### 3.2286 $$\int \frac{(a+b x+c x^2)^3}{(d+e x)^{3/2}} \, dx$$

Optimal. Leaf size=280 $\frac{6 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{6 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{e^7 \sqrt{d+e x}}-\frac{2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7}$

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^3)/(e^7*Sqrt[d + e*x]) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x])
/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)^(3/2))/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)^(5/2))/(5*e^7) + (6*c*(5*c^2*d^2 + b^2*e^2 - c*
e*(5*b*d - a*e))*(d + e*x)^(7/2))/(7*e^7) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(9/2))/(3*e^7) + (2*c^3*(d + e*x)^(
11/2))/(11*e^7)

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Rubi [A]  time = 0.127591, antiderivative size = 280, 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} $\frac{6 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{6 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{e^7 \sqrt{d+e x}}-\frac{2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^3)/(e^7*Sqrt[d + e*x]) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x])
/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)^(3/2))/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)^(5/2))/(5*e^7) + (6*c*(5*c^2*d^2 + b^2*e^2 - c*
e*(5*b*d - a*e))*(d + e*x)^(7/2))/(7*e^7) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(9/2))/(3*e^7) + (2*c^3*(d + e*x)^(
11/2))/(11*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)^{3/2}} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^{3/2}}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 \sqrt{d+e x}}+\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 ) \sqrt{d+e x}}{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)^{3/2}}{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)^{5/2}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{7/2}}{e^6}+\frac{c^3 (d+e x)^{9/2}}{e^6}\right ) \, dx\\ &=-\frac{2 \left (c d^2-b d e+a e^2\right )^3}{e^7 \sqrt{d+e x}}-\frac{6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}{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)^{3/2}}{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)^{5/2}}{5 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)^{7/2}}{7 e^7}-\frac{2 c^2 (2 c d-b e) (d+e x)^{9/2}}{3 e^7}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7}\\ \end{align*}

Mathematica [A]  time = 0.411908, size = 394, normalized size = 1.41 $\frac{-66 c e^2 \left (35 a^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )-42 a b e \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )+3 b^2 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )\right )+462 e^3 \left (15 a^2 b e^2 (2 d+e x)-5 a^3 e^3+5 a b^2 e \left (-8 d^2-4 d e x+e^2 x^2\right )+b^3 \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )+22 c^2 e \left (9 a e \left (16 d^2 e^2 x^2-64 d^3 e x-128 d^4-8 d e^3 x^3+5 e^4 x^4\right )+5 b \left (-32 d^3 e^2 x^2+16 d^2 e^3 x^3+128 d^4 e x+256 d^5-10 d e^4 x^4+7 e^5 x^5\right )\right )-10 c^3 \left (-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+512 d^5 e x+1024 d^6+28 d e^5 x^5-21 e^6 x^6\right )}{1155 e^7 \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-10*c^3*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x
^6) + 462*e^3*(-5*a^3*e^3 + 15*a^2*b*e^2*(2*d + e*x) + 5*a*b^2*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + b^3*(16*d^3 +
8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) - 66*c*e^2*(35*a^2*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) - 42*a*b*e*(16*d^3 + 8*
d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 3*b^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)) +
22*c^2*e*(9*a*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + 5*b*(256*d^5 + 128*d^4*e*
x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)))/(1155*e^7*Sqrt[d + e*x])

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Maple [A]  time = 0.045, size = 495, normalized size = 1.8 \begin{align*} -{\frac{-210\,{c}^{3}{x}^{6}{e}^{6}-770\,b{c}^{2}{e}^{6}{x}^{5}+280\,{c}^{3}d{e}^{5}{x}^{5}-990\,a{c}^{2}{e}^{6}{x}^{4}-990\,{b}^{2}c{e}^{6}{x}^{4}+1100\,b{c}^{2}d{e}^{5}{x}^{4}-400\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-2772\,abc{e}^{6}{x}^{3}+1584\,a{c}^{2}d{e}^{5}{x}^{3}-462\,{b}^{3}{e}^{6}{x}^{3}+1584\,{b}^{2}cd{e}^{5}{x}^{3}-1760\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}-2310\,{a}^{2}c{e}^{6}{x}^{2}-2310\,a{b}^{2}{e}^{6}{x}^{2}+5544\,abcd{e}^{5}{x}^{2}-3168\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+924\,{b}^{3}d{e}^{5}{x}^{2}-3168\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+3520\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-1280\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-6930\,{a}^{2}b{e}^{6}x+9240\,{a}^{2}cd{e}^{5}x+9240\,a{b}^{2}d{e}^{5}x-22176\,abc{d}^{2}{e}^{4}x+12672\,a{c}^{2}{d}^{3}{e}^{3}x-3696\,{b}^{3}{d}^{2}{e}^{4}x+12672\,{b}^{2}c{d}^{3}{e}^{3}x-14080\,b{c}^{2}{d}^{4}{e}^{2}x+5120\,{c}^{3}{d}^{5}ex+2310\,{a}^{3}{e}^{6}-13860\,{a}^{2}bd{e}^{5}+18480\,{a}^{2}c{d}^{2}{e}^{4}+18480\,a{b}^{2}{d}^{2}{e}^{4}-44352\,abc{d}^{3}{e}^{3}+25344\,a{c}^{2}{d}^{4}{e}^{2}-7392\,{b}^{3}{d}^{3}{e}^{3}+25344\,{b}^{2}c{d}^{4}{e}^{2}-28160\,b{c}^{2}{d}^{5}e+10240\,{c}^{3}{d}^{6}}{1155\,{e}^{7}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1155/(e*x+d)^(1/2)*(-105*c^3*e^6*x^6-385*b*c^2*e^6*x^5+140*c^3*d*e^5*x^5-495*a*c^2*e^6*x^4-495*b^2*c*e^6*x^
4+550*b*c^2*d*e^5*x^4-200*c^3*d^2*e^4*x^4-1386*a*b*c*e^6*x^3+792*a*c^2*d*e^5*x^3-231*b^3*e^6*x^3+792*b^2*c*d*e
^5*x^3-880*b*c^2*d^2*e^4*x^3+320*c^3*d^3*e^3*x^3-1155*a^2*c*e^6*x^2-1155*a*b^2*e^6*x^2+2772*a*b*c*d*e^5*x^2-15
84*a*c^2*d^2*e^4*x^2+462*b^3*d*e^5*x^2-1584*b^2*c*d^2*e^4*x^2+1760*b*c^2*d^3*e^3*x^2-640*c^3*d^4*e^2*x^2-3465*
a^2*b*e^6*x+4620*a^2*c*d*e^5*x+4620*a*b^2*d*e^5*x-11088*a*b*c*d^2*e^4*x+6336*a*c^2*d^3*e^3*x-1848*b^3*d^2*e^4*
x+6336*b^2*c*d^3*e^3*x-7040*b*c^2*d^4*e^2*x+2560*c^3*d^5*e*x+1155*a^3*e^6-6930*a^2*b*d*e^5+9240*a^2*c*d^2*e^4+
9240*a*b^2*d^2*e^4-22176*a*b*c*d^3*e^3+12672*a*c^2*d^4*e^2-3696*b^3*d^3*e^3+12672*b^2*c*d^4*e^2-14080*b*c^2*d^
5*e+5120*c^3*d^6)/e^7

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Maxima [A]  time = 1.04633, size = 560, normalized size = 2. \begin{align*} \frac{2 \,{\left (\frac{105 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{3} - 385 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\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{7}{2}} - 231 \,{\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{5}{2}} + 1155 \,{\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{3}{2}} - 3465 \,{\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 )} \sqrt{e x + d}}{e^{6}} - \frac{1155 \,{\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 )}}{\sqrt{e x + d} e^{6}}\right )}}{1155 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/1155*((105*(e*x + d)^(11/2)*c^3 - 385*(2*c^3*d - b*c^2*e)*(e*x + d)^(9/2) + 495*(5*c^3*d^2 - 5*b*c^2*d*e + (
b^2*c + a*c^2)*e^2)*(e*x + d)^(7/2) - 231*(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)^(5/2) + 1155*(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)^(3/2) - 3465*(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)*sqrt(e*x + d))/e^6 - 1155*(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)/
(sqrt(e*x + d)*e^6))/e

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Fricas [A]  time = 2.01709, size = 950, normalized size = 3.39 \begin{align*} \frac{2 \,{\left (105 \, c^{3} e^{6} x^{6} - 5120 \, c^{3} d^{6} + 14080 \, b c^{2} d^{5} e + 6930 \, a^{2} b d e^{5} - 1155 \, a^{3} e^{6} - 12672 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 3696 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 9240 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 35 \,{\left (4 \, c^{3} d e^{5} - 11 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (40 \, c^{3} d^{2} e^{4} - 110 \, b c^{2} d e^{5} + 99 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} -{\left (320 \, c^{3} d^{3} e^{3} - 880 \, b c^{2} d^{2} e^{4} + 792 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} - 231 \,{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} +{\left (640 \, c^{3} d^{4} e^{2} - 1760 \, b c^{2} d^{3} e^{3} + 1584 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 462 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 1155 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} -{\left (2560 \, c^{3} d^{5} e - 7040 \, b c^{2} d^{4} e^{2} - 3465 \, a^{2} b e^{6} + 6336 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 1848 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 4620 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt{e x + d}}{1155 \,{\left (e^{8} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/1155*(105*c^3*e^6*x^6 - 5120*c^3*d^6 + 14080*b*c^2*d^5*e + 6930*a^2*b*d*e^5 - 1155*a^3*e^6 - 12672*(b^2*c +
a*c^2)*d^4*e^2 + 3696*(b^3 + 6*a*b*c)*d^3*e^3 - 9240*(a*b^2 + a^2*c)*d^2*e^4 - 35*(4*c^3*d*e^5 - 11*b*c^2*e^6)
*x^5 + 5*(40*c^3*d^2*e^4 - 110*b*c^2*d*e^5 + 99*(b^2*c + a*c^2)*e^6)*x^4 - (320*c^3*d^3*e^3 - 880*b*c^2*d^2*e^
4 + 792*(b^2*c + a*c^2)*d*e^5 - 231*(b^3 + 6*a*b*c)*e^6)*x^3 + (640*c^3*d^4*e^2 - 1760*b*c^2*d^3*e^3 + 1584*(b
^2*c + a*c^2)*d^2*e^4 - 462*(b^3 + 6*a*b*c)*d*e^5 + 1155*(a*b^2 + a^2*c)*e^6)*x^2 - (2560*c^3*d^5*e - 7040*b*c
^2*d^4*e^2 - 3465*a^2*b*e^6 + 6336*(b^2*c + a*c^2)*d^3*e^3 - 1848*(b^3 + 6*a*b*c)*d^2*e^4 + 4620*(a*b^2 + a^2*
c)*d*e^5)*x)*sqrt(e*x + d)/(e^8*x + d*e^7)

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Sympy [A]  time = 99.2054, size = 428, normalized size = 1.53 \begin{align*} \frac{2 c^{3} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{7}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (6 b c^{2} e - 12 c^{3} d\right )}{9 e^{7}} + \frac{\left (d + e x\right )^{\frac{7}{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 )}{7 e^{7}} + \frac{\left (d + e x\right )^{\frac{5}{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 )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{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 )}{3 e^{7}} + \frac{\sqrt{d + e x} \left (6 a^{2} b e^{5} - 12 a^{2} c d e^{4} - 12 a b^{2} d e^{4} + 36 a b c d^{2} e^{3} - 24 a c^{2} d^{3} e^{2} + 6 b^{3} d^{2} e^{3} - 24 b^{2} c d^{3} e^{2} + 30 b c^{2} d^{4} e - 12 c^{3} d^{5}\right )}{e^{7}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right )^{3}}{e^{7} \sqrt{d + e x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c**3*(d + e*x)**(11/2)/(11*e**7) + (d + e*x)**(9/2)*(6*b*c**2*e - 12*c**3*d)/(9*e**7) + (d + e*x)**(7/2)*(6*
a*c**2*e**2 + 6*b**2*c*e**2 - 30*b*c**2*d*e + 30*c**3*d**2)/(7*e**7) + (d + e*x)**(5/2)*(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)/(5*e**7) + (d + e*x)**(3/2)*(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 - 6
0*b*c**2*d**3*e + 30*c**3*d**4)/(3*e**7) + sqrt(d + e*x)*(6*a**2*b*e**5 - 12*a**2*c*d*e**4 - 12*a*b**2*d*e**4
+ 36*a*b*c*d**2*e**3 - 24*a*c**2*d**3*e**2 + 6*b**3*d**2*e**3 - 24*b**2*c*d**3*e**2 + 30*b*c**2*d**4*e - 12*c*
*3*d**5)/e**7 - 2*(a*e**2 - b*d*e + c*d**2)**3/(e**7*sqrt(d + e*x))

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Giac [B]  time = 1.1384, size = 852, normalized size = 3.04 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*(x*e + d)^(11/2)*c^3*e^70 - 770*(x*e + d)^(9/2)*c^3*d*e^70 + 2475*(x*e + d)^(7/2)*c^3*d^2*e^70 - 4
620*(x*e + d)^(5/2)*c^3*d^3*e^70 + 5775*(x*e + d)^(3/2)*c^3*d^4*e^70 - 6930*sqrt(x*e + d)*c^3*d^5*e^70 + 385*(
x*e + d)^(9/2)*b*c^2*e^71 - 2475*(x*e + d)^(7/2)*b*c^2*d*e^71 + 6930*(x*e + d)^(5/2)*b*c^2*d^2*e^71 - 11550*(x
*e + d)^(3/2)*b*c^2*d^3*e^71 + 17325*sqrt(x*e + d)*b*c^2*d^4*e^71 + 495*(x*e + d)^(7/2)*b^2*c*e^72 + 495*(x*e
+ d)^(7/2)*a*c^2*e^72 - 2772*(x*e + d)^(5/2)*b^2*c*d*e^72 - 2772*(x*e + d)^(5/2)*a*c^2*d*e^72 + 6930*(x*e + d)
^(3/2)*b^2*c*d^2*e^72 + 6930*(x*e + d)^(3/2)*a*c^2*d^2*e^72 - 13860*sqrt(x*e + d)*b^2*c*d^3*e^72 - 13860*sqrt(
x*e + d)*a*c^2*d^3*e^72 + 231*(x*e + d)^(5/2)*b^3*e^73 + 1386*(x*e + d)^(5/2)*a*b*c*e^73 - 1155*(x*e + d)^(3/2
)*b^3*d*e^73 - 6930*(x*e + d)^(3/2)*a*b*c*d*e^73 + 3465*sqrt(x*e + d)*b^3*d^2*e^73 + 20790*sqrt(x*e + d)*a*b*c
*d^2*e^73 + 1155*(x*e + d)^(3/2)*a*b^2*e^74 + 1155*(x*e + d)^(3/2)*a^2*c*e^74 - 6930*sqrt(x*e + d)*a*b^2*d*e^7
4 - 6930*sqrt(x*e + d)*a^2*c*d*e^74 + 3465*sqrt(x*e + d)*a^2*b*e^75)*e^(-77) - 2*(c^3*d^6 - 3*b*c^2*d^5*e + 3*
b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*
d*e^5 + a^3*e^6)*e^(-7)/sqrt(x*e + d)