### 3.483 $$\int \frac{(a+c x^2)^3}{(d+e x)^6} \, dx$$

Optimal. Leaf size=172 $-\frac{3 c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{2 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^2}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{3 c d \left (a e^2+c d^2\right )^2}{2 e^7 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{6 c^3 d \log (d+e x)}{e^7}+\frac{c^3 x}{e^6}$

[Out]

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

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Rubi [A]  time = 0.149449, antiderivative size = 172, 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} $-\frac{3 c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{2 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^2}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{3 c d \left (a e^2+c d^2\right )^2}{2 e^7 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{6 c^3 d \log (d+e x)}{e^7}+\frac{c^3 x}{e^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^3/(d + e*x)^6,x]

[Out]

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

Mathematica [A]  time = 0.0915828, size = 182, normalized size = 1.06 $-\frac{a^2 c e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+2 a^3 e^6+6 a c^2 e^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )+c^3 \left (600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4+375 d^5 e x+87 d^6-50 d e^5 x^5-10 e^6 x^6\right )+60 c^3 d (d+e x)^5 \log (d+e x)}{10 e^7 (d+e x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^3/(d + e*x)^6,x]

[Out]

-(2*a^3*e^6 + a^2*c*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 6*a*c^2*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^
3*x^3 + 5*e^4*x^4) + c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5
*x^5 - 10*e^6*x^6) + 60*c^3*d*(d + e*x)^5*Log[d + e*x])/(10*e^7*(d + e*x)^5)

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Maple [A]  time = 0.053, size = 272, normalized size = 1.6 \begin{align*}{\frac{{c}^{3}x}{{e}^{6}}}+{\frac{3\,cd{a}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}+3\,{\frac{{c}^{2}{d}^{3}a}{{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{3\,{c}^{3}{d}^{5}}{2\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{{a}^{2}c}{{e}^{3} \left ( ex+d \right ) ^{3}}}-6\,{\frac{a{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}-5\,{\frac{{c}^{3}{d}^{4}}{{e}^{7} \left ( ex+d \right ) ^{3}}}-6\,{\frac{{c}^{3}d\ln \left ( ex+d \right ) }{{e}^{7}}}-{\frac{{a}^{3}}{5\,e \left ( ex+d \right ) ^{5}}}-{\frac{3\,{a}^{2}c{d}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{3\,{d}^{4}a{c}^{2}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{d}^{6}{c}^{3}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-3\,{\frac{a{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-15\,{\frac{{c}^{3}{d}^{2}}{{e}^{7} \left ( ex+d \right ) }}+6\,{\frac{a{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{c}^{3}{d}^{3}}{{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)^6,x)

[Out]

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

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Maxima [A]  time = 1.17573, size = 332, normalized size = 1.93 \begin{align*} -\frac{87 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} + 30 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 20 \,{\left (25 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 10 \,{\left (65 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 5 \,{\left (77 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} + \frac{c^{3} x}{e^{6}} - \frac{6 \, c^{3} d \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)^6,x, algorithm="maxima")

[Out]

-1/10*(87*c^3*d^6 + 6*a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + 2*a^3*e^6 + 30*(5*c^3*d^2*e^4 + a*c^2*e^6)*x^4 + 20*(25*
c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 10*(65*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)*x^2 + 5*(77*c^3*d^5*e + 6
*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d
^5*e^7) + c^3*x/e^6 - 6*c^3*d*log(e*x + d)/e^7

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Fricas [A]  time = 2.14604, size = 672, normalized size = 3.91 \begin{align*} \frac{10 \, c^{3} e^{6} x^{6} + 50 \, c^{3} d e^{5} x^{5} - 87 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} - 10 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} - 20 \,{\left (20 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} - 10 \,{\left (60 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} - 5 \,{\left (75 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x - 60 \,{\left (c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{2} e^{4} x^{4} + 10 \, c^{3} d^{3} e^{3} x^{3} + 10 \, c^{3} d^{4} e^{2} x^{2} + 5 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} 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)^6,x, algorithm="fricas")

[Out]

1/10*(10*c^3*e^6*x^6 + 50*c^3*d*e^5*x^5 - 87*c^3*d^6 - 6*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 2*a^3*e^6 - 10*(5*c^3
*d^2*e^4 + 3*a*c^2*e^6)*x^4 - 20*(20*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 - 10*(60*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 +
a^2*c*e^6)*x^2 - 5*(75*c^3*d^5*e + 6*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x - 60*(c^3*d*e^5*x^5 + 5*c^3*d^2*e^4*x^4 +
10*c^3*d^3*e^3*x^3 + 10*c^3*d^4*e^2*x^2 + 5*c^3*d^5*e*x + c^3*d^6)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 1
0*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)

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Sympy [A]  time = 7.58859, size = 257, normalized size = 1.49 \begin{align*} - \frac{6 c^{3} d \log{\left (d + e x \right )}}{e^{7}} + \frac{c^{3} x}{e^{6}} - \frac{2 a^{3} e^{6} + a^{2} c d^{2} e^{4} + 6 a c^{2} d^{4} e^{2} + 87 c^{3} d^{6} + x^{4} \left (30 a c^{2} e^{6} + 150 c^{3} d^{2} e^{4}\right ) + x^{3} \left (60 a c^{2} d e^{5} + 500 c^{3} d^{3} e^{3}\right ) + x^{2} \left (10 a^{2} c e^{6} + 60 a c^{2} d^{2} e^{4} + 650 c^{3} d^{4} e^{2}\right ) + x \left (5 a^{2} c d e^{5} + 30 a c^{2} d^{3} e^{3} + 385 c^{3} d^{5} e\right )}{10 d^{5} e^{7} + 50 d^{4} e^{8} x + 100 d^{3} e^{9} x^{2} + 100 d^{2} e^{10} x^{3} + 50 d e^{11} x^{4} + 10 e^{12} x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**6,x)

[Out]

-6*c**3*d*log(d + e*x)/e**7 + c**3*x/e**6 - (2*a**3*e**6 + a**2*c*d**2*e**4 + 6*a*c**2*d**4*e**2 + 87*c**3*d**
6 + x**4*(30*a*c**2*e**6 + 150*c**3*d**2*e**4) + x**3*(60*a*c**2*d*e**5 + 500*c**3*d**3*e**3) + x**2*(10*a**2*
c*e**6 + 60*a*c**2*d**2*e**4 + 650*c**3*d**4*e**2) + x*(5*a**2*c*d*e**5 + 30*a*c**2*d**3*e**3 + 385*c**3*d**5*
e))/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**
5)

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Giac [A]  time = 1.30574, size = 254, normalized size = 1.48 \begin{align*} -6 \, c^{3} d e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + c^{3} x e^{\left (-6\right )} - \frac{{\left (87 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 30 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 20 \,{\left (25 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 2 \, a^{3} e^{6} + 10 \,{\left (65 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 5 \,{\left (77 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-7\right )}}{10 \,{\left (x e + d\right )}^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^6,x, algorithm="giac")

[Out]

-6*c^3*d*e^(-7)*log(abs(x*e + d)) + c^3*x*e^(-6) - 1/10*(87*c^3*d^6 + 6*a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + 30*(5*
c^3*d^2*e^4 + a*c^2*e^6)*x^4 + 20*(25*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 2*a^3*e^6 + 10*(65*c^3*d^4*e^2 + 6*a*
c^2*d^2*e^4 + a^2*c*e^6)*x^2 + 5*(77*c^3*d^5*e + 6*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)*e^(-7)/(x*e + d)^5