\(\int \frac {(a+c x^2)^3}{d+e x} \, dx\) [478]

Optimal result

Integrand size = 17, antiderivative size = 173 \[ \int \frac {\left (a+c x^2\right )^3}{d+e x} \, dx=-\frac {c d \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right ) x}{e^6}+\frac {c \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right ) x^2}{2 e^5}-\frac {c^2 d \left (c d^2+3 a e^2\right ) x^3}{3 e^4}+\frac {c^2 \left (c d^2+3 a e^2\right ) x^4}{4 e^3}-\frac {c^3 d x^5}{5 e^2}+\frac {c^3 x^6}{6 e}+\frac {\left (c d^2+a e^2\right )^3 \log (d+e x)}{e^7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 173, 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} \[ \int \frac {\left (a+c x^2\right )^3}{d+e x} \, dx=-\frac {c d x \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{e^6}+\frac {c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{2 e^5}-\frac {c^2 d x^3 \left (3 a e^2+c d^2\right )}{3 e^4}+\frac {c^2 x^4 \left (3 a e^2+c d^2\right )}{4 e^3}+\frac {\left (a e^2+c d^2\right )^3 \log (d+e x)}{e^7}-\frac {c^3 d x^5}{5 e^2}+\frac {c^3 x^6}{6 e} \]

[In]

Int[(a + c*x^2)^3/(d + e*x),x]

[Out]

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

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 d \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right )}{e^6}+\frac {c \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right ) x}{e^5}-\frac {c^2 d \left (c d^2+3 a e^2\right ) x^2}{e^4}+\frac {c^2 \left (c d^2+3 a e^2\right ) x^3}{e^3}-\frac {c^3 d x^4}{e^2}+\frac {c^3 x^5}{e}+\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)}\right ) \, dx \\ & = -\frac {c d \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right ) x}{e^6}+\frac {c \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right ) x^2}{2 e^5}-\frac {c^2 d \left (c d^2+3 a e^2\right ) x^3}{3 e^4}+\frac {c^2 \left (c d^2+3 a e^2\right ) x^4}{4 e^3}-\frac {c^3 d x^5}{5 e^2}+\frac {c^3 x^6}{6 e}+\frac {\left (c d^2+a e^2\right )^3 \log (d+e x)}{e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+c x^2\right )^3}{d+e x} \, dx=\frac {c e x \left (90 a^2 e^4 (-2 d+e x)+15 a c e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+c^2 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 \left (c d^2+a e^2\right )^3 \log (d+e x)}{60 e^7} \]

[In]

Integrate[(a + c*x^2)^3/(d + e*x),x]

[Out]

(c*e*x*(90*a^2*e^4*(-2*d + e*x) + 15*a*c*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + c^2*(-60*d^5 +
30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + 60*(c*d^2 + a*e^2)^3*Log[d + e*x]
)/(60*e^7)

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.10

method	result	size
default	\(-\frac {c \left (-\frac {c^{2} x^{6} e^{5}}{6}+\frac {c^{2} d \,x^{5} e^{4}}{5}-\frac {e \left (3 a c \,e^{4}+d^{2} e^{2} c^{2}\right ) x^{4}}{4}+\frac {d \left (3 a c \,e^{4}+d^{2} e^{2} c^{2}\right ) x^{3}}{3}-\frac {\left (3 a^{2} e^{4}+3 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x^{2} e}{2}+d \left (3 a^{2} e^{4}+3 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x \right )}{e^{6}}+\frac {\left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(190\)
norman	\(\frac {c^{3} x^{6}}{6 e}+\frac {c \left (3 a^{2} e^{4}+3 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x^{2}}{2 e^{5}}+\frac {c^{2} \left (3 e^{2} a +c \,d^{2}\right ) x^{4}}{4 e^{3}}-\frac {c^{3} d \,x^{5}}{5 e^{2}}-\frac {c d \left (3 a^{2} e^{4}+3 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{e^{6}}-\frac {c^{2} d \left (3 e^{2} a +c \,d^{2}\right ) x^{3}}{3 e^{4}}+\frac {\left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(190\)
risch	\(\frac {c^{3} x^{6}}{6 e}-\frac {c^{3} d \,x^{5}}{5 e^{2}}+\frac {3 c^{2} a \,x^{4}}{4 e}+\frac {c^{3} d^{2} x^{4}}{4 e^{3}}-\frac {c^{2} a d \,x^{3}}{e^{2}}-\frac {c^{3} d^{3} x^{3}}{3 e^{4}}+\frac {3 c \,a^{2} x^{2}}{2 e}+\frac {3 c^{2} a \,d^{2} x^{2}}{2 e^{3}}+\frac {c^{3} d^{4} x^{2}}{2 e^{5}}-\frac {3 c \,a^{2} d x}{e^{2}}-\frac {3 c^{2} a \,d^{3} x}{e^{4}}-\frac {c^{3} d^{5} x}{e^{6}}+\frac {\ln \left (e x +d \right ) a^{3}}{e}+\frac {3 \ln \left (e x +d \right ) d^{2} a^{2} c}{e^{3}}+\frac {3 \ln \left (e x +d \right ) d^{4} c^{2} a}{e^{5}}+\frac {\ln \left (e x +d \right ) c^{3} d^{6}}{e^{7}}\)	\(220\)
parallelrisch	\(\frac {10 x^{6} c^{3} e^{6}-12 x^{5} c^{3} d \,e^{5}+45 x^{4} a \,c^{2} e^{6}+15 x^{4} c^{3} d^{2} e^{4}-60 x^{3} a \,c^{2} d \,e^{5}-20 x^{3} c^{3} d^{3} e^{3}+90 x^{2} a^{2} c \,e^{6}+90 x^{2} a \,c^{2} d^{2} e^{4}+30 x^{2} c^{3} d^{4} e^{2}+60 \ln \left (e x +d \right ) a^{3} e^{6}+180 \ln \left (e x +d \right ) a^{2} c \,d^{2} e^{4}+180 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}+60 \ln \left (e x +d \right ) c^{3} d^{6}-180 x \,a^{2} c d \,e^{5}-180 x a \,c^{2} d^{3} e^{3}-60 x \,c^{3} d^{5} e}{60 e^{7}}\)	\(222\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.72 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+c x^2\right )^3}{d+e x} \, dx=\frac {10 \, c^{3} e^{6} x^{6} - 12 \, c^{3} d e^{5} x^{5} + 15 \, {\left (c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} - 20 \, {\left (c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 30 \, {\left (c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4} + 3 \, a^{2} c e^{6}\right )} x^{2} - 60 \, {\left (c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x + 60 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \]

[In]

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

[Out]

1/60*(10*c^3*e^6*x^6 - 12*c^3*d*e^5*x^5 + 15*(c^3*d^2*e^4 + 3*a*c^2*e^6)*x^4 - 20*(c^3*d^3*e^3 + 3*a*c^2*d*e^5
)*x^3 + 30*(c^3*d^4*e^2 + 3*a*c^2*d^2*e^4 + 3*a^2*c*e^6)*x^2 - 60*(c^3*d^5*e + 3*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5
)*x + 60*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*log(e*x + d))/e^7

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+c x^2\right )^3}{d+e x} \, dx=- \frac {c^{3} d x^{5}}{5 e^{2}} + \frac {c^{3} x^{6}}{6 e} + x^{4} \cdot \left (\frac {3 a c^{2}}{4 e} + \frac {c^{3} d^{2}}{4 e^{3}}\right ) + x^{3} \left (- \frac {a c^{2} d}{e^{2}} - \frac {c^{3} d^{3}}{3 e^{4}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c}{2 e} + \frac {3 a c^{2} d^{2}}{2 e^{3}} + \frac {c^{3} d^{4}}{2 e^{5}}\right ) + x \left (- \frac {3 a^{2} c d}{e^{2}} - \frac {3 a c^{2} d^{3}}{e^{4}} - \frac {c^{3} d^{5}}{e^{6}}\right ) + \frac {\left (a e^{2} + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{7}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+c x^2\right )^3}{d+e x} \, dx=\frac {10 \, c^{3} e^{5} x^{6} - 12 \, c^{3} d e^{4} x^{5} + 15 \, {\left (c^{3} d^{2} e^{3} + 3 \, a c^{2} e^{5}\right )} x^{4} - 20 \, {\left (c^{3} d^{3} e^{2} + 3 \, a c^{2} d e^{4}\right )} x^{3} + 30 \, {\left (c^{3} d^{4} e + 3 \, a c^{2} d^{2} e^{3} + 3 \, a^{2} c e^{5}\right )} x^{2} - 60 \, {\left (c^{3} d^{5} + 3 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x}{60 \, e^{6}} + \frac {{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \log \left (e x + d\right )}{e^{7}} \]

[In]

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

[Out]

1/60*(10*c^3*e^5*x^6 - 12*c^3*d*e^4*x^5 + 15*(c^3*d^2*e^3 + 3*a*c^2*e^5)*x^4 - 20*(c^3*d^3*e^2 + 3*a*c^2*d*e^4
)*x^3 + 30*(c^3*d^4*e + 3*a*c^2*d^2*e^3 + 3*a^2*c*e^5)*x^2 - 60*(c^3*d^5 + 3*a*c^2*d^3*e^2 + 3*a^2*c*d*e^4)*x)
/e^6 + (c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*log(e*x + d)/e^7

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+c x^2\right )^3}{d+e x} \, dx=\frac {10 \, c^{3} e^{5} x^{6} - 12 \, c^{3} d e^{4} x^{5} + 15 \, c^{3} d^{2} e^{3} x^{4} + 45 \, a c^{2} e^{5} x^{4} - 20 \, c^{3} d^{3} e^{2} x^{3} - 60 \, a c^{2} d e^{4} x^{3} + 30 \, c^{3} d^{4} e x^{2} + 90 \, a c^{2} d^{2} e^{3} x^{2} + 90 \, a^{2} c e^{5} x^{2} - 60 \, c^{3} d^{5} x - 180 \, a c^{2} d^{3} e^{2} x - 180 \, a^{2} c d e^{4} x}{60 \, e^{6}} + \frac {{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} \]

[In]

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

[Out]

1/60*(10*c^3*e^5*x^6 - 12*c^3*d*e^4*x^5 + 15*c^3*d^2*e^3*x^4 + 45*a*c^2*e^5*x^4 - 20*c^3*d^3*e^2*x^3 - 60*a*c^
2*d*e^4*x^3 + 30*c^3*d^4*e*x^2 + 90*a*c^2*d^2*e^3*x^2 + 90*a^2*c*e^5*x^2 - 60*c^3*d^5*x - 180*a*c^2*d^3*e^2*x
- 180*a^2*c*d*e^4*x)/e^6 + (c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*log(abs(e*x + d))/e^7

Mupad [B] (verification not implemented)

Time = 9.43 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+c x^2\right )^3}{d+e x} \, dx=x^2\,\left (\frac {d^2\,\left (\frac {3\,a\,c^2}{e}+\frac {c^3\,d^2}{e^3}\right )}{2\,e^2}+\frac {3\,a^2\,c}{2\,e}\right )+x^4\,\left (\frac {3\,a\,c^2}{4\,e}+\frac {c^3\,d^2}{4\,e^3}\right )+\frac {c^3\,x^6}{6\,e}+\frac {\ln \left (d+e\,x\right )\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}{e^7}-\frac {c^3\,d\,x^5}{5\,e^2}-\frac {d\,x^3\,\left (\frac {3\,a\,c^2}{e}+\frac {c^3\,d^2}{e^3}\right )}{3\,e}-\frac {d\,x\,\left (\frac {d^2\,\left (\frac {3\,a\,c^2}{e}+\frac {c^3\,d^2}{e^3}\right )}{e^2}+\frac {3\,a^2\,c}{e}\right )}{e} \]

[In]

int((a + c*x^2)^3/(d + e*x),x)

[Out]

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