3.19.25 \(\int (a+b x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)
Optimal. Leaf size=158 \[ -\frac {10 b^4 (d+e x)^{17/2} (b d-a e)}{17 e^6}+\frac {4 b^3 (d+e x)^{15/2} (b d-a e)^2}{3 e^6}-\frac {20 b^2 (d+e x)^{13/2} (b d-a e)^3}{13 e^6}+\frac {10 b (d+e x)^{11/2} (b d-a e)^4}{11 e^6}-\frac {2 (d+e x)^{9/2} (b d-a e)^5}{9 e^6}+\frac {2 b^5 (d+e x)^{19/2}}{19 e^6} \]
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Rubi [A] time = 0.07, antiderivative size = 158, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used =
{27, 43} \begin {gather*} -\frac {10 b^4 (d+e x)^{17/2} (b d-a e)}{17 e^6}+\frac {4 b^3 (d+e x)^{15/2} (b d-a e)^2}{3 e^6}-\frac {20 b^2 (d+e x)^{13/2} (b d-a e)^3}{13 e^6}+\frac {10 b (d+e x)^{11/2} (b d-a e)^4}{11 e^6}-\frac {2 (d+e x)^{9/2} (b d-a e)^5}{9 e^6}+\frac {2 b^5 (d+e x)^{19/2}}{19 e^6} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
(-2*(b*d - a*e)^5*(d + e*x)^(9/2))/(9*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(11/2))/(11*e^6) - (20*b^2*(b*d - a
*e)^3*(d + e*x)^(13/2))/(13*e^6) + (4*b^3*(b*d - a*e)^2*(d + e*x)^(15/2))/(3*e^6) - (10*b^4*(b*d - a*e)*(d + e
*x)^(17/2))/(17*e^6) + (2*b^5*(d + e*x)^(19/2))/(19*e^6)
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 (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^5 (d+e x)^{7/2}}{e^5}+\frac {5 b (b d-a e)^4 (d+e x)^{9/2}}{e^5}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{11/2}}{e^5}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{13/2}}{e^5}-\frac {5 b^4 (b d-a e) (d+e x)^{15/2}}{e^5}+\frac {b^5 (d+e x)^{17/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (b d-a e)^5 (d+e x)^{9/2}}{9 e^6}+\frac {10 b (b d-a e)^4 (d+e x)^{11/2}}{11 e^6}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{13/2}}{13 e^6}+\frac {4 b^3 (b d-a e)^2 (d+e x)^{15/2}}{3 e^6}-\frac {10 b^4 (b d-a e) (d+e x)^{17/2}}{17 e^6}+\frac {2 b^5 (d+e x)^{19/2}}{19 e^6}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 123, normalized size = 0.78 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (-122265 b^4 (d+e x)^4 (b d-a e)+277134 b^3 (d+e x)^3 (b d-a e)^2-319770 b^2 (d+e x)^2 (b d-a e)^3+188955 b (d+e x) (b d-a e)^4-46189 (b d-a e)^5+21879 b^5 (d+e x)^5\right )}{415701 e^6} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
(2*(d + e*x)^(9/2)*(-46189*(b*d - a*e)^5 + 188955*b*(b*d - a*e)^4*(d + e*x) - 319770*b^2*(b*d - a*e)^3*(d + e*
x)^2 + 277134*b^3*(b*d - a*e)^2*(d + e*x)^3 - 122265*b^4*(b*d - a*e)*(d + e*x)^4 + 21879*b^5*(d + e*x)^5))/(41
5701*e^6)
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IntegrateAlgebraic [A] time = 0.12, size = 315, normalized size = 1.99 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (46189 a^5 e^5+188955 a^4 b e^4 (d+e x)-230945 a^4 b d e^4+461890 a^3 b^2 d^2 e^3+319770 a^3 b^2 e^3 (d+e x)^2-755820 a^3 b^2 d e^3 (d+e x)-461890 a^2 b^3 d^3 e^2+1133730 a^2 b^3 d^2 e^2 (d+e x)+277134 a^2 b^3 e^2 (d+e x)^3-959310 a^2 b^3 d e^2 (d+e x)^2+230945 a b^4 d^4 e-755820 a b^4 d^3 e (d+e x)+959310 a b^4 d^2 e (d+e x)^2+122265 a b^4 e (d+e x)^4-554268 a b^4 d e (d+e x)^3-46189 b^5 d^5+188955 b^5 d^4 (d+e x)-319770 b^5 d^3 (d+e x)^2+277134 b^5 d^2 (d+e x)^3+21879 b^5 (d+e x)^5-122265 b^5 d (d+e x)^4\right )}{415701 e^6} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
(2*(d + e*x)^(9/2)*(-46189*b^5*d^5 + 230945*a*b^4*d^4*e - 461890*a^2*b^3*d^3*e^2 + 461890*a^3*b^2*d^2*e^3 - 23
0945*a^4*b*d*e^4 + 46189*a^5*e^5 + 188955*b^5*d^4*(d + e*x) - 755820*a*b^4*d^3*e*(d + e*x) + 1133730*a^2*b^3*d
^2*e^2*(d + e*x) - 755820*a^3*b^2*d*e^3*(d + e*x) + 188955*a^4*b*e^4*(d + e*x) - 319770*b^5*d^3*(d + e*x)^2 +
959310*a*b^4*d^2*e*(d + e*x)^2 - 959310*a^2*b^3*d*e^2*(d + e*x)^2 + 319770*a^3*b^2*e^3*(d + e*x)^2 + 277134*b^
5*d^2*(d + e*x)^3 - 554268*a*b^4*d*e*(d + e*x)^3 + 277134*a^2*b^3*e^2*(d + e*x)^3 - 122265*b^5*d*(d + e*x)^4 +
122265*a*b^4*e*(d + e*x)^4 + 21879*b^5*(d + e*x)^5))/(415701*e^6)
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fricas [B] time = 0.44, size = 579, normalized size = 3.66 \begin {gather*} \frac {2 \, {\left (21879 \, b^{5} e^{9} x^{9} - 256 \, b^{5} d^{9} + 2432 \, a b^{4} d^{8} e - 10336 \, a^{2} b^{3} d^{7} e^{2} + 25840 \, a^{3} b^{2} d^{6} e^{3} - 41990 \, a^{4} b d^{5} e^{4} + 46189 \, a^{5} d^{4} e^{5} + 1287 \, {\left (58 \, b^{5} d e^{8} + 95 \, a b^{4} e^{9}\right )} x^{8} + 858 \, {\left (101 \, b^{5} d^{2} e^{7} + 494 \, a b^{4} d e^{8} + 323 \, a^{2} b^{3} e^{9}\right )} x^{7} + 66 \, {\left (524 \, b^{5} d^{3} e^{6} + 7619 \, a b^{4} d^{2} e^{7} + 14858 \, a^{2} b^{3} d e^{8} + 4845 \, a^{3} b^{2} e^{9}\right )} x^{6} + 9 \, {\left (7 \, b^{5} d^{4} e^{5} + 23028 \, a b^{4} d^{3} e^{6} + 133076 \, a^{2} b^{3} d^{2} e^{7} + 129200 \, a^{3} b^{2} d e^{8} + 20995 \, a^{4} b e^{9}\right )} x^{5} - {\left (70 \, b^{5} d^{5} e^{4} - 665 \, a b^{4} d^{4} e^{5} - 516800 \, a^{2} b^{3} d^{3} e^{6} - 1479340 \, a^{3} b^{2} d^{2} e^{7} - 713830 \, a^{4} b d e^{8} - 46189 \, a^{5} e^{9}\right )} x^{4} + 2 \, {\left (40 \, b^{5} d^{6} e^{3} - 380 \, a b^{4} d^{5} e^{4} + 1615 \, a^{2} b^{3} d^{4} e^{5} + 342380 \, a^{3} b^{2} d^{3} e^{6} + 482885 \, a^{4} b d^{2} e^{7} + 92378 \, a^{5} d e^{8}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{7} e^{2} - 152 \, a b^{4} d^{6} e^{3} + 646 \, a^{2} b^{3} d^{5} e^{4} - 1615 \, a^{3} b^{2} d^{4} e^{5} - 83980 \, a^{4} b d^{3} e^{6} - 46189 \, a^{5} d^{2} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{8} e - 1216 \, a b^{4} d^{7} e^{2} + 5168 \, a^{2} b^{3} d^{6} e^{3} - 12920 \, a^{3} b^{2} d^{5} e^{4} + 20995 \, a^{4} b d^{4} e^{5} + 184756 \, a^{5} d^{3} e^{6}\right )} x\right )} \sqrt {e x + d}}{415701 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
[Out]
2/415701*(21879*b^5*e^9*x^9 - 256*b^5*d^9 + 2432*a*b^4*d^8*e - 10336*a^2*b^3*d^7*e^2 + 25840*a^3*b^2*d^6*e^3 -
41990*a^4*b*d^5*e^4 + 46189*a^5*d^4*e^5 + 1287*(58*b^5*d*e^8 + 95*a*b^4*e^9)*x^8 + 858*(101*b^5*d^2*e^7 + 494
*a*b^4*d*e^8 + 323*a^2*b^3*e^9)*x^7 + 66*(524*b^5*d^3*e^6 + 7619*a*b^4*d^2*e^7 + 14858*a^2*b^3*d*e^8 + 4845*a^
3*b^2*e^9)*x^6 + 9*(7*b^5*d^4*e^5 + 23028*a*b^4*d^3*e^6 + 133076*a^2*b^3*d^2*e^7 + 129200*a^3*b^2*d*e^8 + 2099
5*a^4*b*e^9)*x^5 - (70*b^5*d^5*e^4 - 665*a*b^4*d^4*e^5 - 516800*a^2*b^3*d^3*e^6 - 1479340*a^3*b^2*d^2*e^7 - 71
3830*a^4*b*d*e^8 - 46189*a^5*e^9)*x^4 + 2*(40*b^5*d^6*e^3 - 380*a*b^4*d^5*e^4 + 1615*a^2*b^3*d^4*e^5 + 342380*
a^3*b^2*d^3*e^6 + 482885*a^4*b*d^2*e^7 + 92378*a^5*d*e^8)*x^3 - 6*(16*b^5*d^7*e^2 - 152*a*b^4*d^6*e^3 + 646*a^
2*b^3*d^5*e^4 - 1615*a^3*b^2*d^4*e^5 - 83980*a^4*b*d^3*e^6 - 46189*a^5*d^2*e^7)*x^2 + (128*b^5*d^8*e - 1216*a*
b^4*d^7*e^2 + 5168*a^2*b^3*d^6*e^3 - 12920*a^3*b^2*d^5*e^4 + 20995*a^4*b*d^4*e^5 + 184756*a^5*d^3*e^6)*x)*sqrt
(e*x + d)/e^6
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giac [B] time = 0.31, size = 2325, normalized size = 14.72
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
[Out]
2/14549535*(24249225*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*b*d^4*e^(-1) + 9699690*(3*(x*e + d)^(5/2) - 10*
(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*b^2*d^4*e^(-2) + 4157010*(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^3*d^4*e^(-3) + 230945*(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^4*d^4*e^(-4) +
20995*(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^5*d^4*e^(-5) + 19399380*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2
)*d + 15*sqrt(x*e + d)*d^2)*a^4*b*d^3*e^(-1) + 16628040*(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^2*d^3*e^(-2) + 1847560*(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^3*d^3*e^(-3) + 419900*(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^4*d^3*e^(-4) + 19380*(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 + 30
03*sqrt(x*e + d)*d^6)*b^5*d^3*e^(-5) + 14549535*sqrt(x*e + d)*a^5*d^4 + 19399380*((x*e + d)^(3/2) - 3*sqrt(x*e
+ d)*d)*a^5*d^3 + 12471030*(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*d^2*e^(-1) + 2771340*(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^2*d^2*e^(-2) + 1259700*(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^3*d^2*e^(-3) + 145350*(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^
4*d^2*e^(-4) + 13566*(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*sqr
t(x*e + d)*d^7)*b^5*d^2*e^(-5) + 5819814*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^5
*d^2 + 923780*(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*d*e^(-1) + 839800*(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^3*b^2*d*e^(-2) + 19
3800*(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^2*b^3*d*e^(-3) + 45220*(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^4*d*e^(-4
) + 532*(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*sqrt(x*e + d)*d^8)*b^5*d*e^(-5) + 1662804*(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^5*d + 104975*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 99
0*(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^4*b*e^(
-1) + 48450*(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^3*b^2*e^(-2) + 22610*(42
9*(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^2*b^3*
e^(-3) + 665*(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*sqrt(x*e + d)*d^8)*a*b^4*e^(-4) + 63*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(1
7/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(
x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 -
230945*sqrt(x*e + d)*d^9)*b^5*e^(-5) + 46189*(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^5)*e^(-1)
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maple [B] time = 0.05, size = 273, normalized size = 1.73 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (21879 b^{5} e^{5} x^{5}+122265 a \,b^{4} e^{5} x^{4}-12870 b^{5} d \,e^{4} x^{4}+277134 a^{2} b^{3} e^{5} x^{3}-65208 a \,b^{4} d \,e^{4} x^{3}+6864 b^{5} d^{2} e^{3} x^{3}+319770 a^{3} b^{2} e^{5} x^{2}-127908 a^{2} b^{3} d \,e^{4} x^{2}+30096 a \,b^{4} d^{2} e^{3} x^{2}-3168 b^{5} d^{3} e^{2} x^{2}+188955 a^{4} b \,e^{5} x -116280 a^{3} b^{2} d \,e^{4} x +46512 a^{2} b^{3} d^{2} e^{3} x -10944 a \,b^{4} d^{3} e^{2} x +1152 b^{5} d^{4} e x +46189 a^{5} e^{5}-41990 a^{4} b d \,e^{4}+25840 a^{3} b^{2} d^{2} e^{3}-10336 a^{2} b^{3} d^{3} e^{2}+2432 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right )}{415701 e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
2/415701*(e*x+d)^(9/2)*(21879*b^5*e^5*x^5+122265*a*b^4*e^5*x^4-12870*b^5*d*e^4*x^4+277134*a^2*b^3*e^5*x^3-6520
8*a*b^4*d*e^4*x^3+6864*b^5*d^2*e^3*x^3+319770*a^3*b^2*e^5*x^2-127908*a^2*b^3*d*e^4*x^2+30096*a*b^4*d^2*e^3*x^2
-3168*b^5*d^3*e^2*x^2+188955*a^4*b*e^5*x-116280*a^3*b^2*d*e^4*x+46512*a^2*b^3*d^2*e^3*x-10944*a*b^4*d^3*e^2*x+
1152*b^5*d^4*e*x+46189*a^5*e^5-41990*a^4*b*d*e^4+25840*a^3*b^2*d^2*e^3-10336*a^2*b^3*d^3*e^2+2432*a*b^4*d^4*e-
256*b^5*d^5)/e^6
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maxima [A] time = 0.51, size = 259, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left (21879 \, {\left (e x + d\right )}^{\frac {19}{2}} b^{5} - 122265 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 277134 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 319770 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 188955 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 46189 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{415701 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
[Out]
2/415701*(21879*(e*x + d)^(19/2)*b^5 - 122265*(b^5*d - a*b^4*e)*(e*x + d)^(17/2) + 277134*(b^5*d^2 - 2*a*b^4*d
*e + a^2*b^3*e^2)*(e*x + d)^(15/2) - 319770*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d
)^(13/2) + 188955*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d)^(11/2)
- 46189*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*(e*x +
d)^(9/2))/e^6
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mupad [B] time = 0.07, size = 137, normalized size = 0.87 \begin {gather*} \frac {2\,b^5\,{\left (d+e\,x\right )}^{19/2}}{19\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {20\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {4\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{15/2}}{3\,e^6}+\frac {10\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)*(d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
[Out]
(2*b^5*(d + e*x)^(19/2))/(19*e^6) - ((10*b^5*d - 10*a*b^4*e)*(d + e*x)^(17/2))/(17*e^6) + (2*(a*e - b*d)^5*(d
+ e*x)^(9/2))/(9*e^6) + (20*b^2*(a*e - b*d)^3*(d + e*x)^(13/2))/(13*e^6) + (4*b^3*(a*e - b*d)^2*(d + e*x)^(15/
2))/(3*e^6) + (10*b*(a*e - b*d)^4*(d + e*x)^(11/2))/(11*e^6)
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sympy [A] time = 15.96, size = 1187, normalized size = 7.51 \begin {gather*} \begin {cases} \frac {2 a^{5} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 a^{5} d^{3} x \sqrt {d + e x}}{9} + \frac {4 a^{5} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 a^{5} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 a^{5} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {20 a^{4} b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {10 a^{4} b d^{4} x \sqrt {d + e x}}{99 e} + \frac {80 a^{4} b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {460 a^{4} b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {340 a^{4} b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {10 a^{4} b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {160 a^{3} b^{2} d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {80 a^{3} b^{2} d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {20 a^{3} b^{2} d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {4240 a^{3} b^{2} d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {9160 a^{3} b^{2} d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {800 a^{3} b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {20 a^{3} b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {64 a^{2} b^{3} d^{7} \sqrt {d + e x}}{1287 e^{4}} + \frac {32 a^{2} b^{3} d^{6} x \sqrt {d + e x}}{1287 e^{3}} - \frac {8 a^{2} b^{3} d^{5} x^{2} \sqrt {d + e x}}{429 e^{2}} + \frac {20 a^{2} b^{3} d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {3200 a^{2} b^{3} d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {824 a^{2} b^{3} d^{2} e x^{5} \sqrt {d + e x}}{143} + \frac {184 a^{2} b^{3} d e^{2} x^{6} \sqrt {d + e x}}{39} + \frac {4 a^{2} b^{3} e^{3} x^{7} \sqrt {d + e x}}{3} + \frac {256 a b^{4} d^{8} \sqrt {d + e x}}{21879 e^{5}} - \frac {128 a b^{4} d^{7} x \sqrt {d + e x}}{21879 e^{4}} + \frac {32 a b^{4} d^{6} x^{2} \sqrt {d + e x}}{7293 e^{3}} - \frac {80 a b^{4} d^{5} x^{3} \sqrt {d + e x}}{21879 e^{2}} + \frac {70 a b^{4} d^{4} x^{4} \sqrt {d + e x}}{21879 e} + \frac {2424 a b^{4} d^{3} x^{5} \sqrt {d + e x}}{2431} + \frac {1604 a b^{4} d^{2} e x^{6} \sqrt {d + e x}}{663} + \frac {104 a b^{4} d e^{2} x^{7} \sqrt {d + e x}}{51} + \frac {10 a b^{4} e^{3} x^{8} \sqrt {d + e x}}{17} - \frac {512 b^{5} d^{9} \sqrt {d + e x}}{415701 e^{6}} + \frac {256 b^{5} d^{8} x \sqrt {d + e x}}{415701 e^{5}} - \frac {64 b^{5} d^{7} x^{2} \sqrt {d + e x}}{138567 e^{4}} + \frac {160 b^{5} d^{6} x^{3} \sqrt {d + e x}}{415701 e^{3}} - \frac {140 b^{5} d^{5} x^{4} \sqrt {d + e x}}{415701 e^{2}} + \frac {14 b^{5} d^{4} x^{5} \sqrt {d + e x}}{46189 e} + \frac {2096 b^{5} d^{3} x^{6} \sqrt {d + e x}}{12597} + \frac {404 b^{5} d^{2} e x^{7} \sqrt {d + e x}}{969} + \frac {116 b^{5} d e^{2} x^{8} \sqrt {d + e x}}{323} + \frac {2 b^{5} e^{3} x^{9} \sqrt {d + e x}}{19} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (a^{5} x + \frac {5 a^{4} b x^{2}}{2} + \frac {10 a^{3} b^{2} x^{3}}{3} + \frac {5 a^{2} b^{3} x^{4}}{2} + a b^{4} x^{5} + \frac {b^{5} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
Piecewise((2*a**5*d**4*sqrt(d + e*x)/(9*e) + 8*a**5*d**3*x*sqrt(d + e*x)/9 + 4*a**5*d**2*e*x**2*sqrt(d + e*x)/
3 + 8*a**5*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**5*e**3*x**4*sqrt(d + e*x)/9 - 20*a**4*b*d**5*sqrt(d + e*x)/(99*e
**2) + 10*a**4*b*d**4*x*sqrt(d + e*x)/(99*e) + 80*a**4*b*d**3*x**2*sqrt(d + e*x)/33 + 460*a**4*b*d**2*e*x**3*s
qrt(d + e*x)/99 + 340*a**4*b*d*e**2*x**4*sqrt(d + e*x)/99 + 10*a**4*b*e**3*x**5*sqrt(d + e*x)/11 + 160*a**3*b*
*2*d**6*sqrt(d + e*x)/(1287*e**3) - 80*a**3*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 20*a**3*b**2*d**4*x**2*sqr
t(d + e*x)/(429*e) + 4240*a**3*b**2*d**3*x**3*sqrt(d + e*x)/1287 + 9160*a**3*b**2*d**2*e*x**4*sqrt(d + e*x)/12
87 + 800*a**3*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 20*a**3*b**2*e**3*x**6*sqrt(d + e*x)/13 - 64*a**2*b**3*d**7
*sqrt(d + e*x)/(1287*e**4) + 32*a**2*b**3*d**6*x*sqrt(d + e*x)/(1287*e**3) - 8*a**2*b**3*d**5*x**2*sqrt(d + e*
x)/(429*e**2) + 20*a**2*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 3200*a**2*b**3*d**3*x**4*sqrt(d + e*x)/1287 +
824*a**2*b**3*d**2*e*x**5*sqrt(d + e*x)/143 + 184*a**2*b**3*d*e**2*x**6*sqrt(d + e*x)/39 + 4*a**2*b**3*e**3*x*
*7*sqrt(d + e*x)/3 + 256*a*b**4*d**8*sqrt(d + e*x)/(21879*e**5) - 128*a*b**4*d**7*x*sqrt(d + e*x)/(21879*e**4)
+ 32*a*b**4*d**6*x**2*sqrt(d + e*x)/(7293*e**3) - 80*a*b**4*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 70*a*b**4*
d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424*a*b**4*d**3*x**5*sqrt(d + e*x)/2431 + 1604*a*b**4*d**2*e*x**6*sqrt(d
+ e*x)/663 + 104*a*b**4*d*e**2*x**7*sqrt(d + e*x)/51 + 10*a*b**4*e**3*x**8*sqrt(d + e*x)/17 - 512*b**5*d**9*sq
rt(d + e*x)/(415701*e**6) + 256*b**5*d**8*x*sqrt(d + e*x)/(415701*e**5) - 64*b**5*d**7*x**2*sqrt(d + e*x)/(138
567*e**4) + 160*b**5*d**6*x**3*sqrt(d + e*x)/(415701*e**3) - 140*b**5*d**5*x**4*sqrt(d + e*x)/(415701*e**2) +
14*b**5*d**4*x**5*sqrt(d + e*x)/(46189*e) + 2096*b**5*d**3*x**6*sqrt(d + e*x)/12597 + 404*b**5*d**2*e*x**7*sqr
t(d + e*x)/969 + 116*b**5*d*e**2*x**8*sqrt(d + e*x)/323 + 2*b**5*e**3*x**9*sqrt(d + e*x)/19, Ne(e, 0)), (d**(7
/2)*(a**5*x + 5*a**4*b*x**2/2 + 10*a**3*b**2*x**3/3 + 5*a**2*b**3*x**4/2 + a*b**4*x**5 + b**5*x**6/6), True))
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