3.23.83 \(\int (d+e x)^{3/2} (a+b x+c x^2)^3 \, dx\) [2283]

Optimal. Leaf size=286 \[ \frac {2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{3 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{13 e^7}-\frac {2 c^2 (2 c d-b e) (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \]

[Out]

2/5*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(5/2)/e^7-6/7*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(7/2)/e^7+2/3*(a*e^
2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(9/2)/e^7-2/11*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*
c*e*(-3*a*e+5*b*d))*(e*x+d)^(11/2)/e^7+6/13*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(13/2)/e^7-2/5*c^2*
(-b*e+2*c*d)*(e*x+d)^(15/2)/e^7+2/17*c^3*(e*x+d)^(17/2)/e^7

________________________________________________________________________________________

Rubi [A]
time = 0.09, 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 = {712} \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 712

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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.36, size = 396, normalized size = 1.38 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (c^3 \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )+221 e^3 \left (231 a^3 e^3+99 a^2 b e^2 (-2 d+5 e x)+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )+17 c e^2 \left (143 a^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 a b e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+3 b^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )-17 c^2 e \left (-3 a e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+b \left (256 d^5-640 d^4 e x+1120 d^3 e^2 x^2-1680 d^2 e^3 x^3+2310 d e^4 x^4-3003 e^5 x^5\right )\right )\right )}{255255 e^7} \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)*(c^3*(1024*d^6 - 2560*d^5*e*x + 4480*d^4*e^2*x^2 - 6720*d^3*e^3*x^3 + 9240*d^2*e^4*x^4 - 12
012*d*e^5*x^5 + 15015*e^6*x^6) + 221*e^3*(231*a^3*e^3 + 99*a^2*b*e^2*(-2*d + 5*e*x) + 11*a*b^2*e*(8*d^2 - 20*d
*e*x + 35*e^2*x^2) + b^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)) + 17*c*e^2*(143*a^2*e^2*(8*d^2 -
20*d*e*x + 35*e^2*x^2) + 78*a*b*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 3*b^2*(128*d^4 - 320*
d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)) - 17*c^2*e*(-3*a*e*(128*d^4 - 320*d^3*e*x + 560*d^2
*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + b*(256*d^5 - 640*d^4*e*x + 1120*d^3*e^2*x^2 - 1680*d^2*e^3*x^3 + 23
10*d*e^4*x^4 - 3003*e^5*x^5))))/(255255*e^7)

________________________________________________________________________________________

Maple [A]
time = 0.74, size = 357, normalized size = 1.25 Too large to display

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,method=_RETURNVERBOSE)

[Out]

2/e^7*(1/17*c^3*(e*x+d)^(17/2)+1/5*(b*e-2*c*d)*c^2*(e*x+d)^(15/2)+1/13*((a*e^2-b*d*e+c*d^2)*c^2+2*(b*e-2*c*d)^
2*c+c*(2*(a*e^2-b*d*e+c*d^2)*c+(b*e-2*c*d)^2))*(e*x+d)^(13/2)+1/11*(4*(a*e^2-b*d*e+c*d^2)*c*(b*e-2*c*d)+(b*e-2
*c*d)*(2*(a*e^2-b*d*e+c*d^2)*c+(b*e-2*c*d)^2))*(e*x+d)^(11/2)+1/9*((a*e^2-b*d*e+c*d^2)*(2*(a*e^2-b*d*e+c*d^2)*
c+(b*e-2*c*d)^2)+2*(b*e-2*c*d)^2*(a*e^2-b*d*e+c*d^2)+c*(a*e^2-b*d*e+c*d^2)^2)*(e*x+d)^(9/2)+3/7*(a*e^2-b*d*e+c
*d^2)^2*(b*e-2*c*d)*(e*x+d)^(7/2)+1/5*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(5/2))

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 426, normalized size = 1.49 \begin {gather*} \frac {2}{255255} \, {\left (15015 \, {\left (x e + d\right )}^{\frac {17}{2}} c^{3} - 51051 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (x e + d\right )}^{\frac {15}{2}} + 58905 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {13}{2}} - 23205 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e - b^{3} e^{3} - 6 \, a b c e^{3} + 12 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + a b^{2} e^{4} + a^{2} c e^{4} + 6 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{2} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d\right )} {\left (x e + d\right )}^{\frac {9}{2}} - 109395 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{3} - a^{2} b e^{5} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{2} + 2 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 51051 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} - 3 \, a^{2} b d e^{5} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} + a^{3} e^{6} + 3 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2}\right )} {\left (x e + d\right )}^{\frac {5}{2}}\right )} e^{\left (-7\right )} \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*(x*e + d)^(17/2)*c^3 - 51051*(2*c^3*d - b*c^2*e)*(x*e + d)^(15/2) + 58905*(5*c^3*d^2 - 5*b*c^2
*d*e + b^2*c*e^2 + a*c^2*e^2)*(x*e + d)^(13/2) - 23205*(20*c^3*d^3 - 30*b*c^2*d^2*e - b^3*e^3 - 6*a*b*c*e^3 +
12*(b^2*c*e^2 + a*c^2*e^2)*d)*(x*e + d)^(11/2) + 85085*(5*c^3*d^4 - 10*b*c^2*d^3*e + a*b^2*e^4 + a^2*c*e^4 + 6
*(b^2*c*e^2 + a*c^2*e^2)*d^2 - (b^3*e^3 + 6*a*b*c*e^3)*d)*(x*e + d)^(9/2) - 109395*(2*c^3*d^5 - 5*b*c^2*d^4*e
+ 4*(b^2*c*e^2 + a*c^2*e^2)*d^3 - a^2*b*e^5 - (b^3*e^3 + 6*a*b*c*e^3)*d^2 + 2*(a*b^2*e^4 + a^2*c*e^4)*d)*(x*e
+ d)^(7/2) + 51051*(c^3*d^6 - 3*b*c^2*d^5*e + 3*(b^2*c*e^2 + a*c^2*e^2)*d^4 - 3*a^2*b*d*e^5 - (b^3*e^3 + 6*a*b
*c*e^3)*d^3 + a^3*e^6 + 3*(a*b^2*e^4 + a^2*c*e^4)*d^2)*(x*e + d)^(5/2))*e^(-7)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 587 vs. \(2 (264) = 528\).
time = 2.90, size = 587, normalized size = 2.05 \begin {gather*} \frac {2}{255255} \, {\left (1024 \, c^{3} d^{8} + {\left (15015 \, c^{3} x^{8} + 51051 \, b c^{2} x^{7} + 58905 \, {\left (b^{2} c + a c^{2}\right )} x^{6} + 109395 \, a^{2} b x^{3} + 23205 \, {\left (b^{3} + 6 \, a b c\right )} x^{5} + 51051 \, a^{3} x^{2} + 85085 \, {\left (a b^{2} + a^{2} c\right )} x^{4}\right )} e^{8} + 2 \, {\left (9009 \, c^{3} d x^{7} + 31416 \, b c^{2} d x^{6} + 37485 \, {\left (b^{2} c + a c^{2}\right )} d x^{5} + 87516 \, a^{2} b d x^{2} + 15470 \, {\left (b^{3} + 6 \, a b c\right )} d x^{4} + 51051 \, a^{3} d x + 60775 \, {\left (a b^{2} + a^{2} c\right )} d x^{3}\right )} e^{7} + {\left (231 \, c^{3} d^{2} x^{6} + 1071 \, b c^{2} d^{2} x^{5} + 1785 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{4} + 21879 \, a^{2} b d^{2} x + 1105 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x^{3} + 51051 \, a^{3} d^{2} + 7293 \, {\left (a b^{2} + a^{2} c\right )} d^{2} x^{2}\right )} e^{6} - 2 \, {\left (126 \, c^{3} d^{3} x^{5} + 595 \, b c^{2} d^{3} x^{4} + 1020 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x^{3} + 21879 \, a^{2} b d^{3} + 663 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} x^{2} + 4862 \, {\left (a b^{2} + a^{2} c\right )} d^{3} x\right )} e^{5} + 8 \, {\left (35 \, c^{3} d^{4} x^{4} + 170 \, b c^{2} d^{4} x^{3} + 306 \, {\left (b^{2} c + a c^{2}\right )} d^{4} x^{2} + 221 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} x + 2431 \, {\left (a b^{2} + a^{2} c\right )} d^{4}\right )} e^{4} - 16 \, {\left (20 \, c^{3} d^{5} x^{3} + 102 \, b c^{2} d^{5} x^{2} + 204 \, {\left (b^{2} c + a c^{2}\right )} d^{5} x + 221 \, {\left (b^{3} + 6 \, a b c\right )} d^{5}\right )} e^{3} + 128 \, {\left (3 \, c^{3} d^{6} x^{2} + 17 \, b c^{2} d^{6} x + 51 \, {\left (b^{2} c + a c^{2}\right )} d^{6}\right )} e^{2} - 256 \, {\left (2 \, c^{3} d^{7} x + 17 \, b c^{2} d^{7}\right )} e\right )} \sqrt {x e + d} e^{\left (-7\right )} \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*(1024*c^3*d^8 + (15015*c^3*x^8 + 51051*b*c^2*x^7 + 58905*(b^2*c + a*c^2)*x^6 + 109395*a^2*b*x^3 + 232
05*(b^3 + 6*a*b*c)*x^5 + 51051*a^3*x^2 + 85085*(a*b^2 + a^2*c)*x^4)*e^8 + 2*(9009*c^3*d*x^7 + 31416*b*c^2*d*x^
6 + 37485*(b^2*c + a*c^2)*d*x^5 + 87516*a^2*b*d*x^2 + 15470*(b^3 + 6*a*b*c)*d*x^4 + 51051*a^3*d*x + 60775*(a*b
^2 + a^2*c)*d*x^3)*e^7 + (231*c^3*d^2*x^6 + 1071*b*c^2*d^2*x^5 + 1785*(b^2*c + a*c^2)*d^2*x^4 + 21879*a^2*b*d^
2*x + 1105*(b^3 + 6*a*b*c)*d^2*x^3 + 51051*a^3*d^2 + 7293*(a*b^2 + a^2*c)*d^2*x^2)*e^6 - 2*(126*c^3*d^3*x^5 +
595*b*c^2*d^3*x^4 + 1020*(b^2*c + a*c^2)*d^3*x^3 + 21879*a^2*b*d^3 + 663*(b^3 + 6*a*b*c)*d^3*x^2 + 4862*(a*b^2
+ a^2*c)*d^3*x)*e^5 + 8*(35*c^3*d^4*x^4 + 170*b*c^2*d^4*x^3 + 306*(b^2*c + a*c^2)*d^4*x^2 + 221*(b^3 + 6*a*b*
c)*d^4*x + 2431*(a*b^2 + a^2*c)*d^4)*e^4 - 16*(20*c^3*d^5*x^3 + 102*b*c^2*d^5*x^2 + 204*(b^2*c + a*c^2)*d^5*x
+ 221*(b^3 + 6*a*b*c)*d^5)*e^3 + 128*(3*c^3*d^6*x^2 + 17*b*c^2*d^6*x + 51*(b^2*c + a*c^2)*d^6)*e^2 - 256*(2*c^
3*d^7*x + 17*b*c^2*d^7)*e)*sqrt(x*e + d)*e^(-7)

________________________________________________________________________________________

Sympy [A]
time = 27.53, size = 1411, normalized size = 4.93 \begin {gather*} \text {Too large to display} \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)

[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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2092 vs. \(2 (264) = 528\).
time = 1.30, size = 2092, normalized size = 7.31 \begin {gather*} \text {Too large to display} \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="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)

________________________________________________________________________________________

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)

________________________________________________________________________________________