3.14.39 \(\int (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)
Optimal. Leaf size=187 \[ -\frac {12 b^5 (d+e x)^{19/2} (b d-a e)}{19 e^7}+\frac {30 b^4 (d+e x)^{17/2} (b d-a e)^2}{17 e^7}-\frac {8 b^3 (d+e x)^{15/2} (b d-a e)^3}{3 e^7}+\frac {30 b^2 (d+e x)^{13/2} (b d-a e)^4}{13 e^7}-\frac {12 b (d+e x)^{11/2} (b d-a e)^5}{11 e^7}+\frac {2 (d+e x)^{9/2} (b d-a e)^6}{9 e^7}+\frac {2 b^6 (d+e x)^{21/2}}{21 e^7} \]
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Rubi [A] time = 0.08, 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} \begin {gather*} -\frac {12 b^5 (d+e x)^{19/2} (b d-a e)}{19 e^7}+\frac {30 b^4 (d+e x)^{17/2} (b d-a e)^2}{17 e^7}-\frac {8 b^3 (d+e x)^{15/2} (b d-a e)^3}{3 e^7}+\frac {30 b^2 (d+e x)^{13/2} (b d-a e)^4}{13 e^7}-\frac {12 b (d+e x)^{11/2} (b d-a e)^5}{11 e^7}+\frac {2 (d+e x)^{9/2} (b d-a e)^6}{9 e^7}+\frac {2 b^6 (d+e x)^{21/2}}{21 e^7} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(b*d - a*e)^6*(d + e*x)^(9/2))/(9*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^(11/2))/(11*e^7) + (30*b^2*(b*d - a*
e)^4*(d + e*x)^(13/2))/(13*e^7) - (8*b^3*(b*d - a*e)^3*(d + e*x)^(15/2))/(3*e^7) + (30*b^4*(b*d - a*e)^2*(d +
e*x)^(17/2))/(17*e^7) - (12*b^5*(b*d - a*e)*(d + e*x)^(19/2))/(19*e^7) + (2*b^6*(d + e*x)^(21/2))/(21*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)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^6 (d+e x)^{7/2}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{9/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{11/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{13/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{15/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{17/2}}{e^6}+\frac {b^6 (d+e x)^{19/2}}{e^6}\right ) \, dx\\ &=\frac {2 (b d-a e)^6 (d+e x)^{9/2}}{9 e^7}-\frac {12 b (b d-a e)^5 (d+e x)^{11/2}}{11 e^7}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{13/2}}{13 e^7}-\frac {8 b^3 (b d-a e)^3 (d+e x)^{15/2}}{3 e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{17/2}}{17 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{19/2}}{19 e^7}+\frac {2 b^6 (d+e x)^{21/2}}{21 e^7}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 145, normalized size = 0.78 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (-918918 b^5 (d+e x)^5 (b d-a e)+2567565 b^4 (d+e x)^4 (b d-a e)^2-3879876 b^3 (d+e x)^3 (b d-a e)^3+3357585 b^2 (d+e x)^2 (b d-a e)^4-1587222 b (d+e x) (b d-a e)^5+323323 (b d-a e)^6+138567 b^6 (d+e x)^6\right )}{2909907 e^7} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(9/2)*(323323*(b*d - a*e)^6 - 1587222*b*(b*d - a*e)^5*(d + e*x) + 3357585*b^2*(b*d - a*e)^4*(d +
e*x)^2 - 3879876*b^3*(b*d - a*e)^3*(d + e*x)^3 + 2567565*b^4*(b*d - a*e)^2*(d + e*x)^4 - 918918*b^5*(b*d - a*e
)*(d + e*x)^5 + 138567*b^6*(d + e*x)^6))/(2909907*e^7)
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IntegrateAlgebraic [B] time = 0.15, size = 438, normalized size = 2.34 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (323323 a^6 e^6+1587222 a^5 b e^5 (d+e x)-1939938 a^5 b d e^5+4849845 a^4 b^2 d^2 e^4+3357585 a^4 b^2 e^4 (d+e x)^2-7936110 a^4 b^2 d e^4 (d+e x)-6466460 a^3 b^3 d^3 e^3+15872220 a^3 b^3 d^2 e^3 (d+e x)+3879876 a^3 b^3 e^3 (d+e x)^3-13430340 a^3 b^3 d e^3 (d+e x)^2+4849845 a^2 b^4 d^4 e^2-15872220 a^2 b^4 d^3 e^2 (d+e x)+20145510 a^2 b^4 d^2 e^2 (d+e x)^2+2567565 a^2 b^4 e^2 (d+e x)^4-11639628 a^2 b^4 d e^2 (d+e x)^3-1939938 a b^5 d^5 e+7936110 a b^5 d^4 e (d+e x)-13430340 a b^5 d^3 e (d+e x)^2+11639628 a b^5 d^2 e (d+e x)^3+918918 a b^5 e (d+e x)^5-5135130 a b^5 d e (d+e x)^4+323323 b^6 d^6-1587222 b^6 d^5 (d+e x)+3357585 b^6 d^4 (d+e x)^2-3879876 b^6 d^3 (d+e x)^3+2567565 b^6 d^2 (d+e x)^4+138567 b^6 (d+e x)^6-918918 b^6 d (d+e x)^5\right )}{2909907 e^7} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(9/2)*(323323*b^6*d^6 - 1939938*a*b^5*d^5*e + 4849845*a^2*b^4*d^4*e^2 - 6466460*a^3*b^3*d^3*e^3 +
4849845*a^4*b^2*d^2*e^4 - 1939938*a^5*b*d*e^5 + 323323*a^6*e^6 - 1587222*b^6*d^5*(d + e*x) + 7936110*a*b^5*d^
4*e*(d + e*x) - 15872220*a^2*b^4*d^3*e^2*(d + e*x) + 15872220*a^3*b^3*d^2*e^3*(d + e*x) - 7936110*a^4*b^2*d*e^
4*(d + e*x) + 1587222*a^5*b*e^5*(d + e*x) + 3357585*b^6*d^4*(d + e*x)^2 - 13430340*a*b^5*d^3*e*(d + e*x)^2 + 2
0145510*a^2*b^4*d^2*e^2*(d + e*x)^2 - 13430340*a^3*b^3*d*e^3*(d + e*x)^2 + 3357585*a^4*b^2*e^4*(d + e*x)^2 - 3
879876*b^6*d^3*(d + e*x)^3 + 11639628*a*b^5*d^2*e*(d + e*x)^3 - 11639628*a^2*b^4*d*e^2*(d + e*x)^3 + 3879876*a
^3*b^3*e^3*(d + e*x)^3 + 2567565*b^6*d^2*(d + e*x)^4 - 5135130*a*b^5*d*e*(d + e*x)^4 + 2567565*a^2*b^4*e^2*(d
+ e*x)^4 - 918918*b^6*d*(d + e*x)^5 + 918918*a*b^5*e*(d + e*x)^5 + 138567*b^6*(d + e*x)^6))/(2909907*e^7)
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fricas [B] time = 0.42, size = 729, normalized size = 3.90 \begin {gather*} \frac {2 \, {\left (138567 \, b^{6} e^{10} x^{10} + 1024 \, b^{6} d^{10} - 10752 \, a b^{5} d^{9} e + 51072 \, a^{2} b^{4} d^{8} e^{2} - 144704 \, a^{3} b^{3} d^{7} e^{3} + 271320 \, a^{4} b^{2} d^{6} e^{4} - 352716 \, a^{5} b d^{5} e^{5} + 323323 \, a^{6} d^{4} e^{6} + 14586 \, {\left (32 \, b^{6} d e^{9} + 63 \, a b^{5} e^{10}\right )} x^{9} + 3861 \, {\left (138 \, b^{6} d^{2} e^{8} + 812 \, a b^{5} d e^{9} + 665 \, a^{2} b^{4} e^{10}\right )} x^{8} + 1716 \, {\left (121 \, b^{6} d^{3} e^{7} + 2121 \, a b^{5} d^{2} e^{8} + 5187 \, a^{2} b^{4} d e^{9} + 2261 \, a^{3} b^{3} e^{10}\right )} x^{7} + 231 \, {\left (b^{6} d^{4} e^{6} + 6288 \, a b^{5} d^{3} e^{7} + 45714 \, a^{2} b^{4} d^{2} e^{8} + 59432 \, a^{3} b^{3} d e^{9} + 14535 \, a^{4} b^{2} e^{10}\right )} x^{6} - 126 \, {\left (2 \, b^{6} d^{5} e^{5} - 21 \, a b^{5} d^{4} e^{6} - 34542 \, a^{2} b^{4} d^{3} e^{7} - 133076 \, a^{3} b^{3} d^{2} e^{8} - 96900 \, a^{4} b^{2} d e^{9} - 12597 \, a^{5} b e^{10}\right )} x^{5} + 7 \, {\left (40 \, b^{6} d^{6} e^{4} - 420 \, a b^{5} d^{5} e^{5} + 1995 \, a^{2} b^{4} d^{4} e^{6} + 1033600 \, a^{3} b^{3} d^{3} e^{7} + 2219010 \, a^{4} b^{2} d^{2} e^{8} + 856596 \, a^{5} b d e^{9} + 46189 \, a^{6} e^{10}\right )} x^{4} - 4 \, {\left (80 \, b^{6} d^{7} e^{3} - 840 \, a b^{5} d^{6} e^{4} + 3990 \, a^{2} b^{4} d^{5} e^{5} - 11305 \, a^{3} b^{3} d^{4} e^{6} - 1797495 \, a^{4} b^{2} d^{3} e^{7} - 2028117 \, a^{5} b d^{2} e^{8} - 323323 \, a^{6} d e^{9}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{8} e^{2} - 1344 \, a b^{5} d^{7} e^{3} + 6384 \, a^{2} b^{4} d^{6} e^{4} - 18088 \, a^{3} b^{3} d^{5} e^{5} + 33915 \, a^{4} b^{2} d^{4} e^{6} + 1410864 \, a^{5} b d^{3} e^{7} + 646646 \, a^{6} d^{2} e^{8}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{9} e - 2688 \, a b^{5} d^{8} e^{2} + 12768 \, a^{2} b^{4} d^{7} e^{3} - 36176 \, a^{3} b^{3} d^{6} e^{4} + 67830 \, a^{4} b^{2} d^{5} e^{5} - 88179 \, a^{5} b d^{4} e^{6} - 646646 \, a^{6} d^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{2909907 \, e^{7}} \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)^3,x, algorithm="fricas")
[Out]
2/2909907*(138567*b^6*e^10*x^10 + 1024*b^6*d^10 - 10752*a*b^5*d^9*e + 51072*a^2*b^4*d^8*e^2 - 144704*a^3*b^3*d
^7*e^3 + 271320*a^4*b^2*d^6*e^4 - 352716*a^5*b*d^5*e^5 + 323323*a^6*d^4*e^6 + 14586*(32*b^6*d*e^9 + 63*a*b^5*e
^10)*x^9 + 3861*(138*b^6*d^2*e^8 + 812*a*b^5*d*e^9 + 665*a^2*b^4*e^10)*x^8 + 1716*(121*b^6*d^3*e^7 + 2121*a*b^
5*d^2*e^8 + 5187*a^2*b^4*d*e^9 + 2261*a^3*b^3*e^10)*x^7 + 231*(b^6*d^4*e^6 + 6288*a*b^5*d^3*e^7 + 45714*a^2*b^
4*d^2*e^8 + 59432*a^3*b^3*d*e^9 + 14535*a^4*b^2*e^10)*x^6 - 126*(2*b^6*d^5*e^5 - 21*a*b^5*d^4*e^6 - 34542*a^2*
b^4*d^3*e^7 - 133076*a^3*b^3*d^2*e^8 - 96900*a^4*b^2*d*e^9 - 12597*a^5*b*e^10)*x^5 + 7*(40*b^6*d^6*e^4 - 420*a
*b^5*d^5*e^5 + 1995*a^2*b^4*d^4*e^6 + 1033600*a^3*b^3*d^3*e^7 + 2219010*a^4*b^2*d^2*e^8 + 856596*a^5*b*d*e^9 +
46189*a^6*e^10)*x^4 - 4*(80*b^6*d^7*e^3 - 840*a*b^5*d^6*e^4 + 3990*a^2*b^4*d^5*e^5 - 11305*a^3*b^3*d^4*e^6 -
1797495*a^4*b^2*d^3*e^7 - 2028117*a^5*b*d^2*e^8 - 323323*a^6*d*e^9)*x^3 + 3*(128*b^6*d^8*e^2 - 1344*a*b^5*d^7*
e^3 + 6384*a^2*b^4*d^6*e^4 - 18088*a^3*b^3*d^5*e^5 + 33915*a^4*b^2*d^4*e^6 + 1410864*a^5*b*d^3*e^7 + 646646*a^
6*d^2*e^8)*x^2 - 2*(256*b^6*d^9*e - 2688*a*b^5*d^8*e^2 + 12768*a^2*b^4*d^7*e^3 - 36176*a^3*b^3*d^6*e^4 + 67830
*a^4*b^2*d^5*e^5 - 88179*a^5*b*d^4*e^6 - 646646*a^6*d^3*e^7)*x)*sqrt(e*x + d)/e^7
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giac [B] time = 0.38, size = 2950, normalized size = 15.78
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)^3,x, algorithm="giac")
[Out]
2/14549535*(29099070*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*d^4*e^(-1) + 14549535*(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^4*e^(-2) + 8314020*(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^4*e^(-3) + 692835*(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^4*e^(-4)
+ 125970*(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^4*e^(-5) + 4845*(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^4*e^(-6) + 23279256*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*s
qrt(x*e + d)*d^2)*a^5*b*d^3*e^(-1) + 24942060*(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^3*e^(-2) + 3695120*(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*d^3*e^(-3) + 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^4*d^3*e^(-4) + 116280*(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*sqr
t(x*e + d)*d^6)*a*b^5*d^3*e^(-5) + 9044*(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^3*e^(-6) + 14549535*sqrt(x*e + d)*a^6*d^4 + 19399380*((x*e + d)^(3/2)
- 3*sqrt(x*e + d)*d)*a^6*d^3 + 14965236*(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*d^2*e^(-1) + 4157010*(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*d^2*e^(-2) + 2519400*(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^3*d^2*e^(-3) + 436050*(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*d^2*e^(-4) + 81396*(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^5*d^2*e^(-5) + 798*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 2356
20*(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^6*d^2*e^(-6) + 58198
14*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^6*d^2 + 1108536*(35*(x*e + d)^(9/2) - 1
80*(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*b*d*e^(-
1) + 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^4*b^2*d*e^(-2) + 387600*(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^3*d*e^(-3) + 135660*(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^2*b^4*d*e^(-4) + 3192*(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^5*d*e^(-5) + 252*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*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^6*d*e^
(-6) + 1662804*(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^6*
d + 125970*(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^5*b*e^(-1) + 72675*(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^4*b^2*e^(-2) + 45220*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 122
85*(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^3*b^3*e^(-3) + 1995*(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 - 87516
0*(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^
2*b^4*e^(-4) + 378*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*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)*a*b^5*e^(-5) + 15*(4
6189*(x*e + d)^(21/2) - 510510*(x*e + d)^(19/2)*d + 2567565*(x*e + d)^(17/2)*d^2 - 7759752*(x*e + d)^(15/2)*d^
3 + 15668730*(x*e + d)^(13/2)*d^4 - 22221108*(x*e + d)^(11/2)*d^5 + 22632610*(x*e + d)^(9/2)*d^6 - 16628040*(x
*e + d)^(7/2)*d^7 + 8729721*(x*e + d)^(5/2)*d^8 - 3233230*(x*e + d)^(3/2)*d^9 + 969969*sqrt(x*e + d)*d^10)*b^6
*e^(-6) + 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^6)*e^(-1)
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maple [B] time = 0.15, size = 377, normalized size = 2.02 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (138567 b^{6} e^{6} x^{6}+918918 a \,b^{5} e^{6} x^{5}-87516 b^{6} d \,e^{5} x^{5}+2567565 a^{2} b^{4} e^{6} x^{4}-540540 a \,b^{5} d \,e^{5} x^{4}+51480 b^{6} d^{2} e^{4} x^{4}+3879876 a^{3} b^{3} e^{6} x^{3}-1369368 a^{2} b^{4} d \,e^{5} x^{3}+288288 a \,b^{5} d^{2} e^{4} x^{3}-27456 b^{6} d^{3} e^{3} x^{3}+3357585 a^{4} b^{2} e^{6} x^{2}-1790712 a^{3} b^{3} d \,e^{5} x^{2}+632016 a^{2} b^{4} d^{2} e^{4} x^{2}-133056 a \,b^{5} d^{3} e^{3} x^{2}+12672 b^{6} d^{4} e^{2} x^{2}+1587222 a^{5} b \,e^{6} x -1220940 a^{4} b^{2} d \,e^{5} x +651168 a^{3} b^{3} d^{2} e^{4} x -229824 a^{2} b^{4} d^{3} e^{3} x +48384 a \,b^{5} d^{4} e^{2} x -4608 b^{6} d^{5} e x +323323 a^{6} e^{6}-352716 a^{5} b d \,e^{5}+271320 a^{4} b^{2} d^{2} e^{4}-144704 a^{3} b^{3} d^{3} e^{3}+51072 a^{2} b^{4} d^{4} e^{2}-10752 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{2909907 e^{7}} \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)^3,x)
[Out]
2/2909907*(e*x+d)^(9/2)*(138567*b^6*e^6*x^6+918918*a*b^5*e^6*x^5-87516*b^6*d*e^5*x^5+2567565*a^2*b^4*e^6*x^4-5
40540*a*b^5*d*e^5*x^4+51480*b^6*d^2*e^4*x^4+3879876*a^3*b^3*e^6*x^3-1369368*a^2*b^4*d*e^5*x^3+288288*a*b^5*d^2
*e^4*x^3-27456*b^6*d^3*e^3*x^3+3357585*a^4*b^2*e^6*x^2-1790712*a^3*b^3*d*e^5*x^2+632016*a^2*b^4*d^2*e^4*x^2-13
3056*a*b^5*d^3*e^3*x^2+12672*b^6*d^4*e^2*x^2+1587222*a^5*b*e^6*x-1220940*a^4*b^2*d*e^5*x+651168*a^3*b^3*d^2*e^
4*x-229824*a^2*b^4*d^3*e^3*x+48384*a*b^5*d^4*e^2*x-4608*b^6*d^5*e*x+323323*a^6*e^6-352716*a^5*b*d*e^5+271320*a
^4*b^2*d^2*e^4-144704*a^3*b^3*d^3*e^3+51072*a^2*b^4*d^4*e^2-10752*a*b^5*d^5*e+1024*b^6*d^6)/e^7
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maxima [B] time = 1.05, size = 350, normalized size = 1.87 \begin {gather*} \frac {2 \, {\left (138567 \, {\left (e x + d\right )}^{\frac {21}{2}} b^{6} - 918918 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {19}{2}} + 2567565 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {17}{2}} - 3879876 \, {\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 {15}{2}} + 3357585 \, {\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 {13}{2}} - 1587222 \, {\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 {11}{2}} + 323323 \, {\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 {9}{2}}\right )}}{2909907 \, e^{7}} \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)^3,x, algorithm="maxima")
[Out]
2/2909907*(138567*(e*x + d)^(21/2)*b^6 - 918918*(b^6*d - a*b^5*e)*(e*x + d)^(19/2) + 2567565*(b^6*d^2 - 2*a*b^
5*d*e + a^2*b^4*e^2)*(e*x + d)^(17/2) - 3879876*(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)^(15/2) + 3357585*(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)
^(13/2) - 1587222*(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)^(11/2) + 323323*(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)^(9/2))/e^7
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mupad [B] time = 0.58, size = 162, normalized size = 0.87 \begin {gather*} \frac {2\,b^6\,{\left (d+e\,x\right )}^{21/2}}{21\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {30\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {8\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{15/2}}{3\,e^7}+\frac {30\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {12\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7} \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)^3,x)
[Out]
(2*b^6*(d + e*x)^(21/2))/(21*e^7) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(19/2))/(19*e^7) + (2*(a*e - b*d)^6*(d
+ e*x)^(9/2))/(9*e^7) + (30*b^2*(a*e - b*d)^4*(d + e*x)^(13/2))/(13*e^7) + (8*b^3*(a*e - b*d)^3*(d + e*x)^(15/
2))/(3*e^7) + (30*b^4*(a*e - b*d)^2*(d + e*x)^(17/2))/(17*e^7) + (12*b*(a*e - b*d)^5*(d + e*x)^(11/2))/(11*e^7
)
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sympy [A] time = 82.65, size = 2450, normalized size = 13.10
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)**3,x)
[Out]
a**6*d**3*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 6*a**6*d**2*(-d*(d + e*x)**(3/2
)/3 + (d + e*x)**(5/2)/5)/e + 6*a**6*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)
/e + 2*a**6*(-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 + 12*a**5*b*d**3*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 36*a**5*b*d**2*(d**2*(d + e*x)**(3/2
)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 36*a**5*b*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**2 + 12*a**5*b*(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**2 + 30*a*
*4*b**2*d**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 90*a**4*b**2*d**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 + 90
*a**4*b**2*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**3 + 30*a**4*b**2*(-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**3 + 4
0*a**3*b**3*d**3*(-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 + 120*a**3*b**3*d**2*(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 + 120*a**3*b**3*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**4 + 40*a**3*b**3*(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)**(1
5/2)/15)/e**4 + 30*a**2*b**4*d**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**5 + 90*a**2*b**4*d**2*(-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 + 90*a**2*b**4*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**5 + 30*a**2*b**4*(-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**5 + 12*a*b**5*d**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 1
0*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
+ 36*a*b**5*d**2*(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 + 36
*a*b**5*d*(-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**6 + 12*a*b**5*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) -
56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**
(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**6 + 2*b**6*d**3*(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)/1
1 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 6*b**6*d**2*(-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 + 6*b**6*d*(d**8*(d + e*x)**(3/
2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(
11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)*
*(19/2)/19)/e**7 + 2*b**6*(-d**9*(d + e*x)**(3/2)/3 + 9*d**8*(d + e*x)**(5/2)/5 - 36*d**7*(d + e*x)**(7/2)/7 +
28*d**6*(d + e*x)**(9/2)/3 - 126*d**5*(d + e*x)**(11/2)/11 + 126*d**4*(d + e*x)**(13/2)/13 - 28*d**3*(d + e*x
)**(15/2)/5 + 36*d**2*(d + e*x)**(17/2)/17 - 9*d*(d + e*x)**(19/2)/19 + (d + e*x)**(21/2)/21)/e**7
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