### 3.351 $$\int \frac{(b x+c x^2)^2}{(d+e x)^{5/2}} \, dx$$

Optimal. Leaf size=143 $\frac{2 \sqrt{d+e x} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^5}-\frac{2 d^2 (c d-b e)^2}{3 e^5 (d+e x)^{3/2}}-\frac{4 c (d+e x)^{3/2} (2 c d-b e)}{3 e^5}+\frac{4 d (c d-b e) (2 c d-b e)}{e^5 \sqrt{d+e x}}+\frac{2 c^2 (d+e x)^{5/2}}{5 e^5}$

[Out]

(-2*d^2*(c*d - b*e)^2)/(3*e^5*(d + e*x)^(3/2)) + (4*d*(c*d - b*e)*(2*c*d - b*e))/(e^5*Sqrt[d + e*x]) + (2*(6*c
^2*d^2 - 6*b*c*d*e + b^2*e^2)*Sqrt[d + e*x])/e^5 - (4*c*(2*c*d - b*e)*(d + e*x)^(3/2))/(3*e^5) + (2*c^2*(d + e
*x)^(5/2))/(5*e^5)

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Rubi [A]  time = 0.0583636, antiderivative size = 143, 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} $\frac{2 \sqrt{d+e x} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^5}-\frac{2 d^2 (c d-b e)^2}{3 e^5 (d+e x)^{3/2}}-\frac{4 c (d+e x)^{3/2} (2 c d-b e)}{3 e^5}+\frac{4 d (c d-b e) (2 c d-b e)}{e^5 \sqrt{d+e x}}+\frac{2 c^2 (d+e x)^{5/2}}{5 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*d^2*(c*d - b*e)^2)/(3*e^5*(d + e*x)^(3/2)) + (4*d*(c*d - b*e)*(2*c*d - b*e))/(e^5*Sqrt[d + e*x]) + (2*(6*c
^2*d^2 - 6*b*c*d*e + b^2*e^2)*Sqrt[d + e*x])/e^5 - (4*c*(2*c*d - b*e)*(d + e*x)^(3/2))/(3*e^5) + (2*c^2*(d + e
*x)^(5/2))/(5*e^5)

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 )^2}{(d+e x)^{5/2}} \, dx &=\int \left (\frac{d^2 (c d-b e)^2}{e^4 (d+e x)^{5/2}}+\frac{2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^{3/2}}+\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 \sqrt{d+e x}}-\frac{2 c (2 c d-b e) \sqrt{d+e x}}{e^4}+\frac{c^2 (d+e x)^{3/2}}{e^4}\right ) \, dx\\ &=-\frac{2 d^2 (c d-b e)^2}{3 e^5 (d+e x)^{3/2}}+\frac{4 d (c d-b e) (2 c d-b e)}{e^5 \sqrt{d+e x}}+\frac{2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \sqrt{d+e x}}{e^5}-\frac{4 c (2 c d-b e) (d+e x)^{3/2}}{3 e^5}+\frac{2 c^2 (d+e x)^{5/2}}{5 e^5}\\ \end{align*}

Mathematica [A]  time = 0.0662151, size = 123, normalized size = 0.86 $\frac{2 \left (5 b^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+10 b c e \left (-24 d^2 e x-16 d^3-6 d e^2 x^2+e^3 x^3\right )+c^2 \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(5*b^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 10*b*c*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + c^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)))/(15*e^5*(d + e*x)^(3/2))

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Maple [A]  time = 0.049, size = 141, normalized size = 1. \begin{align*}{\frac{6\,{c}^{2}{x}^{4}{e}^{4}+20\,bc{e}^{4}{x}^{3}-16\,{c}^{2}d{e}^{3}{x}^{3}+30\,{b}^{2}{e}^{4}{x}^{2}-120\,bcd{e}^{3}{x}^{2}+96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+120\,{b}^{2}d{e}^{3}x-480\,bc{d}^{2}{e}^{2}x+384\,{c}^{2}{d}^{3}ex+80\,{b}^{2}{d}^{2}{e}^{2}-320\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{15\,{e}^{5}} \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)^2/(e*x+d)^(5/2),x)

[Out]

2/15*(3*c^2*e^4*x^4+10*b*c*e^4*x^3-8*c^2*d*e^3*x^3+15*b^2*e^4*x^2-60*b*c*d*e^3*x^2+48*c^2*d^2*e^2*x^2+60*b^2*d
*e^3*x-240*b*c*d^2*e^2*x+192*c^2*d^3*e*x+40*b^2*d^2*e^2-160*b*c*d^3*e+128*c^2*d^4)/(e*x+d)^(3/2)/e^5

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Maxima [A]  time = 1.03874, size = 196, normalized size = 1.37 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} c^{2} - 10 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{5 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} - 6 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{4}}\right )}}{15 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*((3*(e*x + d)^(5/2)*c^2 - 10*(2*c^2*d - b*c*e)*(e*x + d)^(3/2) + 15*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*sqr
t(e*x + d))/e^4 - 5*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 - 6*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(e*x + d))/
((e*x + d)^(3/2)*e^4))/e

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Fricas [A]  time = 1.94582, size = 343, normalized size = 2.4 \begin{align*} \frac{2 \,{\left (3 \, c^{2} e^{4} x^{4} + 128 \, c^{2} d^{4} - 160 \, b c d^{3} e + 40 \, b^{2} d^{2} e^{2} - 2 \,{\left (4 \, c^{2} d e^{3} - 5 \, b c e^{4}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{2} e^{2} - 20 \, b c d e^{3} + 5 \, b^{2} e^{4}\right )} x^{2} + 12 \,{\left (16 \, c^{2} d^{3} e - 20 \, b c d^{2} e^{2} + 5 \, b^{2} d e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*c^2*e^4*x^4 + 128*c^2*d^4 - 160*b*c*d^3*e + 40*b^2*d^2*e^2 - 2*(4*c^2*d*e^3 - 5*b*c*e^4)*x^3 + 3*(16*c
^2*d^2*e^2 - 20*b*c*d*e^3 + 5*b^2*e^4)*x^2 + 12*(16*c^2*d^3*e - 20*b*c*d^2*e^2 + 5*b^2*d*e^3)*x)*sqrt(e*x + d)
/(e^7*x^2 + 2*d*e^6*x + d^2*e^5)

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Sympy [A]  time = 41.188, size = 139, normalized size = 0.97 \begin{align*} \frac{2 c^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{5}} - \frac{2 d^{2} \left (b e - c d\right )^{2}}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} + \frac{4 d \left (b e - 2 c d\right ) \left (b e - c d\right )}{e^{5} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (4 b c e - 8 c^{2} d\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (2 b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c**2*(d + e*x)**(5/2)/(5*e**5) - 2*d**2*(b*e - c*d)**2/(3*e**5*(d + e*x)**(3/2)) + 4*d*(b*e - 2*c*d)*(b*e -
c*d)/(e**5*sqrt(d + e*x)) + (d + e*x)**(3/2)*(4*b*c*e - 8*c**2*d)/(3*e**5) + sqrt(d + e*x)*(2*b**2*e**2 - 12*b
*c*d*e + 12*c**2*d**2)/e**5

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Giac [A]  time = 1.3858, size = 246, normalized size = 1.72 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} e^{20} - 20 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d e^{20} + 90 \, \sqrt{x e + d} c^{2} d^{2} e^{20} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} b c e^{21} - 90 \, \sqrt{x e + d} b c d e^{21} + 15 \, \sqrt{x e + d} b^{2} e^{22}\right )} e^{\left (-25\right )} + \frac{2 \,{\left (12 \,{\left (x e + d\right )} c^{2} d^{3} - c^{2} d^{4} - 18 \,{\left (x e + d\right )} b c d^{2} e + 2 \, b c d^{3} e + 6 \,{\left (x e + d\right )} b^{2} d e^{2} - b^{2} d^{2} e^{2}\right )} e^{\left (-5\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)^2/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*c^2*e^20 - 20*(x*e + d)^(3/2)*c^2*d*e^20 + 90*sqrt(x*e + d)*c^2*d^2*e^20 + 10*(x*e + d
)^(3/2)*b*c*e^21 - 90*sqrt(x*e + d)*b*c*d*e^21 + 15*sqrt(x*e + d)*b^2*e^22)*e^(-25) + 2/3*(12*(x*e + d)*c^2*d^
3 - c^2*d^4 - 18*(x*e + d)*b*c*d^2*e + 2*b*c*d^3*e + 6*(x*e + d)*b^2*d*e^2 - b^2*d^2*e^2)*e^(-5)/(x*e + d)^(3/
2)