3.486 \(\int \frac{(a+c x^2)^3}{(d+e x)^9} \, dx\)

Optimal. Leaf size=188 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^6}+\frac{6 c d \left (a e^2+c d^2\right )^2}{7 e^7 (d+e x)^7}-\frac{\left (a e^2+c d^2\right )^3}{8 e^7 (d+e x)^8}-\frac{c^3}{2 e^7 (d+e x)^2}+\frac{2 c^3 d}{e^7 (d+e x)^3} \]

[Out]

-(c*d^2 + a*e^2)^3/(8*e^7*(d + e*x)^8) + (6*c*d*(c*d^2 + a*e^2)^2)/(7*e^7*(d + e*x)^7) - (c*(c*d^2 + a*e^2)*(5
*c*d^2 + a*e^2))/(2*e^7*(d + e*x)^6) + (4*c^2*d*(5*c*d^2 + 3*a*e^2))/(5*e^7*(d + e*x)^5) - (3*c^2*(5*c*d^2 + a
*e^2))/(4*e^7*(d + e*x)^4) + (2*c^3*d)/(e^7*(d + e*x)^3) - c^3/(2*e^7*(d + e*x)^2)

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Rubi [A]  time = 0.12541, antiderivative size = 188, 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} \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^6}+\frac{6 c d \left (a e^2+c d^2\right )^2}{7 e^7 (d+e x)^7}-\frac{\left (a e^2+c d^2\right )^3}{8 e^7 (d+e x)^8}-\frac{c^3}{2 e^7 (d+e x)^2}+\frac{2 c^3 d}{e^7 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3/(d + e*x)^9,x]

[Out]

-(c*d^2 + a*e^2)^3/(8*e^7*(d + e*x)^8) + (6*c*d*(c*d^2 + a*e^2)^2)/(7*e^7*(d + e*x)^7) - (c*(c*d^2 + a*e^2)*(5
*c*d^2 + a*e^2))/(2*e^7*(d + e*x)^6) + (4*c^2*d*(5*c*d^2 + 3*a*e^2))/(5*e^7*(d + e*x)^5) - (3*c^2*(5*c*d^2 + a
*e^2))/(4*e^7*(d + e*x)^4) + (2*c^3*d)/(e^7*(d + e*x)^3) - c^3/(2*e^7*(d + e*x)^2)

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)^9} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^9}-\frac{6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^8}+\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)^7}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)^6}+\frac{3 c^2 \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^5}-\frac{6 c^3 d}{e^6 (d+e x)^4}+\frac{c^3}{e^6 (d+e x)^3}\right ) \, dx\\ &=-\frac{\left (c d^2+a e^2\right )^3}{8 e^7 (d+e x)^8}+\frac{6 c d \left (c d^2+a e^2\right )^2}{7 e^7 (d+e x)^7}-\frac{c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^6}+\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{5 e^7 (d+e x)^5}-\frac{3 c^2 \left (5 c d^2+a e^2\right )}{4 e^7 (d+e x)^4}+\frac{2 c^3 d}{e^7 (d+e x)^3}-\frac{c^3}{2 e^7 (d+e x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0616959, size = 163, normalized size = 0.87 \[ -\frac{5 a^2 c e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+35 a^3 e^6+3 a c^2 e^2 \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+5 c^3 \left (28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+8 d^5 e x+d^6+56 d e^5 x^5+28 e^6 x^6\right )}{280 e^7 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^3/(d + e*x)^9,x]

[Out]

-(35*a^3*e^6 + 5*a^2*c*e^4*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*a*c^2*e^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d
*e^3*x^3 + 70*e^4*x^4) + 5*c^3*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*
x^5 + 28*e^6*x^6))/(280*e^7*(d + e*x)^8)

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Maple [A]  time = 0.05, size = 218, normalized size = 1.2 \begin{align*} -{\frac{3\,{c}^{2} \left ( a{e}^{2}+5\,c{d}^{2} \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{c \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}+5\,{c}^{2}{d}^{4} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{{a}^{3}{e}^{6}+3\,{a}^{2}c{d}^{2}{e}^{4}+3\,{d}^{4}{e}^{2}a{c}^{2}+{d}^{6}{c}^{3}}{8\,{e}^{7} \left ( ex+d \right ) ^{8}}}+2\,{\frac{{c}^{3}d}{{e}^{7} \left ( ex+d \right ) ^{3}}}+{\frac{6\,cd \left ({a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}+{\frac{4\,{c}^{2}d \left ( 3\,a{e}^{2}+5\,c{d}^{2} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{{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)^9,x)

[Out]

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

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Maxima [A]  time = 1.2149, size = 381, normalized size = 2.03 \begin{align*} -\frac{140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} + 70 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} + 56 \,{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 28 \,{\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4} + 5 \, a^{2} c e^{6}\right )} x^{2} + 8 \,{\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x}{280 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} 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)^9,x, algorithm="maxima")

[Out]

-1/280*(140*c^3*e^6*x^6 + 280*c^3*d*e^5*x^5 + 5*c^3*d^6 + 3*a*c^2*d^4*e^2 + 5*a^2*c*d^2*e^4 + 35*a^3*e^6 + 70*
(5*c^3*d^2*e^4 + 3*a*c^2*e^6)*x^4 + 56*(5*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 28*(5*c^3*d^4*e^2 + 3*a*c^2*d^2*e
^4 + 5*a^2*c*e^6)*x^2 + 8*(5*c^3*d^5*e + 3*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2
*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

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Fricas [A]  time = 2.06227, size = 590, normalized size = 3.14 \begin{align*} -\frac{140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} + 70 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} + 56 \,{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 28 \,{\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4} + 5 \, a^{2} c e^{6}\right )} x^{2} + 8 \,{\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x}{280 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} 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)^9,x, algorithm="fricas")

[Out]

-1/280*(140*c^3*e^6*x^6 + 280*c^3*d*e^5*x^5 + 5*c^3*d^6 + 3*a*c^2*d^4*e^2 + 5*a^2*c*d^2*e^4 + 35*a^3*e^6 + 70*
(5*c^3*d^2*e^4 + 3*a*c^2*e^6)*x^4 + 56*(5*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 28*(5*c^3*d^4*e^2 + 3*a*c^2*d^2*e
^4 + 5*a^2*c*e^6)*x^2 + 8*(5*c^3*d^5*e + 3*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2
*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

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Sympy [A]  time = 26.2735, size = 296, normalized size = 1.57 \begin{align*} - \frac{35 a^{3} e^{6} + 5 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + 5 c^{3} d^{6} + 280 c^{3} d e^{5} x^{5} + 140 c^{3} e^{6} x^{6} + x^{4} \left (210 a c^{2} e^{6} + 350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (168 a c^{2} d e^{5} + 280 c^{3} d^{3} e^{3}\right ) + x^{2} \left (140 a^{2} c e^{6} + 84 a c^{2} d^{2} e^{4} + 140 c^{3} d^{4} e^{2}\right ) + x \left (40 a^{2} c d e^{5} + 24 a c^{2} d^{3} e^{3} + 40 c^{3} d^{5} e\right )}{280 d^{8} e^{7} + 2240 d^{7} e^{8} x + 7840 d^{6} e^{9} x^{2} + 15680 d^{5} e^{10} x^{3} + 19600 d^{4} e^{11} x^{4} + 15680 d^{3} e^{12} x^{5} + 7840 d^{2} e^{13} x^{6} + 2240 d e^{14} x^{7} + 280 e^{15} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**9,x)

[Out]

-(35*a**3*e**6 + 5*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + 5*c**3*d**6 + 280*c**3*d*e**5*x**5 + 140*c**3*e**6*
x**6 + x**4*(210*a*c**2*e**6 + 350*c**3*d**2*e**4) + x**3*(168*a*c**2*d*e**5 + 280*c**3*d**3*e**3) + x**2*(140
*a**2*c*e**6 + 84*a*c**2*d**2*e**4 + 140*c**3*d**4*e**2) + x*(40*a**2*c*d*e**5 + 24*a*c**2*d**3*e**3 + 40*c**3
*d**5*e))/(280*d**8*e**7 + 2240*d**7*e**8*x + 7840*d**6*e**9*x**2 + 15680*d**5*e**10*x**3 + 19600*d**4*e**11*x
**4 + 15680*d**3*e**12*x**5 + 7840*d**2*e**13*x**6 + 2240*d*e**14*x**7 + 280*e**15*x**8)

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Giac [A]  time = 1.20293, size = 258, normalized size = 1.37 \begin{align*} -\frac{{\left (140 \, c^{3} x^{6} e^{6} + 280 \, c^{3} d x^{5} e^{5} + 350 \, c^{3} d^{2} x^{4} e^{4} + 280 \, c^{3} d^{3} x^{3} e^{3} + 140 \, c^{3} d^{4} x^{2} e^{2} + 40 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 210 \, a c^{2} x^{4} e^{6} + 168 \, a c^{2} d x^{3} e^{5} + 84 \, a c^{2} d^{2} x^{2} e^{4} + 24 \, a c^{2} d^{3} x e^{3} + 3 \, a c^{2} d^{4} e^{2} + 140 \, a^{2} c x^{2} e^{6} + 40 \, a^{2} c d x e^{5} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{280 \,{\left (x e + d\right )}^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/280*(140*c^3*x^6*e^6 + 280*c^3*d*x^5*e^5 + 350*c^3*d^2*x^4*e^4 + 280*c^3*d^3*x^3*e^3 + 140*c^3*d^4*x^2*e^2
+ 40*c^3*d^5*x*e + 5*c^3*d^6 + 210*a*c^2*x^4*e^6 + 168*a*c^2*d*x^3*e^5 + 84*a*c^2*d^2*x^2*e^4 + 24*a*c^2*d^3*x
*e^3 + 3*a*c^2*d^4*e^2 + 140*a^2*c*x^2*e^6 + 40*a^2*c*d*x*e^5 + 5*a^2*c*d^2*e^4 + 35*a^3*e^6)*e^(-7)/(x*e + d)
^8