\(\int \frac {(a+c x^2)^2}{(d+e x)^2} \, dx\) [465]

Optimal result

Integrand size = 17, antiderivative size = 94 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {c \left (3 c d^2+2 a e^2\right ) x}{e^4}-\frac {c^2 d x^2}{e^3}+\frac {c^2 x^3}{3 e^2}-\frac {\left (c d^2+a e^2\right )^2}{e^5 (d+e x)}-\frac {4 c d \left (c d^2+a e^2\right ) \log (d+e x)}{e^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 94, 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} \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=-\frac {\left (a e^2+c d^2\right )^2}{e^5 (d+e x)}-\frac {4 c d \left (a e^2+c d^2\right ) \log (d+e x)}{e^5}+\frac {c x \left (2 a e^2+3 c d^2\right )}{e^4}-\frac {c^2 d x^2}{e^3}+\frac {c^2 x^3}{3 e^2} \]

[In]

Int[(a + c*x^2)^2/(d + e*x)^2,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {3 c e \left (3 c d^2+2 a e^2\right ) x-3 c^2 d e^2 x^2+c^2 e^3 x^3-\frac {3 \left (c d^2+a e^2\right )^2}{d+e x}-12 c d \left (c d^2+a e^2\right ) \log (d+e x)}{3 e^5} \]

[In]

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

[Out]

(3*c*e*(3*c*d^2 + 2*a*e^2)*x - 3*c^2*d*e^2*x^2 + c^2*e^3*x^3 - (3*(c*d^2 + a*e^2)^2)/(d + e*x) - 12*c*d*(c*d^2
+ a*e^2)*Log[d + e*x])/(3*e^5)

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {c \left (\frac {1}{3} c \,e^{2} x^{3}-c d e \,x^{2}+2 a \,e^{2} x +3 c \,d^{2} x \right )}{e^{4}}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{e^{5} \left (e x +d \right )}-\frac {4 c d \left (e^{2} a +c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(100\)
norman	\(\frac {-\frac {a^{2} e^{4}+4 a c \,d^{2} e^{2}+4 c^{2} d^{4}}{e^{5}}+\frac {c^{2} x^{4}}{3 e}+\frac {2 c \left (e^{2} a +c \,d^{2}\right ) x^{2}}{e^{3}}-\frac {2 c^{2} d \,x^{3}}{3 e^{2}}}{e x +d}-\frac {4 c d \left (e^{2} a +c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(109\)
risch	\(\frac {c^{2} x^{3}}{3 e^{2}}-\frac {c^{2} d \,x^{2}}{e^{3}}+\frac {2 c a x}{e^{2}}+\frac {3 c^{2} d^{2} x}{e^{4}}-\frac {a^{2}}{e \left (e x +d \right )}-\frac {2 a c \,d^{2}}{e^{3} \left (e x +d \right )}-\frac {c^{2} d^{4}}{e^{5} \left (e x +d \right )}-\frac {4 c d \ln \left (e x +d \right ) a}{e^{3}}-\frac {4 c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}\)	\(126\)
parallelrisch	\(-\frac {-c^{2} x^{4} e^{4}+2 x^{3} c^{2} d \,e^{3}+12 \ln \left (e x +d \right ) x a c d \,e^{3}+12 \ln \left (e x +d \right ) x \,c^{2} d^{3} e -6 x^{2} a c \,e^{4}-6 x^{2} c^{2} d^{2} e^{2}+12 \ln \left (e x +d \right ) a c \,d^{2} e^{2}+12 \ln \left (e x +d \right ) c^{2} d^{4}+3 a^{2} e^{4}+12 a c \,d^{2} e^{2}+12 c^{2} d^{4}}{3 e^{5} \left (e x +d \right )}\)	\(148\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {c^{2} e^{4} x^{4} - 2 \, c^{2} d e^{3} x^{3} - 3 \, c^{2} d^{4} - 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 6 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 3 \, {\left (3 \, c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x - 12 \, {\left (c^{2} d^{4} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{6} x + d e^{5}\right )}} \]

[In]

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

[Out]

1/3*(c^2*e^4*x^4 - 2*c^2*d*e^3*x^3 - 3*c^2*d^4 - 6*a*c*d^2*e^2 - 3*a^2*e^4 + 6*(c^2*d^2*e^2 + a*c*e^4)*x^2 + 3
*(3*c^2*d^3*e + 2*a*c*d*e^3)*x - 12*(c^2*d^4 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)*log(e*x + d))/(e^6*x +
d*e^5)

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=- \frac {c^{2} d x^{2}}{e^{3}} + \frac {c^{2} x^{3}}{3 e^{2}} - \frac {4 c d \left (a e^{2} + c d^{2}\right ) \log {\left (d + e x \right )}}{e^{5}} + x \left (\frac {2 a c}{e^{2}} + \frac {3 c^{2} d^{2}}{e^{4}}\right ) + \frac {- a^{2} e^{4} - 2 a c d^{2} e^{2} - c^{2} d^{4}}{d e^{5} + e^{6} x} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=-\frac {c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{e^{6} x + d e^{5}} + \frac {c^{2} e^{2} x^{3} - 3 \, c^{2} d e x^{2} + 3 \, {\left (3 \, c^{2} d^{2} + 2 \, a c e^{2}\right )} x}{3 \, e^{4}} - \frac {4 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {{\left (c^{2} - \frac {6 \, c^{2} d}{e x + d} + \frac {6 \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}}\right )} {\left (e x + d\right )}^{3}}{3 \, e^{5}} + \frac {4 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{5}} - \frac {\frac {c^{2} d^{4} e^{3}}{e x + d} + \frac {2 \, a c d^{2} e^{5}}{e x + d} + \frac {a^{2} e^{7}}{e x + d}}{e^{8}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=x\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,a\,c}{e^2}\right )-\frac {\ln \left (d+e\,x\right )\,\left (4\,c^2\,d^3+4\,a\,c\,d\,e^2\right )}{e^5}+\frac {c^2\,x^3}{3\,e^2}-\frac {a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4}{e\,\left (x\,e^5+d\,e^4\right )}-\frac {c^2\,d\,x^2}{e^3} \]

[In]

int((a + c*x^2)^2/(d + e*x)^2,x)

[Out]

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