\(\int \frac {(a+c x^2)^2}{(d+e x)^3} \, dx\) [466]

Optimal result

Integrand size = 17, antiderivative size = 100 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=-\frac {3 c^2 d x}{e^4}+\frac {c^2 x^2}{2 e^3}-\frac {\left (c d^2+a e^2\right )^2}{2 e^5 (d+e x)^2}+\frac {4 c d \left (c d^2+a e^2\right )}{e^5 (d+e x)}+\frac {2 c \left (3 c d^2+a e^2\right ) \log (d+e x)}{e^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {4 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)}-\frac {\left (a e^2+c d^2\right )^2}{2 e^5 (d+e x)^2}+\frac {2 c \left (a e^2+3 c d^2\right ) \log (d+e x)}{e^5}-\frac {3 c^2 d x}{e^4}+\frac {c^2 x^2}{2 e^3} \]

[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 711

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*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {-a^2 e^4+2 a c d e^2 (3 d+4 e x)+c^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+4 c \left (3 c d^2+a e^2\right ) (d+e x)^2 \log (d+e x)}{2 e^5 (d+e x)^2} \]

[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)

Maple [A] (verified)

Time = 2.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07

method	result	size
default	\(-\frac {c^{2} \left (-\frac {1}{2} e \,x^{2}+3 d x \right )}{e^{4}}+\frac {4 c d \left (e^{2} a +c \,d^{2}\right )}{e^{5} \left (e x +d \right )}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{2 e^{5} \left (e x +d \right )^{2}}+\frac {2 c \left (e^{2} a +3 c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(107\)
norman	\(\frac {-\frac {a^{2} e^{4}-6 a c \,d^{2} e^{2}-18 c^{2} d^{4}}{2 e^{5}}+\frac {c^{2} x^{4}}{2 e}-\frac {2 c^{2} d \,x^{3}}{e^{2}}+\frac {2 d \left (2 a c \,e^{2}+6 c^{2} d^{2}\right ) x}{e^{4}}}{\left (e x +d \right )^{2}}+\frac {2 c \left (e^{2} a +3 c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(112\)
risch	\(\frac {c^{2} x^{2}}{2 e^{3}}-\frac {3 c^{2} d x}{e^{4}}+\frac {\left (4 d \,e^{2} a c +4 c^{2} d^{3}\right ) x -\frac {a^{2} e^{4}-6 a c \,d^{2} e^{2}-7 c^{2} d^{4}}{2 e}}{e^{4} \left (e x +d \right )^{2}}+\frac {2 c \ln \left (e x +d \right ) a}{e^{3}}+\frac {6 c^{2} \ln \left (e x +d \right ) d^{2}}{e^{5}}\)	\(115\)
parallelrisch	\(\frac {c^{2} x^{4} e^{4}+4 \ln \left (e x +d \right ) x^{2} a c \,e^{4}+12 \ln \left (e x +d \right ) x^{2} c^{2} d^{2} e^{2}-4 x^{3} c^{2} d \,e^{3}+8 \ln \left (e x +d \right ) x a c d \,e^{3}+24 \ln \left (e x +d \right ) x \,c^{2} d^{3} e +4 \ln \left (e x +d \right ) a c \,d^{2} e^{2}+12 \ln \left (e x +d \right ) c^{2} d^{4}+8 x a c d \,e^{3}+24 x \,c^{2} d^{3} e -a^{2} e^{4}+6 a c \,d^{2} e^{2}+18 c^{2} d^{4}}{2 e^{5} \left (e x +d \right )^{2}}\)	\(178\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {c^{2} e^{4} x^{4} - 4 \, c^{2} d e^{3} x^{3} - 11 \, c^{2} d^{2} e^{2} x^{2} + 7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 2 \, {\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x + 4 \, {\left (3 \, c^{2} d^{4} + a c d^{2} e^{2} + {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 2 \, {\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=- \frac {3 c^{2} d x}{e^{4}} + \frac {c^{2} x^{2}}{2 e^{3}} + \frac {2 c \left (a e^{2} + 3 c d^{2}\right ) \log {\left (d + e x \right )}}{e^{5}} + \frac {- a^{2} e^{4} + 6 a c d^{2} e^{2} + 7 c^{2} d^{4} + x \left (8 a c d e^{3} + 8 c^{2} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac {c^{2} e x^{2} - 6 \, c^{2} d x}{2 \, e^{4}} + \frac {2 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {2 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{5}} + \frac {c^{2} e^{3} x^{2} - 6 \, c^{2} d e^{2} x}{2 \, e^{6}} + \frac {7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{5}} \]

[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)*log(abs(e*x + d))/e^5 + 1/2*(c^2*e^3*x^2 - 6*c^2*d*e^2*x)/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*x + d)^2*e^5)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {x\,\left (4\,c^2\,d^3+4\,a\,c\,d\,e^2\right )+\frac {-a^2\,e^4+6\,a\,c\,d^2\,e^2+7\,c^2\,d^4}{2\,e}}{d^2\,e^4+2\,d\,e^5\,x+e^6\,x^2}+\frac {\ln \left (d+e\,x\right )\,\left (6\,c^2\,d^2+2\,a\,c\,e^2\right )}{e^5}+\frac {c^2\,x^2}{2\,e^3}-\frac {3\,c^2\,d\,x}{e^4} \]

[In]

int((a + c*x^2)^2/(d + e*x)^3,x)

[Out]

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