### 3.2288 $$\int \frac{(a+b x+c x^2)^3}{(d+e x)^{7/2}} \, dx$$

Optimal. Leaf size=278 $\frac{2 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac{6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}$

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^3)/(5*e^7*(d + e*x)^(5/2)) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(e^7*(d +
e*x)^(3/2)) - (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)))/(e^7*Sqrt[d + e*x]) - (2*
(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*Sqrt[d + e*x])/e^7 + (2*c*(5*c^2*d^2 + b^2*e^2 -
c*e*(5*b*d - a*e))*(d + e*x)^(3/2))/e^7 - (6*c^2*(2*c*d - b*e)*(d + e*x)^(5/2))/(5*e^7) + (2*c^3*(d + e*x)^(7/
2))/(7*e^7)

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Rubi [A]  time = 0.134166, antiderivative size = 278, 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} $\frac{2 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac{6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^3)/(5*e^7*(d + e*x)^(5/2)) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(e^7*(d +
e*x)^(3/2)) - (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)))/(e^7*Sqrt[d + e*x]) - (2*
(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*Sqrt[d + e*x])/e^7 + (2*c*(5*c^2*d^2 + b^2*e^2 -
c*e*(5*b*d - a*e))*(d + e*x)^(3/2))/e^7 - (6*c^2*(2*c*d - b*e)*(d + e*x)^(5/2))/(5*e^7) + (2*c^3*(d + e*x)^(7/
2))/(7*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)^{7/2}} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^{7/2}}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^{5/2}}+\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 (d+e x)^{3/2}}+\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 )}{e^6 \sqrt{d+e x}}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \sqrt{d+e x}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{3/2}}{e^6}+\frac{c^3 (d+e x)^{5/2}}{e^6}\right ) \, dx\\ &=-\frac{2 \left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)^{3/2}}-\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 )}{e^7 \sqrt{d+e x}}-\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 ) \sqrt{d+e x}}{e^7}+\frac{2 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^7}-\frac{6 c^2 (2 c d-b e) (d+e x)^{5/2}}{5 e^7}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}\\ \end{align*}

Mathematica [A]  time = 0.364776, size = 391, normalized size = 1.41 $-\frac{2 \left (7 c e^2 \left (a^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 a b e \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )+b^2 \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )\right )+7 e^3 \left (a^2 b e^2 (2 d+5 e x)+a^3 e^3+a b^2 e \left (8 d^2+20 d e x+15 e^2 x^2\right )+b^3 \left (-\left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )\right )\right )-7 c^2 e \left (a e \left (-240 d^2 e^2 x^2-320 d^3 e x-128 d^4-40 d e^3 x^3+5 e^4 x^4\right )+b \left (480 d^3 e^2 x^2+80 d^2 e^3 x^3+640 d^4 e x+256 d^5-10 d e^4 x^4+3 e^5 x^5\right )\right )+c^3 \left (1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+2560 d^5 e x+1024 d^6+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (d+e x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(-2*(c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6
*x^6) + 7*e^3*(a^3*e^3 + a^2*b*e^2*(2*d + 5*e*x) + a*b^2*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - b^3*(16*d^3 + 40*
d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)) + 7*c*e^2*(a^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*a*b*e*(16*d^3 + 40
*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + b^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4
)) - 7*c^2*e*(a*e*(-128*d^4 - 320*d^3*e*x - 240*d^2*e^2*x^2 - 40*d*e^3*x^3 + 5*e^4*x^4) + b*(256*d^5 + 640*d^4
*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5))))/(35*e^7*(d + e*x)^(5/2))

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Maple [A]  time = 0.047, size = 495, normalized size = 1.8 \begin{align*} -{\frac{-10\,{c}^{3}{x}^{6}{e}^{6}-42\,b{c}^{2}{e}^{6}{x}^{5}+24\,{c}^{3}d{e}^{5}{x}^{5}-70\,a{c}^{2}{e}^{6}{x}^{4}-70\,{b}^{2}c{e}^{6}{x}^{4}+140\,b{c}^{2}d{e}^{5}{x}^{4}-80\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-420\,abc{e}^{6}{x}^{3}+560\,a{c}^{2}d{e}^{5}{x}^{3}-70\,{b}^{3}{e}^{6}{x}^{3}+560\,{b}^{2}cd{e}^{5}{x}^{3}-1120\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+210\,{a}^{2}c{e}^{6}{x}^{2}+210\,a{b}^{2}{e}^{6}{x}^{2}-2520\,abcd{e}^{5}{x}^{2}+3360\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-420\,{b}^{3}d{e}^{5}{x}^{2}+3360\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}-6720\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}+3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+70\,{a}^{2}b{e}^{6}x+280\,{a}^{2}cd{e}^{5}x+280\,a{b}^{2}d{e}^{5}x-3360\,abc{d}^{2}{e}^{4}x+4480\,a{c}^{2}{d}^{3}{e}^{3}x-560\,{b}^{3}{d}^{2}{e}^{4}x+4480\,{b}^{2}c{d}^{3}{e}^{3}x-8960\,b{c}^{2}{d}^{4}{e}^{2}x+5120\,{c}^{3}{d}^{5}ex+14\,{a}^{3}{e}^{6}+28\,{a}^{2}bd{e}^{5}+112\,{a}^{2}c{d}^{2}{e}^{4}+112\,a{b}^{2}{d}^{2}{e}^{4}-1344\,abc{d}^{3}{e}^{3}+1792\,a{c}^{2}{d}^{4}{e}^{2}-224\,{b}^{3}{d}^{3}{e}^{3}+1792\,{b}^{2}c{d}^{4}{e}^{2}-3584\,b{c}^{2}{d}^{5}e+2048\,{c}^{3}{d}^{6}}{35\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{5}{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)^(7/2),x)

[Out]

-2/35/(e*x+d)^(5/2)*(-5*c^3*e^6*x^6-21*b*c^2*e^6*x^5+12*c^3*d*e^5*x^5-35*a*c^2*e^6*x^4-35*b^2*c*e^6*x^4+70*b*c
^2*d*e^5*x^4-40*c^3*d^2*e^4*x^4-210*a*b*c*e^6*x^3+280*a*c^2*d*e^5*x^3-35*b^3*e^6*x^3+280*b^2*c*d*e^5*x^3-560*b
*c^2*d^2*e^4*x^3+320*c^3*d^3*e^3*x^3+105*a^2*c*e^6*x^2+105*a*b^2*e^6*x^2-1260*a*b*c*d*e^5*x^2+1680*a*c^2*d^2*e
^4*x^2-210*b^3*d*e^5*x^2+1680*b^2*c*d^2*e^4*x^2-3360*b*c^2*d^3*e^3*x^2+1920*c^3*d^4*e^2*x^2+35*a^2*b*e^6*x+140
*a^2*c*d*e^5*x+140*a*b^2*d*e^5*x-1680*a*b*c*d^2*e^4*x+2240*a*c^2*d^3*e^3*x-280*b^3*d^2*e^4*x+2240*b^2*c*d^3*e^
3*x-4480*b*c^2*d^4*e^2*x+2560*c^3*d^5*e*x+7*a^3*e^6+14*a^2*b*d*e^5+56*a^2*c*d^2*e^4+56*a*b^2*d^2*e^4-672*a*b*c
*d^3*e^3+896*a*c^2*d^4*e^2-112*b^3*d^3*e^3+896*b^2*c*d^4*e^2-1792*b*c^2*d^5*e+1024*c^3*d^6)/e^7

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Maxima [A]  time = 1.03108, size = 558, normalized size = 2.01 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} - 21 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\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{3}{2}} - 35 \,{\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 )} \sqrt{e x + d}}{e^{6}} - \frac{7 \,{\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} + 15 \,{\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 )}^{2} - 5 \,{\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{5}{2}} e^{6}}\right )}}{35 \, 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)^(7/2),x, algorithm="maxima")

[Out]

2/35*((5*(e*x + d)^(7/2)*c^3 - 21*(2*c^3*d - b*c^2*e)*(e*x + d)^(5/2) + 35*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c +
a*c^2)*e^2)*(e*x + d)^(3/2) - 35*(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)*sqrt(e*x + d))/e^6 - 7*(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 + 15*(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)^2 - 5*(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)^(5/2)*e^6))/e

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Fricas [A]  time = 1.99361, size = 950, normalized size = 3.42 \begin{align*} \frac{2 \,{\left (5 \, c^{3} e^{6} x^{6} - 1024 \, c^{3} d^{6} + 1792 \, b c^{2} d^{5} e - 14 \, a^{2} b d e^{5} - 7 \, a^{3} e^{6} - 896 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 112 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 56 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \,{\left (4 \, c^{3} d e^{5} - 7 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (8 \, c^{3} d^{2} e^{4} - 14 \, b c^{2} d e^{5} + 7 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{3} e^{3} - 112 \, b c^{2} d^{2} e^{4} + 56 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} - 7 \,{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 15 \,{\left (128 \, c^{3} d^{4} e^{2} - 224 \, b c^{2} d^{3} e^{3} + 112 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 14 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 7 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 5 \,{\left (512 \, c^{3} d^{5} e - 896 \, b c^{2} d^{4} e^{2} + 7 \, a^{2} b e^{6} + 448 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 56 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 28 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} 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)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*c^3*e^6*x^6 - 1024*c^3*d^6 + 1792*b*c^2*d^5*e - 14*a^2*b*d*e^5 - 7*a^3*e^6 - 896*(b^2*c + a*c^2)*d^4*e
^2 + 112*(b^3 + 6*a*b*c)*d^3*e^3 - 56*(a*b^2 + a^2*c)*d^2*e^4 - 3*(4*c^3*d*e^5 - 7*b*c^2*e^6)*x^5 + 5*(8*c^3*d
^2*e^4 - 14*b*c^2*d*e^5 + 7*(b^2*c + a*c^2)*e^6)*x^4 - 5*(64*c^3*d^3*e^3 - 112*b*c^2*d^2*e^4 + 56*(b^2*c + a*c
^2)*d*e^5 - 7*(b^3 + 6*a*b*c)*e^6)*x^3 - 15*(128*c^3*d^4*e^2 - 224*b*c^2*d^3*e^3 + 112*(b^2*c + a*c^2)*d^2*e^4
- 14*(b^3 + 6*a*b*c)*d*e^5 + 7*(a*b^2 + a^2*c)*e^6)*x^2 - 5*(512*c^3*d^5*e - 896*b*c^2*d^4*e^2 + 7*a^2*b*e^6
+ 448*(b^2*c + a*c^2)*d^3*e^3 - 56*(b^3 + 6*a*b*c)*d^2*e^4 + 28*(a*b^2 + a^2*c)*d*e^5)*x)*sqrt(e*x + d)/(e^10*
x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3/(e*x+d)**(7/2),x)

[Out]

Timed out

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Giac [B]  time = 1.13184, size = 822, normalized size = 2.96 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} e^{42} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d e^{42} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt{x e + d} c^{3} d^{3} e^{42} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} e^{43} - 175 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d e^{43} + 1050 \, \sqrt{x e + d} b c^{2} d^{2} e^{43} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c e^{44} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} e^{44} - 420 \, \sqrt{x e + d} b^{2} c d e^{44} - 420 \, \sqrt{x e + d} a c^{2} d e^{44} + 35 \, \sqrt{x e + d} b^{3} e^{45} + 210 \, \sqrt{x e + d} a b c e^{45}\right )} e^{\left (-49\right )} - \frac{2 \,{\left (75 \,{\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \,{\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} - 150 \,{\left (x e + d\right )}^{2} b c^{2} d^{3} e + 25 \,{\left (x e + d\right )} b c^{2} d^{4} e - 3 \, b c^{2} d^{5} e + 90 \,{\left (x e + d\right )}^{2} b^{2} c d^{2} e^{2} + 90 \,{\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \,{\left (x e + d\right )} b^{2} c d^{3} e^{2} - 20 \,{\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} - 15 \,{\left (x e + d\right )}^{2} b^{3} d e^{3} - 90 \,{\left (x e + d\right )}^{2} a b c d e^{3} + 5 \,{\left (x e + d\right )} b^{3} d^{2} e^{3} + 30 \,{\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} + 15 \,{\left (x e + d\right )}^{2} a b^{2} e^{4} + 15 \,{\left (x e + d\right )}^{2} a^{2} c e^{4} - 10 \,{\left (x e + d\right )} a b^{2} d e^{4} - 10 \,{\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} + 5 \,{\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 )}}{5 \,{\left (x e + d\right )}^{\frac{5}{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)^(7/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*c^3*e^42 - 42*(x*e + d)^(5/2)*c^3*d*e^42 + 175*(x*e + d)^(3/2)*c^3*d^2*e^42 - 700*sqrt
(x*e + d)*c^3*d^3*e^42 + 21*(x*e + d)^(5/2)*b*c^2*e^43 - 175*(x*e + d)^(3/2)*b*c^2*d*e^43 + 1050*sqrt(x*e + d)
*b*c^2*d^2*e^43 + 35*(x*e + d)^(3/2)*b^2*c*e^44 + 35*(x*e + d)^(3/2)*a*c^2*e^44 - 420*sqrt(x*e + d)*b^2*c*d*e^
44 - 420*sqrt(x*e + d)*a*c^2*d*e^44 + 35*sqrt(x*e + d)*b^3*e^45 + 210*sqrt(x*e + d)*a*b*c*e^45)*e^(-49) - 2/5*
(75*(x*e + d)^2*c^3*d^4 - 10*(x*e + d)*c^3*d^5 + c^3*d^6 - 150*(x*e + d)^2*b*c^2*d^3*e + 25*(x*e + d)*b*c^2*d^
4*e - 3*b*c^2*d^5*e + 90*(x*e + d)^2*b^2*c*d^2*e^2 + 90*(x*e + d)^2*a*c^2*d^2*e^2 - 20*(x*e + d)*b^2*c*d^3*e^2
- 20*(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 - 15*(x*e + d)^2*b^3*d*e^3 - 90*(x*e + d)^2*
a*b*c*d*e^3 + 5*(x*e + d)*b^3*d^2*e^3 + 30*(x*e + d)*a*b*c*d^2*e^3 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 15*(x*e +
d)^2*a*b^2*e^4 + 15*(x*e + d)^2*a^2*c*e^4 - 10*(x*e + d)*a*b^2*d*e^4 - 10*(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 + 5*(x*e + d)*a^2*b*e^5 - 3*a^2*b*d*e^5 + a^3*e^6)*e^(-7)/(x*e + d)^(5/2)