### 3.478 $$\int \frac{(a+c x^2)^3}{d+e x} \, dx$$

Optimal. Leaf size=173 $\frac{c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{2 e^5}-\frac{c d x \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{e^6}+\frac{c^2 x^4 \left (3 a e^2+c d^2\right )}{4 e^3}-\frac{c^2 d x^3 \left (3 a e^2+c d^2\right )}{3 e^4}+\frac{\left (a e^2+c d^2\right )^3 \log (d+e x)}{e^7}-\frac{c^3 d x^5}{5 e^2}+\frac{c^3 x^6}{6 e}$

[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

________________________________________________________________________________________

Rubi [A]  time = 0.144567, antiderivative size = 173, 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} $\frac{c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{2 e^5}-\frac{c d x \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{e^6}+\frac{c^2 x^4 \left (3 a e^2+c d^2\right )}{4 e^3}-\frac{c^2 d x^3 \left (3 a e^2+c d^2\right )}{3 e^4}+\frac{\left (a e^2+c d^2\right )^3 \log (d+e x)}{e^7}-\frac{c^3 d x^5}{5 e^2}+\frac{c^3 x^6}{6 e}$

Antiderivative was successfully veriﬁed.

[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 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 )^3}{d+e x} \, dx &=\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]  time = 0.0539716, size = 142, normalized size = 0.82 $\frac{c e x \left (90 a^2 e^4 (e x-2 d)+15 a c e^2 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+c^2 \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 \left (a e^2+c d^2\right )^3 \log (d+e x)}{60 e^7}$

Antiderivative was successfully veriﬁed.

[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]  time = 0.046, size = 220, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}{x}^{6}}{6\,e}}-{\frac{{c}^{3}d{x}^{5}}{5\,{e}^{2}}}+{\frac{3\,{c}^{2}{x}^{4}a}{4\,e}}+{\frac{{c}^{3}{x}^{4}{d}^{2}}{4\,{e}^{3}}}-{\frac{{c}^{2}{x}^{3}ad}{{e}^{2}}}-{\frac{{x}^{3}{c}^{3}{d}^{3}}{3\,{e}^{4}}}+{\frac{3\,c{x}^{2}{a}^{2}}{2\,e}}+{\frac{3\,{c}^{2}{x}^{2}a{d}^{2}}{2\,{e}^{3}}}+{\frac{{c}^{3}{x}^{2}{d}^{4}}{2\,{e}^{5}}}-3\,{\frac{cd{a}^{2}x}{{e}^{2}}}-3\,{\frac{a{c}^{2}{d}^{3}x}{{e}^{4}}}-{\frac{{c}^{3}{d}^{5}x}{{e}^{6}}}+{\frac{\ln \left ( ex+d \right ){a}^{3}}{e}}+3\,{\frac{\ln \left ( ex+d \right ){a}^{2}c{d}^{2}}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ){d}^{4}a{c}^{2}}{{e}^{5}}}+{\frac{\ln \left ( ex+d \right ){d}^{6}{c}^{3}}{{e}^{7}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.14583, size = 267, normalized size = 1.54 \begin{align*} \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]  time = 2.06353, size = 410, normalized size = 2.37 \begin{align*} \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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]  time = 0.719874, size = 173, normalized size = 1. \begin{align*} - \frac{c^{3} d x^{5}}{5 e^{2}} + \frac{c^{3} x^{6}}{6 e} + \frac{x^{4} \left (3 a c^{2} e^{2} + c^{3} d^{2}\right )}{4 e^{3}} - \frac{x^{3} \left (3 a c^{2} d e^{2} + c^{3} d^{3}\right )}{3 e^{4}} + \frac{x^{2} \left (3 a^{2} c e^{4} + 3 a c^{2} d^{2} e^{2} + c^{3} d^{4}\right )}{2 e^{5}} - \frac{x \left (3 a^{2} c d e^{4} + 3 a c^{2} d^{3} e^{2} + c^{3} d^{5}\right )}{e^{6}} + \frac{\left (a e^{2} + c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Giac [A]  time = 1.32853, size = 259, normalized size = 1.5 \begin{align*}{\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 )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, c^{3} x^{6} e^{5} - 12 \, c^{3} d x^{5} e^{4} + 15 \, c^{3} d^{2} x^{4} e^{3} - 20 \, c^{3} d^{3} x^{3} e^{2} + 30 \, c^{3} d^{4} x^{2} e - 60 \, c^{3} d^{5} x + 45 \, a c^{2} x^{4} e^{5} - 60 \, a c^{2} d x^{3} e^{4} + 90 \, a c^{2} d^{2} x^{2} e^{3} - 180 \, a c^{2} d^{3} x e^{2} + 90 \, a^{2} c x^{2} e^{5} - 180 \, a^{2} c d x e^{4}\right )} e^{\left (-6\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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