3.20.52 \(\int (d+e x)^{3/2} (a+b x+c x^2)^3 \, dx\)
Optimal. Leaf size=286 \[ \frac {6 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^7}-\frac {2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^7}+\frac {2 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac {6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac {2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \]
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Rubi [A] time = 0.13, antiderivative size = 286, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used =
{698} \begin {gather*} \frac {6 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^7}-\frac {2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^7}+\frac {2 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac {6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac {2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^(3/2)*(a + b*x + c*x^2)^3,x]
[Out]
(2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(5/2))/(5*e^7) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(
7/2))/(7*e^7) + (2*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(9/2))/(3*e^7)
- (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(11/2))/(11*e^7) + (6*c*(5*c^2*d^2
+ b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(13/2))/(13*e^7) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15/2))/(5*e^7) +
(2*c^3*(d + e*x)^(17/2))/(17*e^7)
Rule 698
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}{e^6}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{e^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{7/2}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{9/2}}{e^6}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{13/2}}{e^6}+\frac {c^3 (d+e x)^{15/2}}{e^6}\right ) \, dx\\ &=\frac {2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{3 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{13 e^7}-\frac {2 c^2 (2 c d-b e) (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 320, normalized size = 1.12 \begin {gather*} \frac {2 \left ((d+e x)^{5/2} (a+x (b+c x))^3-\frac {2 (d+e x)^{7/2} \left (-34 c e^2 \left (143 a^2 e^2 (2 d-7 e x)-39 a b e \left (8 d^2-28 d e x+63 e^2 x^2\right )+6 b^2 \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )+221 b e^3 \left (99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+17 c^2 e \left (12 a e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )-2 c^3 \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )}{51051 e^6}\right )}{5 e} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^(3/2)*(a + b*x + c*x^2)^3,x]
[Out]
(2*((d + e*x)^(5/2)*(a + x*(b + c*x))^3 - (2*(d + e*x)^(7/2)*(-2*c^3*(256*d^5 - 896*d^4*e*x + 2016*d^3*e^2*x^2
- 3696*d^2*e^3*x^3 + 6006*d*e^4*x^4 - 9009*e^5*x^5) + 221*b*e^3*(99*a^2*e^2 + 22*a*b*e*(-2*d + 7*e*x) + b^2*(
8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 34*c*e^2*(143*a^2*e^2*(2*d - 7*e*x) - 39*a*b*e*(8*d^2 - 28*d*e*x + 63*e^2*x^
2) + 6*b^2*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3)) + 17*c^2*e*(12*a*e*(-16*d^3 + 56*d^2*e*x - 126
*d*e^2*x^2 + 231*e^3*x^3) + b*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4))))/(5
1051*e^6)))/(5*e)
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IntegrateAlgebraic [B] time = 0.23, size = 592, normalized size = 2.07 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (51051 a^3 e^6+109395 a^2 b e^5 (d+e x)-153153 a^2 b d e^5+153153 a^2 c d^2 e^4-218790 a^2 c d e^4 (d+e x)+85085 a^2 c e^4 (d+e x)^2+153153 a b^2 d^2 e^4-218790 a b^2 d e^4 (d+e x)+85085 a b^2 e^4 (d+e x)^2-306306 a b c d^3 e^3+656370 a b c d^2 e^3 (d+e x)-510510 a b c d e^3 (d+e x)^2+139230 a b c e^3 (d+e x)^3+153153 a c^2 d^4 e^2-437580 a c^2 d^3 e^2 (d+e x)+510510 a c^2 d^2 e^2 (d+e x)^2-278460 a c^2 d e^2 (d+e x)^3+58905 a c^2 e^2 (d+e x)^4-51051 b^3 d^3 e^3+109395 b^3 d^2 e^3 (d+e x)-85085 b^3 d e^3 (d+e x)^2+23205 b^3 e^3 (d+e x)^3+153153 b^2 c d^4 e^2-437580 b^2 c d^3 e^2 (d+e x)+510510 b^2 c d^2 e^2 (d+e x)^2-278460 b^2 c d e^2 (d+e x)^3+58905 b^2 c e^2 (d+e x)^4-153153 b c^2 d^5 e+546975 b c^2 d^4 e (d+e x)-850850 b c^2 d^3 e (d+e x)^2+696150 b c^2 d^2 e (d+e x)^3-294525 b c^2 d e (d+e x)^4+51051 b c^2 e (d+e x)^5+51051 c^3 d^6-218790 c^3 d^5 (d+e x)+425425 c^3 d^4 (d+e x)^2-464100 c^3 d^3 (d+e x)^3+294525 c^3 d^2 (d+e x)^4-102102 c^3 d (d+e x)^5+15015 c^3 (d+e x)^6\right )}{255255 e^7} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(d + e*x)^(3/2)*(a + b*x + c*x^2)^3,x]
[Out]
(2*(d + e*x)^(5/2)*(51051*c^3*d^6 - 153153*b*c^2*d^5*e + 153153*b^2*c*d^4*e^2 + 153153*a*c^2*d^4*e^2 - 51051*b
^3*d^3*e^3 - 306306*a*b*c*d^3*e^3 + 153153*a*b^2*d^2*e^4 + 153153*a^2*c*d^2*e^4 - 153153*a^2*b*d*e^5 + 51051*a
^3*e^6 - 218790*c^3*d^5*(d + e*x) + 546975*b*c^2*d^4*e*(d + e*x) - 437580*b^2*c*d^3*e^2*(d + e*x) - 437580*a*c
^2*d^3*e^2*(d + e*x) + 109395*b^3*d^2*e^3*(d + e*x) + 656370*a*b*c*d^2*e^3*(d + e*x) - 218790*a*b^2*d*e^4*(d +
e*x) - 218790*a^2*c*d*e^4*(d + e*x) + 109395*a^2*b*e^5*(d + e*x) + 425425*c^3*d^4*(d + e*x)^2 - 850850*b*c^2*
d^3*e*(d + e*x)^2 + 510510*b^2*c*d^2*e^2*(d + e*x)^2 + 510510*a*c^2*d^2*e^2*(d + e*x)^2 - 85085*b^3*d*e^3*(d +
e*x)^2 - 510510*a*b*c*d*e^3*(d + e*x)^2 + 85085*a*b^2*e^4*(d + e*x)^2 + 85085*a^2*c*e^4*(d + e*x)^2 - 464100*
c^3*d^3*(d + e*x)^3 + 696150*b*c^2*d^2*e*(d + e*x)^3 - 278460*b^2*c*d*e^2*(d + e*x)^3 - 278460*a*c^2*d*e^2*(d
+ e*x)^3 + 23205*b^3*e^3*(d + e*x)^3 + 139230*a*b*c*e^3*(d + e*x)^3 + 294525*c^3*d^2*(d + e*x)^4 - 294525*b*c^
2*d*e*(d + e*x)^4 + 58905*b^2*c*e^2*(d + e*x)^4 + 58905*a*c^2*e^2*(d + e*x)^4 - 102102*c^3*d*(d + e*x)^5 + 510
51*b*c^2*e*(d + e*x)^5 + 15015*c^3*(d + e*x)^6))/(255255*e^7)
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fricas [B] time = 0.40, size = 619, normalized size = 2.16 \begin {gather*} \frac {2 \, {\left (15015 \, c^{3} e^{8} x^{8} + 1024 \, c^{3} d^{8} - 4352 \, b c^{2} d^{7} e - 43758 \, a^{2} b d^{3} e^{5} + 51051 \, a^{3} d^{2} e^{6} + 6528 \, {\left (b^{2} c + a c^{2}\right )} d^{6} e^{2} - 3536 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} e^{3} + 19448 \, {\left (a b^{2} + a^{2} c\right )} d^{4} e^{4} + 3003 \, {\left (6 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \, {\left (c^{3} d^{2} e^{6} + 272 \, b c^{2} d e^{7} + 255 \, {\left (b^{2} c + a c^{2}\right )} e^{8}\right )} x^{6} - 21 \, {\left (12 \, c^{3} d^{3} e^{5} - 51 \, b c^{2} d^{2} e^{6} - 3570 \, {\left (b^{2} c + a c^{2}\right )} d e^{7} - 1105 \, {\left (b^{3} + 6 \, a b c\right )} e^{8}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{4} e^{4} - 34 \, b c^{2} d^{3} e^{5} + 51 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{6} + 884 \, {\left (b^{3} + 6 \, a b c\right )} d e^{7} + 2431 \, {\left (a b^{2} + a^{2} c\right )} e^{8}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{5} e^{3} - 272 \, b c^{2} d^{4} e^{4} - 21879 \, a^{2} b e^{8} + 408 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{5} - 221 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{6} - 24310 \, {\left (a b^{2} + a^{2} c\right )} d e^{7}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{6} e^{2} - 544 \, b c^{2} d^{5} e^{3} + 58344 \, a^{2} b d e^{7} + 17017 \, a^{3} e^{8} + 816 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{4} - 442 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{5} + 2431 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{6}\right )} x^{2} - {\left (512 \, c^{3} d^{7} e - 2176 \, b c^{2} d^{6} e^{2} - 21879 \, a^{2} b d^{2} e^{6} - 102102 \, a^{3} d e^{7} + 3264 \, {\left (b^{2} c + a c^{2}\right )} d^{5} e^{3} - 1768 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} e^{4} + 9724 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e^{5}\right )} x\right )} \sqrt {e x + d}}{255255 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x, algorithm="fricas")
[Out]
2/255255*(15015*c^3*e^8*x^8 + 1024*c^3*d^8 - 4352*b*c^2*d^7*e - 43758*a^2*b*d^3*e^5 + 51051*a^3*d^2*e^6 + 6528
*(b^2*c + a*c^2)*d^6*e^2 - 3536*(b^3 + 6*a*b*c)*d^5*e^3 + 19448*(a*b^2 + a^2*c)*d^4*e^4 + 3003*(6*c^3*d*e^7 +
17*b*c^2*e^8)*x^7 + 231*(c^3*d^2*e^6 + 272*b*c^2*d*e^7 + 255*(b^2*c + a*c^2)*e^8)*x^6 - 21*(12*c^3*d^3*e^5 - 5
1*b*c^2*d^2*e^6 - 3570*(b^2*c + a*c^2)*d*e^7 - 1105*(b^3 + 6*a*b*c)*e^8)*x^5 + 35*(8*c^3*d^4*e^4 - 34*b*c^2*d^
3*e^5 + 51*(b^2*c + a*c^2)*d^2*e^6 + 884*(b^3 + 6*a*b*c)*d*e^7 + 2431*(a*b^2 + a^2*c)*e^8)*x^4 - 5*(64*c^3*d^5
*e^3 - 272*b*c^2*d^4*e^4 - 21879*a^2*b*e^8 + 408*(b^2*c + a*c^2)*d^3*e^5 - 221*(b^3 + 6*a*b*c)*d^2*e^6 - 24310
*(a*b^2 + a^2*c)*d*e^7)*x^3 + 3*(128*c^3*d^6*e^2 - 544*b*c^2*d^5*e^3 + 58344*a^2*b*d*e^7 + 17017*a^3*e^8 + 816
*(b^2*c + a*c^2)*d^4*e^4 - 442*(b^3 + 6*a*b*c)*d^3*e^5 + 2431*(a*b^2 + a^2*c)*d^2*e^6)*x^2 - (512*c^3*d^7*e -
2176*b*c^2*d^6*e^2 - 21879*a^2*b*d^2*e^6 - 102102*a^3*d*e^7 + 3264*(b^2*c + a*c^2)*d^5*e^3 - 1768*(b^3 + 6*a*b
*c)*d^4*e^4 + 9724*(a*b^2 + a^2*c)*d^3*e^5)*x)*sqrt(e*x + d)/e^7
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giac [B] time = 0.62, size = 2092, normalized size = 7.31
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x, algorithm="giac")
[Out]
2/765765*(765765*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*d^2*e^(-1) + 153153*(3*(x*e + d)^(5/2) - 10*(x*e
+ d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^2*d^2*e^(-2) + 153153*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*
sqrt(x*e + d)*d^2)*a^2*c*d^2*e^(-2) + 21879*(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)*b^3*d^2*e^(-3) + 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*b*c*d^2*e^(-3) + 7293*(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^2*c*d^2*e^(-4) + 7293*(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*c^2*d^2*
e^(-4) + 3315*(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*c^2*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)*c^3*d^2*e^(-6) + 306306*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*
sqrt(x*e + d)*d^2)*a^2*b*d*e^(-1) + 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*b^2*d*e^(-2) + 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^2*c*d*e^(-2) + 4862*(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^3*d*e^(-3) + 29172*(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*c*d*e^(-3) +
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)*b^2*c*d*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*
c^2*d*e^(-4) + 1530*(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*c^2*d*e^(-5) + 2
38*(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)*c^
3*d*e^(-6) + 765765*sqrt(x*e + d)*a^3*d^2 + 510510*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*d + 65637*(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*e^(-1) + 7293*(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^2*e^(-2) + 7293*(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*c*e^(-2) + 1105*(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^3*e^(-3) + 6
630*(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*c*e^(-3) + 765*(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^2*c*e^(-4) + 765*(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*c^2*e^(-4) + 357*(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*c^2*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*sqrt(x*e + d)*d^8)*c^3*e^(-6) + 51051*(3*(x*e + d)^(5/2) -
10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3)*e^(-1)
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maple [A] time = 0.05, size = 495, normalized size = 1.73 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (15015 c^{3} x^{6} e^{6}+51051 b \,c^{2} e^{6} x^{5}-12012 c^{3} d \,e^{5} x^{5}+58905 a \,c^{2} e^{6} x^{4}+58905 b^{2} c \,e^{6} x^{4}-39270 b \,c^{2} d \,e^{5} x^{4}+9240 c^{3} d^{2} e^{4} x^{4}+139230 a b c \,e^{6} x^{3}-42840 a \,c^{2} d \,e^{5} x^{3}+23205 b^{3} e^{6} x^{3}-42840 b^{2} c d \,e^{5} x^{3}+28560 b \,c^{2} d^{2} e^{4} x^{3}-6720 c^{3} d^{3} e^{3} x^{3}+85085 a^{2} c \,e^{6} x^{2}+85085 a \,b^{2} e^{6} x^{2}-92820 a b c d \,e^{5} x^{2}+28560 a \,c^{2} d^{2} e^{4} x^{2}-15470 b^{3} d \,e^{5} x^{2}+28560 b^{2} c \,d^{2} e^{4} x^{2}-19040 b \,c^{2} d^{3} e^{3} x^{2}+4480 c^{3} d^{4} e^{2} x^{2}+109395 a^{2} b \,e^{6} x -48620 a^{2} c d \,e^{5} x -48620 a \,b^{2} d \,e^{5} x +53040 a b c \,d^{2} e^{4} x -16320 a \,c^{2} d^{3} e^{3} x +8840 b^{3} d^{2} e^{4} x -16320 b^{2} c \,d^{3} e^{3} x +10880 b \,c^{2} d^{4} e^{2} x -2560 c^{3} d^{5} e x +51051 a^{3} e^{6}-43758 a^{2} b d \,e^{5}+19448 a^{2} c \,d^{2} e^{4}+19448 a \,b^{2} d^{2} e^{4}-21216 a b c \,d^{3} e^{3}+6528 a \,c^{2} d^{4} e^{2}-3536 b^{3} d^{3} e^{3}+6528 b^{2} c \,d^{4} e^{2}-4352 b \,c^{2} d^{5} e +1024 c^{3} d^{6}\right )}{255255 e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x)
[Out]
2/255255*(e*x+d)^(5/2)*(15015*c^3*e^6*x^6+51051*b*c^2*e^6*x^5-12012*c^3*d*e^5*x^5+58905*a*c^2*e^6*x^4+58905*b^
2*c*e^6*x^4-39270*b*c^2*d*e^5*x^4+9240*c^3*d^2*e^4*x^4+139230*a*b*c*e^6*x^3-42840*a*c^2*d*e^5*x^3+23205*b^3*e^
6*x^3-42840*b^2*c*d*e^5*x^3+28560*b*c^2*d^2*e^4*x^3-6720*c^3*d^3*e^3*x^3+85085*a^2*c*e^6*x^2+85085*a*b^2*e^6*x
^2-92820*a*b*c*d*e^5*x^2+28560*a*c^2*d^2*e^4*x^2-15470*b^3*d*e^5*x^2+28560*b^2*c*d^2*e^4*x^2-19040*b*c^2*d^3*e
^3*x^2+4480*c^3*d^4*e^2*x^2+109395*a^2*b*e^6*x-48620*a^2*c*d*e^5*x-48620*a*b^2*d*e^5*x+53040*a*b*c*d^2*e^4*x-1
6320*a*c^2*d^3*e^3*x+8840*b^3*d^2*e^4*x-16320*b^2*c*d^3*e^3*x+10880*b*c^2*d^4*e^2*x-2560*c^3*d^5*e*x+51051*a^3
*e^6-43758*a^2*b*d*e^5+19448*a^2*c*d^2*e^4+19448*a*b^2*d^2*e^4-21216*a*b*c*d^3*e^3+6528*a*c^2*d^4*e^2-3536*b^3
*d^3*e^3+6528*b^2*c*d^4*e^2-4352*b*c^2*d^5*e+1024*c^3*d^6)/e^7
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maxima [A] time = 0.95, size = 407, normalized size = 1.42 \begin {gather*} \frac {2 \, {\left (15015 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{3} - 51051 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 58905 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 23205 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 109395 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 51051 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{255255 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x, algorithm="maxima")
[Out]
2/255255*(15015*(e*x + d)^(17/2)*c^3 - 51051*(2*c^3*d - b*c^2*e)*(e*x + d)^(15/2) + 58905*(5*c^3*d^2 - 5*b*c^2
*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(13/2) - 23205*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 -
(b^3 + 6*a*b*c)*e^3)*(e*x + d)^(11/2) + 85085*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2 - (b^3
+ 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*(e*x + d)^(9/2) - 109395*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b
^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d)^(7/2) + 51051*(c^3*d^6 -
3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2
*c)*d^2*e^4)*(e*x + d)^(5/2))/e^7
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mupad [B] time = 0.85, size = 297, normalized size = 1.04 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{9\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{13\,e^7}+\frac {2\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{5\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{11\,e^7}+\frac {6\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{7\,e^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x)^(3/2)*(a + b*x + c*x^2)^3,x)
[Out]
((d + e*x)^(9/2)*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^4 - 6*b^3*d*e^3 + 36*a*c^2*d^2*e^2 + 36*b^2*c*d^2*e^2 -
60*b*c^2*d^3*e - 36*a*b*c*d*e^3))/(9*e^7) + (2*c^3*(d + e*x)^(17/2))/(17*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d +
e*x)^(15/2))/(15*e^7) + ((d + e*x)^(13/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(13*e^7) +
(2*(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d*e)^3)/(5*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(11/2)*(b^2*e^2 + 10*c^2*d^
2 + 6*a*c*e^2 - 10*b*c*d*e))/(11*e^7) + (6*(b*e - 2*c*d)*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(7*e^7)
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sympy [A] time = 54.12, size = 1411, normalized size = 4.93
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**(3/2)*(c*x**2+b*x+a)**3,x)
[Out]
a**3*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**3*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 6*a**2*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 6*a**2*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 + 6*a**2*c*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 + 6*a**2*c*(-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 + 6*a*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 + 6*a*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 + 12*a*b*c*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 + 12*a*b*c*(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 + 6*a*c**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)/1
1)/e**5 + 6*a*c**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 + 2*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 + 2*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 + 6*b**2*c*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 + 6*b**2*c*(-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 + 6
*b*c**2*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 + 6*b*c**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 + 2*c**3*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*c**3*(-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)/1
3 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7
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