### 3.482 $$\int \frac{(a+c x^2)^3}{(d+e x)^5} \, dx$$

Optimal. Leaf size=171 $\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{3 c^2 \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac{2 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac{5 c^3 d x}{e^6}+\frac{c^3 x^2}{2 e^5}$

[Out]

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

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Rubi [A]  time = 0.157142, antiderivative size = 171, 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} $\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{3 c^2 \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac{2 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac{5 c^3 d x}{e^6}+\frac{c^3 x^2}{2 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^3/(d + e*x)^5,x]

[Out]

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

Mathematica [A]  time = 0.0647442, size = 185, normalized size = 1.08 $\frac{-a^2 c e^4 \left (d^2+4 d e x+6 e^2 x^2\right )-a^3 e^6+a c^2 d e^2 \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+12 c^2 (d+e x)^4 \left (a e^2+5 c d^2\right ) \log (d+e x)+c^3 \left (132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4+168 d^5 e x+57 d^6-12 d e^5 x^5+2 e^6 x^6\right )}{4 e^7 (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^3/(d + e*x)^5,x]

[Out]

(-(a^3*e^6) - a^2*c*e^4*(d^2 + 4*d*e*x + 6*e^2*x^2) + a*c^2*d*e^2*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^
3*x^3) + c^3*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*
x^6) + 12*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^4*Log[d + e*x])/(4*e^7*(d + e*x)^4)

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Maple [A]  time = 0.053, size = 268, normalized size = 1.6 \begin{align*}{\frac{{c}^{3}{x}^{2}}{2\,{e}^{5}}}-5\,{\frac{{c}^{3}dx}{{e}^{6}}}-{\frac{{a}^{3}}{4\,e \left ( ex+d \right ) ^{4}}}-{\frac{3\,{a}^{2}c{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{3\,{d}^{4}a{c}^{2}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{{d}^{6}{c}^{3}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}+2\,{\frac{cd{a}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}+4\,{\frac{{c}^{2}{d}^{3}a}{{e}^{5} \left ( ex+d \right ) ^{3}}}+2\,{\frac{{c}^{3}{d}^{5}}{{e}^{7} \left ( ex+d \right ) ^{3}}}+3\,{\frac{{c}^{2}\ln \left ( ex+d \right ) a}{{e}^{5}}}+15\,{\frac{{c}^{3}\ln \left ( ex+d \right ){d}^{2}}{{e}^{7}}}+12\,{\frac{a{c}^{2}d}{{e}^{5} \left ( ex+d \right ) }}+20\,{\frac{{c}^{3}{d}^{3}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{3\,{a}^{2}c}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-9\,{\frac{a{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{c}^{3}{d}^{4}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3/(e*x+d)^5,x)

[Out]

1/2*c^3*x^2/e^5-5*c^3*d*x/e^6-1/4/e/(e*x+d)^4*a^3-3/4/e^3/(e*x+d)^4*d^2*a^2*c-3/4/e^5/(e*x+d)^4*d^4*a*c^2-1/4/
e^7/(e*x+d)^4*d^6*c^3+2*c*d/e^3/(e*x+d)^3*a^2+4*c^2*d^3/e^5/(e*x+d)^3*a+2*c^3*d^5/e^7/(e*x+d)^3+3*c^2/e^5*ln(e
*x+d)*a+15*c^3/e^7*ln(e*x+d)*d^2+12*c^2*d/e^5/(e*x+d)*a+20*c^3*d^3/e^7/(e*x+d)-3/2*c/e^3/(e*x+d)^2*a^2-9*c^2/e
^5/(e*x+d)^2*a*d^2-15/2*c^3/e^7/(e*x+d)^2*d^4

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Maxima [A]  time = 1.20839, size = 323, normalized size = 1.89 \begin{align*} \frac{57 \, c^{3} d^{6} + 25 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - a^{3} e^{6} + 16 \,{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 6 \,{\left (35 \, c^{3} d^{4} e^{2} + 18 \, a c^{2} d^{2} e^{4} - a^{2} c e^{6}\right )} x^{2} + 4 \,{\left (47 \, c^{3} d^{5} e + 22 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac{c^{3} e x^{2} - 10 \, c^{3} d x}{2 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\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)^5,x, algorithm="maxima")

[Out]

1/4*(57*c^3*d^6 + 25*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - a^3*e^6 + 16*(5*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 6*(35*
c^3*d^4*e^2 + 18*a*c^2*d^2*e^4 - a^2*c*e^6)*x^2 + 4*(47*c^3*d^5*e + 22*a*c^2*d^3*e^3 - a^2*c*d*e^5)*x)/(e^11*x
^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7) + 1/2*(c^3*e*x^2 - 10*c^3*d*x)/e^6 + 3*(5*c^3*d^2 +
a*c^2*e^2)*log(e*x + d)/e^7

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Fricas [B]  time = 2.08485, size = 716, normalized size = 4.19 \begin{align*} \frac{2 \, c^{3} e^{6} x^{6} - 12 \, c^{3} d e^{5} x^{5} - 68 \, c^{3} d^{2} e^{4} x^{4} + 57 \, c^{3} d^{6} + 25 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - a^{3} e^{6} - 16 \,{\left (2 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 6 \,{\left (22 \, c^{3} d^{4} e^{2} + 18 \, a c^{2} d^{2} e^{4} - a^{2} c e^{6}\right )} x^{2} + 4 \,{\left (42 \, c^{3} d^{5} e + 22 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} +{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 4 \,{\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 6 \,{\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^5,x, algorithm="fricas")

[Out]

1/4*(2*c^3*e^6*x^6 - 12*c^3*d*e^5*x^5 - 68*c^3*d^2*e^4*x^4 + 57*c^3*d^6 + 25*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - a
^3*e^6 - 16*(2*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 6*(22*c^3*d^4*e^2 + 18*a*c^2*d^2*e^4 - a^2*c*e^6)*x^2 + 4*(4
2*c^3*d^5*e + 22*a*c^2*d^3*e^3 - a^2*c*d*e^5)*x + 12*(5*c^3*d^6 + a*c^2*d^4*e^2 + (5*c^3*d^2*e^4 + a*c^2*e^6)*
x^4 + 4*(5*c^3*d^3*e^3 + a*c^2*d*e^5)*x^3 + 6*(5*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + 4*(5*c^3*d^5*e + a*c^2*d^3
*e^3)*x)*log(e*x + d))/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)

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Sympy [A]  time = 4.59792, size = 243, normalized size = 1.42 \begin{align*} - \frac{5 c^{3} d x}{e^{6}} + \frac{c^{3} x^{2}}{2 e^{5}} + \frac{3 c^{2} \left (a e^{2} + 5 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} + \frac{- a^{3} e^{6} - a^{2} c d^{2} e^{4} + 25 a c^{2} d^{4} e^{2} + 57 c^{3} d^{6} + x^{3} \left (48 a c^{2} d e^{5} + 80 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 6 a^{2} c e^{6} + 108 a c^{2} d^{2} e^{4} + 210 c^{3} d^{4} e^{2}\right ) + x \left (- 4 a^{2} c d e^{5} + 88 a c^{2} d^{3} e^{3} + 188 c^{3} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**5,x)

[Out]

-5*c**3*d*x/e**6 + c**3*x**2/(2*e**5) + 3*c**2*(a*e**2 + 5*c*d**2)*log(d + e*x)/e**7 + (-a**3*e**6 - a**2*c*d*
*2*e**4 + 25*a*c**2*d**4*e**2 + 57*c**3*d**6 + x**3*(48*a*c**2*d*e**5 + 80*c**3*d**3*e**3) + x**2*(-6*a**2*c*e
**6 + 108*a*c**2*d**2*e**4 + 210*c**3*d**4*e**2) + x*(-4*a**2*c*d*e**5 + 88*a*c**2*d**3*e**3 + 188*c**3*d**5*e
))/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4)

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Giac [A]  time = 1.32245, size = 389, normalized size = 2.27 \begin{align*} \frac{1}{2} \,{\left (c^{3} - \frac{12 \, c^{3} d}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-7\right )} - 3 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} e^{\left (-7\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{4} \,{\left (\frac{80 \, c^{3} d^{3} e^{29}}{x e + d} - \frac{30 \, c^{3} d^{4} e^{29}}{{\left (x e + d\right )}^{2}} + \frac{8 \, c^{3} d^{5} e^{29}}{{\left (x e + d\right )}^{3}} - \frac{c^{3} d^{6} e^{29}}{{\left (x e + d\right )}^{4}} + \frac{48 \, a c^{2} d e^{31}}{x e + d} - \frac{36 \, a c^{2} d^{2} e^{31}}{{\left (x e + d\right )}^{2}} + \frac{16 \, a c^{2} d^{3} e^{31}}{{\left (x e + d\right )}^{3}} - \frac{3 \, a c^{2} d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac{6 \, a^{2} c e^{33}}{{\left (x e + d\right )}^{2}} + \frac{8 \, a^{2} c d e^{33}}{{\left (x e + d\right )}^{3}} - \frac{3 \, a^{2} c d^{2} e^{33}}{{\left (x e + d\right )}^{4}} - \frac{a^{3} e^{35}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-36\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^5,x, algorithm="giac")

[Out]

1/2*(c^3 - 12*c^3*d/(x*e + d))*(x*e + d)^2*e^(-7) - 3*(5*c^3*d^2 + a*c^2*e^2)*e^(-7)*log(abs(x*e + d)*e^(-1)/(
x*e + d)^2) + 1/4*(80*c^3*d^3*e^29/(x*e + d) - 30*c^3*d^4*e^29/(x*e + d)^2 + 8*c^3*d^5*e^29/(x*e + d)^3 - c^3*
d^6*e^29/(x*e + d)^4 + 48*a*c^2*d*e^31/(x*e + d) - 36*a*c^2*d^2*e^31/(x*e + d)^2 + 16*a*c^2*d^3*e^31/(x*e + d)
^3 - 3*a*c^2*d^4*e^31/(x*e + d)^4 - 6*a^2*c*e^33/(x*e + d)^2 + 8*a^2*c*d*e^33/(x*e + d)^3 - 3*a^2*c*d^2*e^33/(
x*e + d)^4 - a^3*e^35/(x*e + d)^4)*e^(-36)