3.358 \(\int \frac{(b x+c x^2)^3}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=242 \[ \frac{6 c (d+e x)^{7/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7}+\frac{2 d (d+e x)^{3/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}-\frac{6 d^2 \sqrt{d+e x} (c d-b e)^2 (2 c d-b e)}{e^7}-\frac{2 d^3 (c d-b e)^3}{e^7 \sqrt{d+e x}}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7} \]

[Out]

(-2*d^3*(c*d - b*e)^3)/(e^7*Sqrt[d + e*x]) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*e)*Sqrt[d + e*x])/e^7 + (2*d*(c*d
- b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(3/2))/e^7 - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b
^2*e^2)*(d + e*x)^(5/2))/(5*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(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 = 0.103651, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {698} \[ \frac{6 c (d+e x)^{7/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7}+\frac{2 d (d+e x)^{3/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}-\frac{6 d^2 \sqrt{d+e x} (c d-b e)^2 (2 c d-b e)}{e^7}-\frac{2 d^3 (c d-b e)^3}{e^7 \sqrt{d+e x}}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^3*(c*d - b*e)^3)/(e^7*Sqrt[d + e*x]) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*e)*Sqrt[d + e*x])/e^7 + (2*d*(c*d
- b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(3/2))/e^7 - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b
^2*e^2)*(d + e*x)^(5/2))/(5*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(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 (b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{d^3 (c d-b e)^3}{e^6 (d+e x)^{3/2}}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 \sqrt{d+e x}}+\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) \sqrt{d+e x}}{e^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{3/2}}{e^6}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\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 d^3 (c d-b e)^3}{e^7 \sqrt{d+e x}}-\frac{6 d^2 (c d-b e)^2 (2 c d-b e) \sqrt{d+e x}}{e^7}+\frac{2 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{e^7}-\frac{2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^7}+\frac{6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\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 = 0.142685, size = 206, normalized size = 0.85 \[ \frac{2 \left (495 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-231 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+1155 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-385 c^2 (d+e x)^5 (2 c d-b e)-3465 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)-1155 d^3 (c d-b e)^3+105 c^3 (d+e x)^6\right )}{1155 e^7 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1155*d^3*(c*d - b*e)^3 - 3465*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 1155*d*(c*d - b*e)*(5*c^2*d^2 -
5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 231*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 495*c*
(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 385*c^2*(2*c*d - b*e)*(d + e*x)^5 + 105*c^3*(d + e*x)^6))/(115
5*e^7*Sqrt[d + e*x])

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Maple [A]  time = 0.047, size = 286, normalized size = 1.2 \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\,{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}+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}-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}+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+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)^3/(e*x+d)^(3/2),x)

[Out]

2/1155*(105*c^3*e^6*x^6+385*b*c^2*e^6*x^5-140*c^3*d*e^5*x^5+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+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^3-462*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+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+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*x+d)^(1/2)
/e^7

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Maxima [A]  time = 1.15697, size = 377, normalized size = 1.56 \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 + b^{2} c 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 \, b^{2} c d e^{2} - b^{3} 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 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 3465 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} \sqrt{e x + d}}{e^{6}} - \frac{1155 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\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)^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*e^2)*(e*x + d)^(7/2) - 231*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e*x + d)^(5/2) + 115
5*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e*x + d)^(3/2) - 3465*(2*c^3*d^5 - 5*b*c^2*d^4*e
+ 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*sqrt(e*x + d))/e^6 - 1155*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d
^3*e^3)/(sqrt(e*x + d)*e^6))/e

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Fricas [A]  time = 2.30352, size = 629, normalized size = 2.6 \begin{align*} \frac{2 \,{\left (105 \, c^{3} e^{6} x^{6} - 5120 \, c^{3} d^{6} + 14080 \, b c^{2} d^{5} e - 12672 \, b^{2} c d^{4} e^{2} + 3696 \, b^{3} d^{3} e^{3} - 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 \, b^{2} c e^{6}\right )} x^{4} -{\left (320 \, c^{3} d^{3} e^{3} - 880 \, b c^{2} d^{2} e^{4} + 792 \, b^{2} c d e^{5} - 231 \, b^{3} e^{6}\right )} x^{3} + 2 \,{\left (320 \, c^{3} d^{4} e^{2} - 880 \, b c^{2} d^{3} e^{3} + 792 \, b^{2} c d^{2} e^{4} - 231 \, b^{3} d e^{5}\right )} x^{2} - 8 \,{\left (320 \, c^{3} d^{5} e - 880 \, b c^{2} d^{4} e^{2} + 792 \, b^{2} c d^{3} e^{3} - 231 \, b^{3} d^{2} e^{4}\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)^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 - 12672*b^2*c*d^4*e^2 + 3696*b^3*d^3*e^3 - 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*e^6)*x^4 - (320*c^3*d^3*e^3 - 880
*b*c^2*d^2*e^4 + 792*b^2*c*d*e^5 - 231*b^3*e^6)*x^3 + 2*(320*c^3*d^4*e^2 - 880*b*c^2*d^3*e^3 + 792*b^2*c*d^2*e
^4 - 231*b^3*d*e^5)*x^2 - 8*(320*c^3*d^5*e - 880*b*c^2*d^4*e^2 + 792*b^2*c*d^3*e^3 - 231*b^3*d^2*e^4)*x)*sqrt(
e*x + d)/(e^8*x + d*e^7)

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Sympy [A]  time = 52.0278, size = 284, normalized size = 1.17 \begin{align*} \frac{2 c^{3} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{7}} + \frac{2 d^{3} \left (b e - c d\right )^{3}}{e^{7} \sqrt{d + e x}} + \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 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 (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 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 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c**3*(d + e*x)**(11/2)/(11*e**7) + 2*d**3*(b*e - c*d)**3/(e**7*sqrt(d + e*x)) + (d + e*x)**(9/2)*(6*b*c**2*e
- 12*c**3*d)/(9*e**7) + (d + e*x)**(7/2)*(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)*(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*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)/(3*e**7) + sqrt(d + e*x)*(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

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Giac [A]  time = 1.36876, size = 501, normalized size = 2.07 \begin{align*} \frac{2}{1155} \,{\left (105 \,{\left (x e + d\right )}^{\frac{11}{2}} c^{3} e^{70} - 770 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} d e^{70} + 2475 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d^{2} e^{70} - 4620 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{3} e^{70} + 5775 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{4} e^{70} - 6930 \, \sqrt{x e + d} c^{3} d^{5} e^{70} + 385 \,{\left (x e + d\right )}^{\frac{9}{2}} b c^{2} e^{71} - 2475 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{2} d e^{71} + 6930 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} d^{2} e^{71} - 11550 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{3} e^{71} + 17325 \, \sqrt{x e + d} b c^{2} d^{4} e^{71} + 495 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{2} c e^{72} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c d e^{72} + 6930 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d^{2} e^{72} - 13860 \, \sqrt{x e + d} b^{2} c d^{3} e^{72} + 231 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} e^{73} - 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d e^{73} + 3465 \, \sqrt{x e + d} b^{3} d^{2} e^{73}\right )} e^{\left (-77\right )} - \frac{2 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{\sqrt{x e + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^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 - 2772*(x*e
+ d)^(5/2)*b^2*c*d*e^72 + 6930*(x*e + d)^(3/2)*b^2*c*d^2*e^72 - 13860*sqrt(x*e + d)*b^2*c*d^3*e^72 + 231*(x*e
+ d)^(5/2)*b^3*e^73 - 1155*(x*e + d)^(3/2)*b^3*d*e^73 + 3465*sqrt(x*e + d)*b^3*d^2*e^73)*e^(-77) - 2*(c^3*d^6
- 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*e^(-7)/sqrt(x*e + d)