### 3.481 $$\int \frac{(a+c x^2)^3}{(d+e x)^4} \, dx$$

Optimal. Leaf size=165 $\frac{c^2 x \left (3 a e^2+10 c d^2\right )}{e^6}-\frac{4 c^2 d \left (3 a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{3 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^3}-\frac{2 c^3 d x^2}{e^5}+\frac{c^3 x^3}{3 e^4}$

[Out]

(c^2*(10*c*d^2 + 3*a*e^2)*x)/e^6 - (2*c^3*d*x^2)/e^5 + (c^3*x^3)/(3*e^4) - (c*d^2 + a*e^2)^3/(3*e^7*(d + e*x)^
3) + (3*c*d*(c*d^2 + a*e^2)^2)/(e^7*(d + e*x)^2) - (3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2))/(e^7*(d + e*x)) - (
4*c^2*d*(5*c*d^2 + 3*a*e^2)*Log[d + e*x])/e^7

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Rubi [A]  time = 0.164905, antiderivative size = 165, 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} $\frac{c^2 x \left (3 a e^2+10 c d^2\right )}{e^6}-\frac{4 c^2 d \left (3 a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{3 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^3}-\frac{2 c^3 d x^2}{e^5}+\frac{c^3 x^3}{3 e^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^3/(d + e*x)^4,x]

[Out]

(c^2*(10*c*d^2 + 3*a*e^2)*x)/e^6 - (2*c^3*d*x^2)/e^5 + (c^3*x^3)/(3*e^4) - (c*d^2 + a*e^2)^3/(3*e^7*(d + e*x)^
3) + (3*c*d*(c*d^2 + a*e^2)^2)/(e^7*(d + e*x)^2) - (3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2))/(e^7*(d + e*x)) - (
4*c^2*d*(5*c*d^2 + 3*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)^4} \, dx &=\int \left (\frac{c^2 \left (10 c d^2+3 a e^2\right )}{e^6}-\frac{4 c^3 d x}{e^5}+\frac{c^3 x^2}{e^4}+\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^4}-\frac{6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^3}+\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)^2}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{c^2 \left (10 c d^2+3 a e^2\right ) x}{e^6}-\frac{2 c^3 d x^2}{e^5}+\frac{c^3 x^3}{3 e^4}-\frac{\left (c d^2+a e^2\right )^3}{3 e^7 (d+e x)^3}+\frac{3 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)^2}-\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^7 (d+e x)}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.0657386, size = 197, normalized size = 1.19 $\frac{-3 a^2 c e^4 \left (d^2+3 d e x+3 e^2 x^2\right )-a^3 e^6+3 a c^2 e^2 \left (-9 d^2 e^2 x^2-27 d^3 e x-13 d^4+9 d e^3 x^3+3 e^4 x^4\right )-12 c^2 d (d+e x)^3 \left (3 a e^2+5 c d^2\right ) \log (d+e x)+c^3 \left (39 d^4 e^2 x^2+73 d^3 e^3 x^3+15 d^2 e^4 x^4-51 d^5 e x-37 d^6-3 d e^5 x^5+e^6 x^6\right )}{3 e^7 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^3/(d + e*x)^4,x]

[Out]

(-(a^3*e^6) - 3*a^2*c*e^4*(d^2 + 3*d*e*x + 3*e^2*x^2) + 3*a*c^2*e^2*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*
d*e^3*x^3 + 3*e^4*x^4) + c^3*(-37*d^6 - 51*d^5*e*x + 39*d^4*e^2*x^2 + 73*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 3*d*e^
5*x^5 + e^6*x^6) - 12*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^3*Log[d + e*x])/(3*e^7*(d + e*x)^3)

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Maple [A]  time = 0.056, size = 258, normalized size = 1.6 \begin{align*}{\frac{{x}^{3}{c}^{3}}{3\,{e}^{4}}}-2\,{\frac{{c}^{3}d{x}^{2}}{{e}^{5}}}+3\,{\frac{a{c}^{2}x}{{e}^{4}}}+10\,{\frac{x{c}^{3}{d}^{2}}{{e}^{6}}}-{\frac{{a}^{3}}{3\,e \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{2}c{d}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{4}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{6}{c}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}-12\,{\frac{{c}^{2}d\ln \left ( ex+d \right ) a}{{e}^{5}}}-20\,{\frac{{c}^{3}{d}^{3}\ln \left ( ex+d \right ) }{{e}^{7}}}-3\,{\frac{{a}^{2}c}{{e}^{3} \left ( ex+d \right ) }}-18\,{\frac{a{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }}-15\,{\frac{{c}^{3}{d}^{4}}{{e}^{7} \left ( ex+d \right ) }}+3\,{\frac{cd{a}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}+6\,{\frac{a{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{c}^{3}{d}^{5}}{{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)^4,x)

[Out]

1/3*c^3*x^3/e^4-2*c^3*d*x^2/e^5+3*c^2/e^4*a*x+10*c^3/e^6*d^2*x-1/3/e/(e*x+d)^3*a^3-1/e^3/(e*x+d)^3*d^2*a^2*c-1
/e^5/(e*x+d)^3*d^4*a*c^2-1/3/e^7/(e*x+d)^3*d^6*c^3-12*c^2*d/e^5*ln(e*x+d)*a-20*c^3*d^3/e^7*ln(e*x+d)-3*c/e^3/(
e*x+d)*a^2-18*c^2/e^5/(e*x+d)*a*d^2-15*c^3/e^7/(e*x+d)*d^4+3*c*d/e^3/(e*x+d)^2*a^2+6*c^2*d^3/e^5/(e*x+d)^2*a+3
*c^3*d^5/e^7/(e*x+d)^2

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Maxima [A]  time = 1.102, size = 305, normalized size = 1.85 \begin{align*} -\frac{37 \, c^{3} d^{6} + 39 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 9 \,{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (9 \, c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac{c^{3} e^{2} x^{3} - 6 \, c^{3} d e x^{2} + 3 \,{\left (10 \, c^{3} d^{2} + 3 \, a c^{2} e^{2}\right )} x}{3 \, e^{6}} - \frac{4 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d 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)^4,x, algorithm="maxima")

[Out]

-1/3*(37*c^3*d^6 + 39*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6 + 9*(5*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e
^6)*x^2 + 9*(9*c^3*d^5*e + 10*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)
+ 1/3*(c^3*e^2*x^3 - 6*c^3*d*e*x^2 + 3*(10*c^3*d^2 + 3*a*c^2*e^2)*x)/e^6 - 4*(5*c^3*d^3 + 3*a*c^2*d*e^2)*log(
e*x + d)/e^7

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Fricas [B]  time = 2.10702, size = 679, normalized size = 4.12 \begin{align*} \frac{c^{3} e^{6} x^{6} - 3 \, c^{3} d e^{5} x^{5} - 37 \, c^{3} d^{6} - 39 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 3 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} +{\left (73 \, c^{3} d^{3} e^{3} + 27 \, a c^{2} d e^{5}\right )} x^{3} + 3 \,{\left (13 \, c^{3} d^{4} e^{2} - 9 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} - 3 \,{\left (17 \, c^{3} d^{5} e + 27 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 12 \,{\left (5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} +{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 3 \,{\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \,{\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} 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)^4,x, algorithm="fricas")

[Out]

1/3*(c^3*e^6*x^6 - 3*c^3*d*e^5*x^5 - 37*c^3*d^6 - 39*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - a^3*e^6 + 3*(5*c^3*d^2*
e^4 + 3*a*c^2*e^6)*x^4 + (73*c^3*d^3*e^3 + 27*a*c^2*d*e^5)*x^3 + 3*(13*c^3*d^4*e^2 - 9*a*c^2*d^2*e^4 - 3*a^2*c
*e^6)*x^2 - 3*(17*c^3*d^5*e + 27*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x - 12*(5*c^3*d^6 + 3*a*c^2*d^4*e^2 + (5*c^3*d
^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(5*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(5*c^3*d^5*e + 3*a*c^2*d^3*e^3)*x)*l
og(e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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Sympy [A]  time = 2.88631, size = 235, normalized size = 1.42 \begin{align*} - \frac{2 c^{3} d x^{2}}{e^{5}} + \frac{c^{3} x^{3}}{3 e^{4}} - \frac{4 c^{2} d \left (3 a e^{2} + 5 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 39 a c^{2} d^{4} e^{2} + 37 c^{3} d^{6} + x^{2} \left (9 a^{2} c e^{6} + 54 a c^{2} d^{2} e^{4} + 45 c^{3} d^{4} e^{2}\right ) + x \left (9 a^{2} c d e^{5} + 90 a c^{2} d^{3} e^{3} + 81 c^{3} d^{5} e\right )}{3 d^{3} e^{7} + 9 d^{2} e^{8} x + 9 d e^{9} x^{2} + 3 e^{10} x^{3}} + \frac{x \left (3 a c^{2} e^{2} + 10 c^{3} d^{2}\right )}{e^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**4,x)

[Out]

-2*c**3*d*x**2/e**5 + c**3*x**3/(3*e**4) - 4*c**2*d*(3*a*e**2 + 5*c*d**2)*log(d + e*x)/e**7 - (a**3*e**6 + 3*a
**2*c*d**2*e**4 + 39*a*c**2*d**4*e**2 + 37*c**3*d**6 + x**2*(9*a**2*c*e**6 + 54*a*c**2*d**2*e**4 + 45*c**3*d**
4*e**2) + x*(9*a**2*c*d*e**5 + 90*a*c**2*d**3*e**3 + 81*c**3*d**5*e))/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*
x**2 + 3*e**10*x**3) + x*(3*a*c**2*e**2 + 10*c**3*d**2)/e**6

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Giac [A]  time = 1.30372, size = 259, normalized size = 1.57 \begin{align*} -4 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{3} \,{\left (c^{3} x^{3} e^{8} - 6 \, c^{3} d x^{2} e^{7} + 30 \, c^{3} d^{2} x e^{6} + 9 \, a c^{2} x e^{8}\right )} e^{\left (-12\right )} - \frac{{\left (37 \, c^{3} d^{6} + 39 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 9 \,{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (9 \, c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^4,x, algorithm="giac")

[Out]

-4*(5*c^3*d^3 + 3*a*c^2*d*e^2)*e^(-7)*log(abs(x*e + d)) + 1/3*(c^3*x^3*e^8 - 6*c^3*d*x^2*e^7 + 30*c^3*d^2*x*e^
6 + 9*a*c^2*x*e^8)*e^(-12) - 1/3*(37*c^3*d^6 + 39*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6 + 9*(5*c^3*d^4*e^2
+ 6*a*c^2*d^2*e^4 + a^2*c*e^6)*x^2 + 9*(9*c^3*d^5*e + 10*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)*e^(-7)/(x*e + d)^3