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

Optimal. Leaf size=163 \[ \frac{3 c^2 x^2 \left (a e^2+2 c d^2\right )}{2 e^5}-\frac{c^2 d x \left (9 a e^2+10 c d^2\right )}{e^6}+\frac{6 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)}-\frac{\left (a e^2+c d^2\right )^3}{2 e^7 (d+e x)^2}+\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{c^3 d x^3}{e^4}+\frac{c^3 x^4}{4 e^3} \]

[Out]

-((c^2*d*(10*c*d^2 + 9*a*e^2)*x)/e^6) + (3*c^2*(2*c*d^2 + a*e^2)*x^2)/(2*e^5) - (c^3*d*x^3)/e^4 + (c^3*x^4)/(4
*e^3) - (c*d^2 + a*e^2)^3/(2*e^7*(d + e*x)^2) + (6*c*d*(c*d^2 + a*e^2)^2)/(e^7*(d + e*x)) + (3*c*(c*d^2 + a*e^
2)*(5*c*d^2 + a*e^2)*Log[d + e*x])/e^7

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Rubi [A]  time = 0.167547, antiderivative size = 163, 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} \[ \frac{3 c^2 x^2 \left (a e^2+2 c d^2\right )}{2 e^5}-\frac{c^2 d x \left (9 a e^2+10 c d^2\right )}{e^6}+\frac{6 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)}-\frac{\left (a e^2+c d^2\right )^3}{2 e^7 (d+e x)^2}+\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{c^3 d x^3}{e^4}+\frac{c^3 x^4}{4 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3/(d + e*x)^3,x]

[Out]

-((c^2*d*(10*c*d^2 + 9*a*e^2)*x)/e^6) + (3*c^2*(2*c*d^2 + a*e^2)*x^2)/(2*e^5) - (c^3*d*x^3)/e^4 + (c^3*x^4)/(4
*e^3) - (c*d^2 + a*e^2)^3/(2*e^7*(d + e*x)^2) + (6*c*d*(c*d^2 + a*e^2)^2)/(e^7*(d + e*x)) + (3*c*(c*d^2 + a*e^
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)^3} \, dx &=\int \left (-\frac{c^2 d \left (10 c d^2+9 a e^2\right )}{e^6}+\frac{3 c^2 \left (2 c d^2+a e^2\right ) x}{e^5}-\frac{3 c^3 d x^2}{e^4}+\frac{c^3 x^3}{e^3}+\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^3}-\frac{6 c d \left (c d^2+a e^2\right )^2}{e^6 (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^6 (d+e x)}\right ) \, dx\\ &=-\frac{c^2 d \left (10 c d^2+9 a e^2\right ) x}{e^6}+\frac{3 c^2 \left (2 c d^2+a e^2\right ) x^2}{2 e^5}-\frac{c^3 d x^3}{e^4}+\frac{c^3 x^4}{4 e^3}-\frac{\left (c d^2+a e^2\right )^3}{2 e^7 (d+e x)^2}+\frac{6 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.0690721, size = 198, normalized size = 1.21 \[ \frac{12 c (d+e x)^2 \left (a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \log (d+e x)+6 a^2 c d e^4 (3 d+4 e x)-2 a^3 e^6+6 a c^2 e^2 \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )+c^3 \left (-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-16 d^5 e x+22 d^6-2 d e^5 x^5+e^6 x^6\right )}{4 e^7 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*e^6 + 6*a^2*c*d*e^4*(3*d + 4*e*x) + 6*a*c^2*e^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^
4*x^4) + c^3*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6) +
12*c*(5*c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*(d + e*x)^2*Log[d + e*x])/(4*e^7*(d + e*x)^2)

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Maple [A]  time = 0.052, size = 249, normalized size = 1.5 \begin{align*}{\frac{{c}^{3}{x}^{4}}{4\,{e}^{3}}}-{\frac{{c}^{3}d{x}^{3}}{{e}^{4}}}+{\frac{3\,{c}^{2}{x}^{2}a}{2\,{e}^{3}}}+3\,{\frac{{c}^{3}{x}^{2}{d}^{2}}{{e}^{5}}}-9\,{\frac{a{c}^{2}dx}{{e}^{4}}}-10\,{\frac{{c}^{3}{d}^{3}x}{{e}^{6}}}+3\,{\frac{c\ln \left ( ex+d \right ){a}^{2}}{{e}^{3}}}+18\,{\frac{{c}^{2}\ln \left ( ex+d \right ) a{d}^{2}}{{e}^{5}}}+15\,{\frac{{c}^{3}\ln \left ( ex+d \right ){d}^{4}}{{e}^{7}}}+6\,{\frac{cd{a}^{2}}{{e}^{3} \left ( ex+d \right ) }}+12\,{\frac{a{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}+6\,{\frac{{c}^{3}{d}^{5}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{{a}^{3}}{2\,e \left ( ex+d \right ) ^{2}}}-{\frac{3\,{a}^{2}c{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{d}^{4}a{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{6}{c}^{3}}{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)^3,x)

[Out]

1/4*c^3*x^4/e^3-c^3*d*x^3/e^4+3/2*c^2/e^3*x^2*a+3*c^3/e^5*x^2*d^2-9*c^2/e^4*a*d*x-10*c^3/e^6*d^3*x+3*c/e^3*ln(
e*x+d)*a^2+18*c^2/e^5*ln(e*x+d)*a*d^2+15*c^3/e^7*ln(e*x+d)*d^4+6*c*d/e^3/(e*x+d)*a^2+12*c^2*d^3/e^5/(e*x+d)*a+
6*c^3*d^5/e^7/(e*x+d)-1/2/e/(e*x+d)^2*a^3-3/2/e^3/(e*x+d)^2*d^2*a^2*c-3/2/e^5/(e*x+d)^2*d^4*a*c^2-1/2/e^7/(e*x
+d)^2*d^6*c^3

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Maxima [A]  time = 1.49776, size = 289, normalized size = 1.77 \begin{align*} \frac{11 \, c^{3} d^{6} + 21 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 12 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{c^{3} e^{3} x^{4} - 4 \, c^{3} d e^{2} x^{3} + 6 \,{\left (2 \, c^{3} d^{2} e + a c^{2} e^{3}\right )} x^{2} - 4 \,{\left (10 \, c^{3} d^{3} + 9 \, a c^{2} d e^{2}\right )} x}{4 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\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)^3,x, algorithm="maxima")

[Out]

1/2*(11*c^3*d^6 + 21*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - a^3*e^6 + 12*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + a^2*c*d*e^5
)*x)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/4*(c^3*e^3*x^4 - 4*c^3*d*e^2*x^3 + 6*(2*c^3*d^2*e + a*c^2*e^3)*x^2 -
4*(10*c^3*d^3 + 9*a*c^2*d*e^2)*x)/e^6 + 3*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*log(e*x + d)/e^7

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Fricas [B]  time = 2.17337, size = 640, normalized size = 3.93 \begin{align*} \frac{c^{3} e^{6} x^{6} - 2 \, c^{3} d e^{5} x^{5} + 22 \, c^{3} d^{6} + 42 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} +{\left (5 \, c^{3} d^{2} e^{4} + 6 \, a c^{2} e^{6}\right )} x^{4} - 4 \,{\left (5 \, c^{3} d^{3} e^{3} + 6 \, a c^{2} d e^{5}\right )} x^{3} - 2 \,{\left (34 \, c^{3} d^{4} e^{2} + 33 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 4 \,{\left (4 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3} - 6 \, a^{2} c d e^{5}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} +{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 2 \,{\left (5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/4*(c^3*e^6*x^6 - 2*c^3*d*e^5*x^5 + 22*c^3*d^6 + 42*a*c^2*d^4*e^2 + 18*a^2*c*d^2*e^4 - 2*a^3*e^6 + (5*c^3*d^2
*e^4 + 6*a*c^2*e^6)*x^4 - 4*(5*c^3*d^3*e^3 + 6*a*c^2*d*e^5)*x^3 - 2*(34*c^3*d^4*e^2 + 33*a*c^2*d^2*e^4)*x^2 -
4*(4*c^3*d^5*e - 3*a*c^2*d^3*e^3 - 6*a^2*c*d*e^5)*x + 12*(5*c^3*d^6 + 6*a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + (5*c^3
*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)*x^2 + 2*(5*c^3*d^5*e + 6*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)*log(e*x + d))
/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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Sympy [A]  time = 1.92448, size = 216, normalized size = 1.33 \begin{align*} - \frac{c^{3} d x^{3}}{e^{4}} + \frac{c^{3} x^{4}}{4 e^{3}} + \frac{3 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} + \frac{- a^{3} e^{6} + 9 a^{2} c d^{2} e^{4} + 21 a c^{2} d^{4} e^{2} + 11 c^{3} d^{6} + x \left (12 a^{2} c d e^{5} + 24 a c^{2} d^{3} e^{3} + 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac{x^{2} \left (3 a c^{2} e^{2} + 6 c^{3} d^{2}\right )}{2 e^{5}} - \frac{x \left (9 a c^{2} d e^{2} + 10 c^{3} d^{3}\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)**3,x)

[Out]

-c**3*d*x**3/e**4 + c**3*x**4/(4*e**3) + 3*c*(a*e**2 + c*d**2)*(a*e**2 + 5*c*d**2)*log(d + e*x)/e**7 + (-a**3*
e**6 + 9*a**2*c*d**2*e**4 + 21*a*c**2*d**4*e**2 + 11*c**3*d**6 + x*(12*a**2*c*d*e**5 + 24*a*c**2*d**3*e**3 + 1
2*c**3*d**5*e))/(2*d**2*e**7 + 4*d*e**8*x + 2*e**9*x**2) + x**2*(3*a*c**2*e**2 + 6*c**3*d**2)/(2*e**5) - x*(9*
a*c**2*d*e**2 + 10*c**3*d**3)/e**6

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Giac [A]  time = 1.22611, size = 259, normalized size = 1.59 \begin{align*} 3 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{4} \,{\left (c^{3} x^{4} e^{9} - 4 \, c^{3} d x^{3} e^{8} + 12 \, c^{3} d^{2} x^{2} e^{7} - 40 \, c^{3} d^{3} x e^{6} + 6 \, a c^{2} x^{2} e^{9} - 36 \, a c^{2} d x e^{8}\right )} e^{\left (-12\right )} + \frac{{\left (11 \, c^{3} d^{6} + 21 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 12 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*e^(-7)*log(abs(x*e + d)) + 1/4*(c^3*x^4*e^9 - 4*c^3*d*x^3*e^8 + 12
*c^3*d^2*x^2*e^7 - 40*c^3*d^3*x*e^6 + 6*a*c^2*x^2*e^9 - 36*a*c^2*d*x*e^8)*e^(-12) + 1/2*(11*c^3*d^6 + 21*a*c^2
*d^4*e^2 + 9*a^2*c*d^2*e^4 - a^3*e^6 + 12*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)*e^(-7)/(x*e + d)^2