Out]
2/5*(-a*e+b*d)^6*(e*x+d)^(5/2)/e^7-12/7*b*(-a*e+b*d)^5*(e*x+d)^(7/2)/e^7+10/3*b^2*(-a*e+b*d)^4*(e*x+d)^(9/2)/e
^7-40/11*b^3*(-a*e+b*d)^3*(e*x+d)^(11/2)/e^7+30/13*b^4*(-a*e+b*d)^2*(e*x+d)^(13/2)/e^7-4/5*b^5*(-a*e+b*d)*(e*x
+d)^(15/2)/e^7+2/17*b^6*(e*x+d)^(17/2)/e^7
________________________________________________________________________________________
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} \[ -\frac {4 b^5 (d+e x)^{15/2} (b d-a e)}{5 e^7}+\frac {30 b^4 (d+e x)^{13/2} (b d-a e)^2}{13 e^7}-\frac {40 b^3 (d+e x)^{11/2} (b d-a e)^3}{11 e^7}+\frac {10 b^2 (d+e x)^{9/2} (b d-a e)^4}{3 e^7}-\frac {12 b (d+e x)^{7/2} (b d-a e)^5}{7 e^7}+\frac {2 (d+e x)^{5/2} (b d-a e)^6}{5 e^7}+\frac {2 b^6 (d+e x)^{17/2}}{17 e^7} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(b*d - a*e)^6*(d + e*x)^(5/2))/(5*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^(7/2))/(7*e^7) + (10*b^2*(b*d - a*e)
^4*(d + e*x)^(9/2))/(3*e^7) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(11/2))/(11*e^7) + (30*b^4*(b*d - a*e)^2*(d + e*
x)^(13/2))/(13*e^7) - (4*b^5*(b*d - a*e)*(d + e*x)^(15/2))/(5*e^7) + (2*b^6*(d + e*x)^(17/2))/(17*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 (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^{3/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^6 (d+e x)^{3/2}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{5/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{7/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{9/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{11/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{13/2}}{e^6}+\frac {b^6 (d+e x)^{15/2}}{e^6}\right ) \, dx\\ &=\frac {2 (b d-a e)^6 (d+e x)^{5/2}}{5 e^7}-\frac {12 b (b d-a e)^5 (d+e x)^{7/2}}{7 e^7}+\frac {10 b^2 (b d-a e)^4 (d+e x)^{9/2}}{3 e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{11/2}}{11 e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{13/2}}{13 e^7}-\frac {4 b^5 (b d-a e) (d+e x)^{15/2}}{5 e^7}+\frac {2 b^6 (d+e x)^{17/2}}{17 e^7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 145, normalized size = 0.78 \[ \frac {2 (d+e x)^{5/2} \left (-102102 b^5 (d+e x)^5 (b d-a e)+294525 b^4 (d+e x)^4 (b d-a e)^2-464100 b^3 (d+e x)^3 (b d-a e)^3+425425 b^2 (d+e x)^2 (b d-a e)^4-218790 b (d+e x) (b d-a e)^5+51051 (b d-a e)^6+15015 b^6 (d+e x)^6\right )}{255255 e^7} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(5/2)*(51051*(b*d - a*e)^6 - 218790*b*(b*d - a*e)^5*(d + e*x) + 425425*b^2*(b*d - a*e)^4*(d + e*x
)^2 - 464100*b^3*(b*d - a*e)^3*(d + e*x)^3 + 294525*b^4*(b*d - a*e)^2*(d + e*x)^4 - 102102*b^5*(b*d - a*e)*(d
+ e*x)^5 + 15015*b^6*(d + e*x)^6))/(255255*e^7)
________________________________________________________________________________________
fricas [B] time = 0.92, size = 541, normalized size = 2.89 \[ \frac {2 \, {\left (15015 \, b^{6} e^{8} x^{8} + 1024 \, b^{6} d^{8} - 8704 \, a b^{5} d^{7} e + 32640 \, a^{2} b^{4} d^{6} e^{2} - 70720 \, a^{3} b^{3} d^{5} e^{3} + 97240 \, a^{4} b^{2} d^{4} e^{4} - 87516 \, a^{5} b d^{3} e^{5} + 51051 \, a^{6} d^{2} e^{6} + 6006 \, {\left (3 \, b^{6} d e^{7} + 17 \, a b^{5} e^{8}\right )} x^{7} + 231 \, {\left (b^{6} d^{2} e^{6} + 544 \, a b^{5} d e^{7} + 1275 \, a^{2} b^{4} e^{8}\right )} x^{6} - 42 \, {\left (6 \, b^{6} d^{3} e^{5} - 51 \, a b^{5} d^{2} e^{6} - 8925 \, a^{2} b^{4} d e^{7} - 11050 \, a^{3} b^{3} e^{8}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{4} e^{4} - 68 \, a b^{5} d^{3} e^{5} + 255 \, a^{2} b^{4} d^{2} e^{6} + 17680 \, a^{3} b^{3} d e^{7} + 12155 \, a^{4} b^{2} e^{8}\right )} x^{4} - 10 \, {\left (32 \, b^{6} d^{5} e^{3} - 272 \, a b^{5} d^{4} e^{4} + 1020 \, a^{2} b^{4} d^{3} e^{5} - 2210 \, a^{3} b^{3} d^{2} e^{6} - 60775 \, a^{4} b^{2} d e^{7} - 21879 \, a^{5} b e^{8}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{6} e^{2} - 1088 \, a b^{5} d^{5} e^{3} + 4080 \, a^{2} b^{4} d^{4} e^{4} - 8840 \, a^{3} b^{3} d^{3} e^{5} + 12155 \, a^{4} b^{2} d^{2} e^{6} + 116688 \, a^{5} b d e^{7} + 17017 \, a^{6} e^{8}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{7} e - 2176 \, a b^{5} d^{6} e^{2} + 8160 \, a^{2} b^{4} d^{5} e^{3} - 17680 \, a^{3} b^{3} d^{4} e^{4} + 24310 \, a^{4} b^{2} d^{3} e^{5} - 21879 \, a^{5} b d^{2} e^{6} - 51051 \, a^{6} d e^{7}\right )} x\right )} \sqrt {e x + d}}{255255 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
[Out]
2/255255*(15015*b^6*e^8*x^8 + 1024*b^6*d^8 - 8704*a*b^5*d^7*e + 32640*a^2*b^4*d^6*e^2 - 70720*a^3*b^3*d^5*e^3
+ 97240*a^4*b^2*d^4*e^4 - 87516*a^5*b*d^3*e^5 + 51051*a^6*d^2*e^6 + 6006*(3*b^6*d*e^7 + 17*a*b^5*e^8)*x^7 + 23
1*(b^6*d^2*e^6 + 544*a*b^5*d*e^7 + 1275*a^2*b^4*e^8)*x^6 - 42*(6*b^6*d^3*e^5 - 51*a*b^5*d^2*e^6 - 8925*a^2*b^4
*d*e^7 - 11050*a^3*b^3*e^8)*x^5 + 35*(8*b^6*d^4*e^4 - 68*a*b^5*d^3*e^5 + 255*a^2*b^4*d^2*e^6 + 17680*a^3*b^3*d
*e^7 + 12155*a^4*b^2*e^8)*x^4 - 10*(32*b^6*d^5*e^3 - 272*a*b^5*d^4*e^4 + 1020*a^2*b^4*d^3*e^5 - 2210*a^3*b^3*d
^2*e^6 - 60775*a^4*b^2*d*e^7 - 21879*a^5*b*e^8)*x^3 + 3*(128*b^6*d^6*e^2 - 1088*a*b^5*d^5*e^3 + 4080*a^2*b^4*d
^4*e^4 - 8840*a^3*b^3*d^3*e^5 + 12155*a^4*b^2*d^2*e^6 + 116688*a^5*b*d*e^7 + 17017*a^6*e^8)*x^2 - 2*(256*b^6*d
^7*e - 2176*a*b^5*d^6*e^2 + 8160*a^2*b^4*d^5*e^3 - 17680*a^3*b^3*d^4*e^4 + 24310*a^4*b^2*d^3*e^5 - 21879*a^5*b
*d^2*e^6 - 51051*a^6*d*e^7)*x)*sqrt(e*x + d)/e^7
________________________________________________________________________________________
giac [B] time = 0.26, size = 1483, normalized size = 7.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
[Out]
2/765765*(1531530*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*d^2*e^(-1) + 765765*(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^2*e^(-2) + 437580*(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^2*e^(-3) + 36465*(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^4*d^2*e^(-4) + 6630
*(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^2*e^(-5) + 255*(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^2*e^(-6) + 612612*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)
*d^2)*a^5*b*d*e^(-1) + 656370*(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*d*e^(-2) + 97240*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 42
0*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^3*b^3*d*e^(-3) + 33150*(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*d*e^(-4) + 3060*(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*d*e^(-
5) + 238*(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*d*e^(-6) + 765765*sqrt(x*e + d)*a^6*d^2 + 510510*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^6*d + 131274*
(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^5*b*e^(-1) + 3646
5*(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*b^2*e^(-2) + 22100*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1
386*(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^3*e^(-3) + 3825*(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^2*b^4*e^(-4) + 714*(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 - 2702
7*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*a*b^5*e^(-5) + 7*(6435*(x*e + d)^(
17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(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*sq
rt(x*e + d)*d^8)*b^6*e^(-6) + 51051*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^6)*e^(
-1)
________________________________________________________________________________________
maple [B] time = 0.07, size = 377, normalized size = 2.02 \[ \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (15015 b^{6} e^{6} x^{6}+102102 a \,b^{5} e^{6} x^{5}-12012 b^{6} d \,e^{5} x^{5}+294525 a^{2} b^{4} e^{6} x^{4}-78540 a \,b^{5} d \,e^{5} x^{4}+9240 b^{6} d^{2} e^{4} x^{4}+464100 a^{3} b^{3} e^{6} x^{3}-214200 a^{2} b^{4} d \,e^{5} x^{3}+57120 a \,b^{5} d^{2} e^{4} x^{3}-6720 b^{6} d^{3} e^{3} x^{3}+425425 a^{4} b^{2} e^{6} x^{2}-309400 a^{3} b^{3} d \,e^{5} x^{2}+142800 a^{2} b^{4} d^{2} e^{4} x^{2}-38080 a \,b^{5} d^{3} e^{3} x^{2}+4480 b^{6} d^{4} e^{2} x^{2}+218790 a^{5} b \,e^{6} x -243100 a^{4} b^{2} d \,e^{5} x +176800 a^{3} b^{3} d^{2} e^{4} x -81600 a^{2} b^{4} d^{3} e^{3} x +21760 a \,b^{5} d^{4} e^{2} x -2560 b^{6} d^{5} e x +51051 a^{6} e^{6}-87516 a^{5} b d \,e^{5}+97240 a^{4} b^{2} d^{2} e^{4}-70720 a^{3} b^{3} d^{3} e^{3}+32640 a^{2} b^{4} d^{4} e^{2}-8704 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{255255 e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
2/255255*(e*x+d)^(5/2)*(15015*b^6*e^6*x^6+102102*a*b^5*e^6*x^5-12012*b^6*d*e^5*x^5+294525*a^2*b^4*e^6*x^4-7854
0*a*b^5*d*e^5*x^4+9240*b^6*d^2*e^4*x^4+464100*a^3*b^3*e^6*x^3-214200*a^2*b^4*d*e^5*x^3+57120*a*b^5*d^2*e^4*x^3
-6720*b^6*d^3*e^3*x^3+425425*a^4*b^2*e^6*x^2-309400*a^3*b^3*d*e^5*x^2+142800*a^2*b^4*d^2*e^4*x^2-38080*a*b^5*d
^3*e^3*x^2+4480*b^6*d^4*e^2*x^2+218790*a^5*b*e^6*x-243100*a^4*b^2*d*e^5*x+176800*a^3*b^3*d^2*e^4*x-81600*a^2*b
^4*d^3*e^3*x+21760*a*b^5*d^4*e^2*x-2560*b^6*d^5*e*x+51051*a^6*e^6-87516*a^5*b*d*e^5+97240*a^4*b^2*d^2*e^4-7072
0*a^3*b^3*d^3*e^3+32640*a^2*b^4*d^4*e^2-8704*a*b^5*d^5*e+1024*b^6*d^6)/e^7
________________________________________________________________________________________
maxima [B] time = 1.15, size = 350, normalized size = 1.87 \[ \frac {2 \, {\left (15015 \, {\left (e x + d\right )}^{\frac {17}{2}} b^{6} - 102102 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 294525 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 464100 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 425425 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 218790 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 51051 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{255255 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
[Out]
2/255255*(15015*(e*x + d)^(17/2)*b^6 - 102102*(b^6*d - a*b^5*e)*(e*x + d)^(15/2) + 294525*(b^6*d^2 - 2*a*b^5*d
*e + a^2*b^4*e^2)*(e*x + d)^(13/2) - 464100*(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
)^(11/2) + 425425*(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)^(9/2
) - 218790*(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)^(7/2) + 51051*(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)^(5/2))/e^7
________________________________________________________________________________________
mupad [B] time = 0.55, size = 162, normalized size = 0.87 \[ \frac {2\,b^6\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {30\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {12\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
[Out]
(2*b^6*(d + e*x)^(17/2))/(17*e^7) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(15/2))/(15*e^7) + (2*(a*e - b*d)^6*(d
+ e*x)^(5/2))/(5*e^7) + (10*b^2*(a*e - b*d)^4*(d + e*x)^(9/2))/(3*e^7) + (40*b^3*(a*e - b*d)^3*(d + e*x)^(11/2
))/(11*e^7) + (30*b^4*(a*e - b*d)^2*(d + e*x)^(13/2))/(13*e^7) + (12*b*(a*e - b*d)^5*(d + e*x)^(7/2))/(7*e^7)
________________________________________________________________________________________
sympy [A] time = 35.95, size = 1000, normalized size = 5.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
a**6*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**6*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 12*a**5*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*a**5*b*(d**2*(d + e*x
)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 30*a**4*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d
*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 30*a**4*b**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**3 + 40*a**3*b**3*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**4 + 40*a**3*b**3*(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
+ 30*a**2*b**4*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**5 + 30*a**2*b**4*(-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 + 12*a*b**5*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**6 + 12*a*b**5*(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**6 + 2*b**6*d*(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**7 + 2*b**6*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/
5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(
13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7
________________________________________________________________________________________