### 3.2287 $$\int \frac{(a+b x+c x^2)^3}{(d+e x)^{5/2}} \, dx$$

Optimal. Leaf size=282 $\frac{6 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac{6 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 \sqrt{d+e x}}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^{3/2}}-\frac{6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}$

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^7*(d + e*x)^(3/2)) + (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(e^7*Sqrt
[d + e*x]) + (6*(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))*Sqrt[d + e*x])/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)^(3/2))/(3*e^7) + (6*c*(5*c^2*d^2 + b^2*e^2
- c*e*(5*b*d - a*e))*(d + e*x)^(5/2))/(5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^7) + (2*c^3*(d + e*
x)^(9/2))/(9*e^7)

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Rubi [A]  time = 0.131565, antiderivative size = 282, normalized size of antiderivative = 1., 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} $\frac{6 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac{6 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 \sqrt{d+e x}}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^{3/2}}-\frac{6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^(5/2),x]

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^7*(d + e*x)^(3/2)) + (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(e^7*Sqrt
[d + e*x]) + (6*(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))*Sqrt[d + e*x])/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)^(3/2))/(3*e^7) + (6*c*(5*c^2*d^2 + b^2*e^2
- c*e*(5*b*d - a*e))*(d + e*x)^(5/2))/(5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^7) + (2*c^3*(d + e*
x)^(9/2))/(9*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 \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^{5/2}}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^{3/2}}+\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 )}{e^6 \sqrt{d+e x}}+\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 ) \sqrt{d+e x}}{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)^{3/2}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{5/2}}{e^6}+\frac{c^3 (d+e x)^{7/2}}{e^6}\right ) \, dx\\ &=-\frac{2 \left (c d^2-b d e+a e^2\right )^3}{3 e^7 (d+e x)^{3/2}}+\frac{6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 \sqrt{d+e x}}+\frac{6 \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 ) \sqrt{d+e x}}{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)^{3/2}}{3 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)^{5/2}}{5 e^7}-\frac{6 c^2 (2 c d-b e) (d+e x)^{7/2}}{7 e^7}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}\\ \end{align*}

Mathematica [A]  time = 0.381602, size = 395, normalized size = 1.4 $\frac{2 \left (63 c e^2 \left (5 a^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+10 a b e \left (-24 d^2 e x-16 d^3-6 d e^2 x^2+e^3 x^3\right )+b^2 \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )\right )-105 e^3 \left (3 a^2 b e^2 (2 d+3 e x)+a^3 e^3-3 a b^2 e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b^3 \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )\right )-9 c^2 e \left (5 b \left (96 d^3 e^2 x^2-16 d^2 e^3 x^3+384 d^4 e x+256 d^5+6 d e^4 x^4-3 e^5 x^5\right )-7 a e \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )\right )+5 c^3 \left (384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4+1536 d^5 e x+1024 d^6-12 d e^5 x^5+7 e^6 x^6\right )\right )}{315 e^7 (d+e x)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^(5/2),x]

[Out]

(2*(5*c^3*(1024*d^6 + 1536*d^5*e*x + 384*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 24*d^2*e^4*x^4 - 12*d*e^5*x^5 + 7*e^6*
x^6) - 105*e^3*(a^3*e^3 + 3*a^2*b*e^2*(2*d + 3*e*x) - 3*a*b^2*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + b^3*(16*d^3 +
24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)) + 63*c*e^2*(5*a^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 10*a*b*e*(-16*d^3
- 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + b^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4
)) - 9*c^2*e*(-7*a*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + 5*b*(256*d^5 + 384*d
^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5))))/(315*e^7*(d + e*x)^(3/2))

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Maple [A]  time = 0.044, size = 495, normalized size = 1.8 \begin{align*} -{\frac{-70\,{c}^{3}{x}^{6}{e}^{6}-270\,b{c}^{2}{e}^{6}{x}^{5}+120\,{c}^{3}d{e}^{5}{x}^{5}-378\,a{c}^{2}{e}^{6}{x}^{4}-378\,{b}^{2}c{e}^{6}{x}^{4}+540\,b{c}^{2}d{e}^{5}{x}^{4}-240\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-1260\,abc{e}^{6}{x}^{3}+1008\,a{c}^{2}d{e}^{5}{x}^{3}-210\,{b}^{3}{e}^{6}{x}^{3}+1008\,{b}^{2}cd{e}^{5}{x}^{3}-1440\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}-1890\,{a}^{2}c{e}^{6}{x}^{2}-1890\,a{b}^{2}{e}^{6}{x}^{2}+7560\,abcd{e}^{5}{x}^{2}-6048\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+1260\,{b}^{3}d{e}^{5}{x}^{2}-6048\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+8640\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+1890\,{a}^{2}b{e}^{6}x-7560\,{a}^{2}cd{e}^{5}x-7560\,a{b}^{2}d{e}^{5}x+30240\,abc{d}^{2}{e}^{4}x-24192\,a{c}^{2}{d}^{3}{e}^{3}x+5040\,{b}^{3}{d}^{2}{e}^{4}x-24192\,{b}^{2}c{d}^{3}{e}^{3}x+34560\,b{c}^{2}{d}^{4}{e}^{2}x-15360\,{c}^{3}{d}^{5}ex+210\,{a}^{3}{e}^{6}+1260\,{a}^{2}bd{e}^{5}-5040\,{a}^{2}c{d}^{2}{e}^{4}-5040\,a{b}^{2}{d}^{2}{e}^{4}+20160\,abc{d}^{3}{e}^{3}-16128\,a{c}^{2}{d}^{4}{e}^{2}+3360\,{b}^{3}{d}^{3}{e}^{3}-16128\,{b}^{2}c{d}^{4}{e}^{2}+23040\,b{c}^{2}{d}^{5}e-10240\,{c}^{3}{d}^{6}}{315\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(e*x+d)^(5/2),x)

[Out]

-2/315/(e*x+d)^(3/2)*(-35*c^3*e^6*x^6-135*b*c^2*e^6*x^5+60*c^3*d*e^5*x^5-189*a*c^2*e^6*x^4-189*b^2*c*e^6*x^4+2
70*b*c^2*d*e^5*x^4-120*c^3*d^2*e^4*x^4-630*a*b*c*e^6*x^3+504*a*c^2*d*e^5*x^3-105*b^3*e^6*x^3+504*b^2*c*d*e^5*x
^3-720*b*c^2*d^2*e^4*x^3+320*c^3*d^3*e^3*x^3-945*a^2*c*e^6*x^2-945*a*b^2*e^6*x^2+3780*a*b*c*d*e^5*x^2-3024*a*c
^2*d^2*e^4*x^2+630*b^3*d*e^5*x^2-3024*b^2*c*d^2*e^4*x^2+4320*b*c^2*d^3*e^3*x^2-1920*c^3*d^4*e^2*x^2+945*a^2*b*
e^6*x-3780*a^2*c*d*e^5*x-3780*a*b^2*d*e^5*x+15120*a*b*c*d^2*e^4*x-12096*a*c^2*d^3*e^3*x+2520*b^3*d^2*e^4*x-120
96*b^2*c*d^3*e^3*x+17280*b*c^2*d^4*e^2*x-7680*c^3*d^5*e*x+105*a^3*e^6+630*a^2*b*d*e^5-2520*a^2*c*d^2*e^4-2520*
a*b^2*d^2*e^4+10080*a*b*c*d^3*e^3-8064*a*c^2*d^4*e^2+1680*b^3*d^3*e^3-8064*b^2*c*d^4*e^2+11520*b*c^2*d^5*e-512
0*c^3*d^6)/e^7

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Maxima [A]  time = 1.08639, size = 558, normalized size = 1.98 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{3} - 135 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\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{5}{2}} - 105 \,{\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{3}{2}} + 945 \,{\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 )} \sqrt{e x + d}}{e^{6}} - \frac{105 \,{\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} - 9 \,{\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 )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{6}}\right )}}{315 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/315*((35*(e*x + d)^(9/2)*c^3 - 135*(2*c^3*d - b*c^2*e)*(e*x + d)^(7/2) + 189*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2
*c + a*c^2)*e^2)*(e*x + d)^(5/2) - 105*(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)^(3/2) + 945*(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)*sqrt(e*x + d))/e^6 - 105*(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 - 9*(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))/((e*x + d)^(3/
2)*e^6))/e

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Fricas [A]  time = 2.09166, size = 964, normalized size = 3.42 \begin{align*} \frac{2 \,{\left (35 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 11520 \, b c^{2} d^{5} e - 630 \, a^{2} b d e^{5} - 105 \, a^{3} e^{6} + 8064 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 1680 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2520 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 15 \,{\left (4 \, c^{3} d e^{5} - 9 \, b c^{2} e^{6}\right )} x^{5} + 3 \,{\left (40 \, c^{3} d^{2} e^{4} - 90 \, b c^{2} d e^{5} + 63 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} -{\left (320 \, c^{3} d^{3} e^{3} - 720 \, b c^{2} d^{2} e^{4} + 504 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} - 105 \,{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \,{\left (640 \, c^{3} d^{4} e^{2} - 1440 \, b c^{2} d^{3} e^{3} + 1008 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 210 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 315 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 3 \,{\left (2560 \, c^{3} d^{5} e - 5760 \, b c^{2} d^{4} e^{2} - 315 \, a^{2} b e^{6} + 4032 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 840 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 1260 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt{e x + d}}{315 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/315*(35*c^3*e^6*x^6 + 5120*c^3*d^6 - 11520*b*c^2*d^5*e - 630*a^2*b*d*e^5 - 105*a^3*e^6 + 8064*(b^2*c + a*c^2
)*d^4*e^2 - 1680*(b^3 + 6*a*b*c)*d^3*e^3 + 2520*(a*b^2 + a^2*c)*d^2*e^4 - 15*(4*c^3*d*e^5 - 9*b*c^2*e^6)*x^5 +
3*(40*c^3*d^2*e^4 - 90*b*c^2*d*e^5 + 63*(b^2*c + a*c^2)*e^6)*x^4 - (320*c^3*d^3*e^3 - 720*b*c^2*d^2*e^4 + 504
*(b^2*c + a*c^2)*d*e^5 - 105*(b^3 + 6*a*b*c)*e^6)*x^3 + 3*(640*c^3*d^4*e^2 - 1440*b*c^2*d^3*e^3 + 1008*(b^2*c
+ a*c^2)*d^2*e^4 - 210*(b^3 + 6*a*b*c)*d*e^5 + 315*(a*b^2 + a^2*c)*e^6)*x^2 + 3*(2560*c^3*d^5*e - 5760*b*c^2*d
^4*e^2 - 315*a^2*b*e^6 + 4032*(b^2*c + a*c^2)*d^3*e^3 - 840*(b^3 + 6*a*b*c)*d^2*e^4 + 1260*(a*b^2 + a^2*c)*d*e
^5)*x)*sqrt(e*x + d)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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Sympy [A]  time = 117.06, size = 348, normalized size = 1.23 \begin{align*} \frac{2 c^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{7}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (6 b c^{2} e - 12 c^{3} d\right )}{7 e^{7}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (6 a c^{2} e^{2} + 6 b^{2} c e^{2} - 30 b c^{2} d e + 30 c^{3} d^{2}\right )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (12 a b c e^{3} - 24 a c^{2} d e^{2} + 2 b^{3} e^{3} - 24 b^{2} c d e^{2} + 60 b c^{2} d^{2} e - 40 c^{3} d^{3}\right )}{3 e^{7}} + \frac{\sqrt{d + e x} \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 )}{e^{7}} - \frac{6 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{7} \sqrt{d + e x}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right )^{3}}{3 e^{7} \left (d + e x\right )^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3/(e*x+d)**(5/2),x)

[Out]

2*c**3*(d + e*x)**(9/2)/(9*e**7) + (d + e*x)**(7/2)*(6*b*c**2*e - 12*c**3*d)/(7*e**7) + (d + e*x)**(5/2)*(6*a*
c**2*e**2 + 6*b**2*c*e**2 - 30*b*c**2*d*e + 30*c**3*d**2)/(5*e**7) + (d + e*x)**(3/2)*(12*a*b*c*e**3 - 24*a*c*
*2*d*e**2 + 2*b**3*e**3 - 24*b**2*c*d*e**2 + 60*b*c**2*d**2*e - 40*c**3*d**3)/(3*e**7) + sqrt(d + e*x)*(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)/e**7 - 6*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(e**7*sqrt(d + e*x)) - 2*(a*e**
2 - b*d*e + c*d**2)**3/(3*e**7*(d + e*x)**(3/2))

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Giac [B]  time = 1.14024, size = 826, normalized size = 2.93 \begin{align*} \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} e^{56} - 270 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d e^{56} + 945 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{2} e^{56} - 2100 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e^{56} + 4725 \, \sqrt{x e + d} c^{3} d^{4} e^{56} + 135 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{2} e^{57} - 945 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} d e^{57} + 3150 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{2} e^{57} - 9450 \, \sqrt{x e + d} b c^{2} d^{3} e^{57} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c e^{58} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} a c^{2} e^{58} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d e^{58} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} d e^{58} + 5670 \, \sqrt{x e + d} b^{2} c d^{2} e^{58} + 5670 \, \sqrt{x e + d} a c^{2} d^{2} e^{58} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{59} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} a b c e^{59} - 945 \, \sqrt{x e + d} b^{3} d e^{59} - 5670 \, \sqrt{x e + d} a b c d e^{59} + 945 \, \sqrt{x e + d} a b^{2} e^{60} + 945 \, \sqrt{x e + d} a^{2} c e^{60}\right )} e^{\left (-63\right )} + \frac{2 \,{\left (18 \,{\left (x e + d\right )} c^{3} d^{5} - c^{3} d^{6} - 45 \,{\left (x e + d\right )} b c^{2} d^{4} e + 3 \, b c^{2} d^{5} e + 36 \,{\left (x e + d\right )} b^{2} c d^{3} e^{2} + 36 \,{\left (x e + d\right )} a c^{2} d^{3} e^{2} - 3 \, b^{2} c d^{4} e^{2} - 3 \, a c^{2} d^{4} e^{2} - 9 \,{\left (x e + d\right )} b^{3} d^{2} e^{3} - 54 \,{\left (x e + d\right )} a b c d^{2} e^{3} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 18 \,{\left (x e + d\right )} a b^{2} d e^{4} + 18 \,{\left (x e + d\right )} a^{2} c d e^{4} - 3 \, a b^{2} d^{2} e^{4} - 3 \, a^{2} c d^{2} e^{4} - 9 \,{\left (x e + d\right )} a^{2} b e^{5} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/315*(35*(x*e + d)^(9/2)*c^3*e^56 - 270*(x*e + d)^(7/2)*c^3*d*e^56 + 945*(x*e + d)^(5/2)*c^3*d^2*e^56 - 2100*
(x*e + d)^(3/2)*c^3*d^3*e^56 + 4725*sqrt(x*e + d)*c^3*d^4*e^56 + 135*(x*e + d)^(7/2)*b*c^2*e^57 - 945*(x*e + d
)^(5/2)*b*c^2*d*e^57 + 3150*(x*e + d)^(3/2)*b*c^2*d^2*e^57 - 9450*sqrt(x*e + d)*b*c^2*d^3*e^57 + 189*(x*e + d)
^(5/2)*b^2*c*e^58 + 189*(x*e + d)^(5/2)*a*c^2*e^58 - 1260*(x*e + d)^(3/2)*b^2*c*d*e^58 - 1260*(x*e + d)^(3/2)*
a*c^2*d*e^58 + 5670*sqrt(x*e + d)*b^2*c*d^2*e^58 + 5670*sqrt(x*e + d)*a*c^2*d^2*e^58 + 105*(x*e + d)^(3/2)*b^3
*e^59 + 630*(x*e + d)^(3/2)*a*b*c*e^59 - 945*sqrt(x*e + d)*b^3*d*e^59 - 5670*sqrt(x*e + d)*a*b*c*d*e^59 + 945*
sqrt(x*e + d)*a*b^2*e^60 + 945*sqrt(x*e + d)*a^2*c*e^60)*e^(-63) + 2/3*(18*(x*e + d)*c^3*d^5 - c^3*d^6 - 45*(x
*e + d)*b*c^2*d^4*e + 3*b*c^2*d^5*e + 36*(x*e + d)*b^2*c*d^3*e^2 + 36*(x*e + d)*a*c^2*d^3*e^2 - 3*b^2*c*d^4*e^
2 - 3*a*c^2*d^4*e^2 - 9*(x*e + d)*b^3*d^2*e^3 - 54*(x*e + d)*a*b*c*d^2*e^3 + b^3*d^3*e^3 + 6*a*b*c*d^3*e^3 + 1
8*(x*e + d)*a*b^2*d*e^4 + 18*(x*e + d)*a^2*c*d*e^4 - 3*a*b^2*d^2*e^4 - 3*a^2*c*d^2*e^4 - 9*(x*e + d)*a^2*b*e^5
+ 3*a^2*b*d*e^5 - a^3*e^6)*e^(-7)/(x*e + d)^(3/2)