3.466 \(\int \frac{(a+c x^2)^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=100 \[ \frac{4 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)}-\frac{\left (a e^2+c d^2\right )^2}{2 e^5 (d+e x)^2}+\frac{2 c \left (a e^2+3 c d^2\right ) \log (d+e x)}{e^5}-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^2}{2 e^3} \]

[Out]

(-3*c^2*d*x)/e^4 + (c^2*x^2)/(2*e^3) - (c*d^2 + a*e^2)^2/(2*e^5*(d + e*x)^2) + (4*c*d*(c*d^2 + a*e^2))/(e^5*(d
+ e*x)) + (2*c*(3*c*d^2 + a*e^2)*Log[d + e*x])/e^5

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Rubi [A]  time = 0.0810327, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ \frac{4 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)}-\frac{\left (a e^2+c d^2\right )^2}{2 e^5 (d+e x)^2}+\frac{2 c \left (a e^2+3 c d^2\right ) \log (d+e x)}{e^5}-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^2}{2 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2/(d + e*x)^3,x]

[Out]

(-3*c^2*d*x)/e^4 + (c^2*x^2)/(2*e^3) - (c*d^2 + a*e^2)^2/(2*e^5*(d + e*x)^2) + (4*c*d*(c*d^2 + a*e^2))/(e^5*(d
+ e*x)) + (2*c*(3*c*d^2 + a*e^2)*Log[d + e*x])/e^5

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+c x^2\right )^2}{(d+e x)^3} \, dx &=\int \left (-\frac{3 c^2 d}{e^4}+\frac{c^2 x}{e^3}+\frac{\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^3}-\frac{4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^2}+\frac{2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^2}{2 e^3}-\frac{\left (c d^2+a e^2\right )^2}{2 e^5 (d+e x)^2}+\frac{4 c d \left (c d^2+a e^2\right )}{e^5 (d+e x)}+\frac{2 c \left (3 c d^2+a e^2\right ) \log (d+e x)}{e^5}\\ \end{align*}

Mathematica [A]  time = 0.0374773, size = 111, normalized size = 1.11 \[ \frac{-a^2 e^4+4 c (d+e x)^2 \left (a e^2+3 c d^2\right ) \log (d+e x)+2 a c d e^2 (3 d+4 e x)+c^2 \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )}{2 e^5 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^2*e^4) + 2*a*c*d*e^2*(3*d + 4*e*x) + c^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + 4
*c*(3*c*d^2 + a*e^2)*(d + e*x)^2*Log[d + e*x])/(2*e^5*(d + e*x)^2)

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Maple [A]  time = 0.049, size = 136, normalized size = 1.4 \begin{align*}{\frac{{c}^{2}{x}^{2}}{2\,{e}^{3}}}-3\,{\frac{x{c}^{2}d}{{e}^{4}}}+2\,{\frac{c\ln \left ( ex+d \right ) a}{{e}^{3}}}+6\,{\frac{{c}^{2}\ln \left ( ex+d \right ){d}^{2}}{{e}^{5}}}+4\,{\frac{acd}{{e}^{3} \left ( ex+d \right ) }}+4\,{\frac{{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{a}^{2}}{2\,e \left ( ex+d \right ) ^{2}}}-{\frac{ac{d}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2}{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c^2*x^2/e^3-3*c^2*d*x/e^4+2*c/e^3*ln(e*x+d)*a+6*c^2/e^5*ln(e*x+d)*d^2+4*c*d/e^3/(e*x+d)*a+4*c^2*d^3/e^5/(e
*x+d)-1/2/e/(e*x+d)^2*a^2-1/e^3/(e*x+d)^2*a*c*d^2-1/2/e^5/(e*x+d)^2*c^2*d^4

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Maxima [A]  time = 1.32838, size = 162, normalized size = 1.62 \begin{align*} \frac{7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac{c^{2} e x^{2} - 6 \, c^{2} d x}{2 \, e^{4}} + \frac{2 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(7*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x)/(e^7*x^2 + 2*d*e^6*x + d^2*e^5) + 1/2*
(c^2*e*x^2 - 6*c^2*d*x)/e^4 + 2*(3*c^2*d^2 + a*c*e^2)*log(e*x + d)/e^5

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Fricas [A]  time = 1.93023, size = 360, normalized size = 3.6 \begin{align*} \frac{c^{2} e^{4} x^{4} - 4 \, c^{2} d e^{3} x^{3} - 11 \, c^{2} d^{2} e^{2} x^{2} + 7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 2 \,{\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x + 4 \,{\left (3 \, c^{2} d^{4} + a c d^{2} e^{2} +{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 2 \,{\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\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+a)^2/(e*x+d)^3,x, algorithm="fricas")

[Out]

1/2*(c^2*e^4*x^4 - 4*c^2*d*e^3*x^3 - 11*c^2*d^2*e^2*x^2 + 7*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4 + 2*(c^2*d^3*e +
4*a*c*d*e^3)*x + 4*(3*c^2*d^4 + a*c*d^2*e^2 + (3*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(3*c^2*d^3*e + a*c*d*e^3)*x)*
log(e*x + d))/(e^7*x^2 + 2*d*e^6*x + d^2*e^5)

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Sympy [A]  time = 1.48875, size = 122, normalized size = 1.22 \begin{align*} - \frac{3 c^{2} d x}{e^{4}} + \frac{c^{2} x^{2}}{2 e^{3}} + \frac{2 c \left (a e^{2} + 3 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{5}} + \frac{- a^{2} e^{4} + 6 a c d^{2} e^{2} + 7 c^{2} d^{4} + x \left (8 a c d e^{3} + 8 c^{2} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*c**2*d*x/e**4 + c**2*x**2/(2*e**3) + 2*c*(a*e**2 + 3*c*d**2)*log(d + e*x)/e**5 + (-a**2*e**4 + 6*a*c*d**2*e
**2 + 7*c**2*d**4 + x*(8*a*c*d*e**3 + 8*c**2*d**3*e))/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2)

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Giac [A]  time = 1.30426, size = 143, normalized size = 1.43 \begin{align*} 2 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (c^{2} x^{2} e^{3} - 6 \, c^{2} d x e^{2}\right )} e^{\left (-6\right )} + \frac{{\left (7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} e^{\left (-5\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(3*c^2*d^2 + a*c*e^2)*e^(-5)*log(abs(x*e + d)) + 1/2*(c^2*x^2*e^3 - 6*c^2*d*x*e^2)*e^(-6) + 1/2*(7*c^2*d^4 +
6*a*c*d^2*e^2 - a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x)*e^(-5)/(x*e + d)^2