### 3.485 $$\int \frac{(a+c x^2)^3}{(d+e x)^8} \, dx$$

Optimal. Leaf size=178 $-\frac{c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^4}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac{c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^3}{7 e^7 (d+e x)^7}-\frac{c^3}{e^7 (d+e x)}+\frac{3 c^3 d}{e^7 (d+e x)^2}$

[Out]

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

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Rubi [A]  time = 0.121561, antiderivative size = 178, 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} $-\frac{c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^4}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac{c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^3}{7 e^7 (d+e x)^7}-\frac{c^3}{e^7 (d+e x)}+\frac{3 c^3 d}{e^7 (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^3/(d + e*x)^8,x]

[Out]

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

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

Mathematica [A]  time = 0.052798, size = 161, normalized size = 0.9 $-\frac{a^2 c e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+5 a^3 e^6+a c^2 e^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )}{35 e^7 (d+e x)^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^3/(d + e*x)^8,x]

[Out]

-(5*a^3*e^6 + a^2*c*e^4*(d^2 + 7*d*e*x + 21*e^2*x^2) + a*c^2*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*
x^3 + 35*e^4*x^4) + 5*c^3*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 +
7*e^6*x^6))/(35*e^7*(d + e*x)^7)

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Maple [A]  time = 0.052, size = 216, normalized size = 1.2 \begin{align*}{\frac{{c}^{2}d \left ( 3\,a{e}^{2}+5\,c{d}^{2} \right ) }{{e}^{7} \left ( ex+d \right ) ^{4}}}+{\frac{cd \left ({a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{2} \left ( a{e}^{2}+5\,c{d}^{2} \right ) }{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\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}}{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{3\,c \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}+5\,{c}^{2}{d}^{4} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{3}}{{e}^{7} \left ( ex+d \right ) }}+3\,{\frac{{c}^{3}d}{{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)^8,x)

[Out]

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

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Maxima [A]  time = 1.21563, size = 355, normalized size = 1.99 \begin{align*} -\frac{35 \, c^{3} e^{6} x^{6} + 105 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 35 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 35 \,{\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 21 \,{\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 7 \,{\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{35 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} 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)^8,x, algorithm="maxima")

[Out]

-1/35*(35*c^3*e^6*x^6 + 105*c^3*d*e^5*x^5 + 5*c^3*d^6 + a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + 5*a^3*e^6 + 35*(5*c^3*
d^2*e^4 + a*c^2*e^6)*x^4 + 35*(5*c^3*d^3*e^3 + a*c^2*d*e^5)*x^3 + 21*(5*c^3*d^4*e^2 + a*c^2*d^2*e^4 + a^2*c*e^
6)*x^2 + 7*(5*c^3*d^5*e + a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*
e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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Fricas [A]  time = 2.0345, size = 540, normalized size = 3.03 \begin{align*} -\frac{35 \, c^{3} e^{6} x^{6} + 105 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 35 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 35 \,{\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 21 \,{\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 7 \,{\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{35 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} 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)^8,x, algorithm="fricas")

[Out]

-1/35*(35*c^3*e^6*x^6 + 105*c^3*d*e^5*x^5 + 5*c^3*d^6 + a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + 5*a^3*e^6 + 35*(5*c^3*
d^2*e^4 + a*c^2*e^6)*x^4 + 35*(5*c^3*d^3*e^3 + a*c^2*d*e^5)*x^3 + 21*(5*c^3*d^4*e^2 + a*c^2*d^2*e^4 + a^2*c*e^
6)*x^2 + 7*(5*c^3*d^5*e + a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*
e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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Sympy [A]  time = 15.2541, size = 280, normalized size = 1.57 \begin{align*} - \frac{5 a^{3} e^{6} + a^{2} c d^{2} e^{4} + a c^{2} d^{4} e^{2} + 5 c^{3} d^{6} + 105 c^{3} d e^{5} x^{5} + 35 c^{3} e^{6} x^{6} + x^{4} \left (35 a c^{2} e^{6} + 175 c^{3} d^{2} e^{4}\right ) + x^{3} \left (35 a c^{2} d e^{5} + 175 c^{3} d^{3} e^{3}\right ) + x^{2} \left (21 a^{2} c e^{6} + 21 a c^{2} d^{2} e^{4} + 105 c^{3} d^{4} e^{2}\right ) + x \left (7 a^{2} c d e^{5} + 7 a c^{2} d^{3} e^{3} + 35 c^{3} d^{5} e\right )}{35 d^{7} e^{7} + 245 d^{6} e^{8} x + 735 d^{5} e^{9} x^{2} + 1225 d^{4} e^{10} x^{3} + 1225 d^{3} e^{11} x^{4} + 735 d^{2} e^{12} x^{5} + 245 d e^{13} x^{6} + 35 e^{14} x^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**8,x)

[Out]

-(5*a**3*e**6 + a**2*c*d**2*e**4 + a*c**2*d**4*e**2 + 5*c**3*d**6 + 105*c**3*d*e**5*x**5 + 35*c**3*e**6*x**6 +
x**4*(35*a*c**2*e**6 + 175*c**3*d**2*e**4) + x**3*(35*a*c**2*d*e**5 + 175*c**3*d**3*e**3) + x**2*(21*a**2*c*e
**6 + 21*a*c**2*d**2*e**4 + 105*c**3*d**4*e**2) + x*(7*a**2*c*d*e**5 + 7*a*c**2*d**3*e**3 + 35*c**3*d**5*e))/(
35*d**7*e**7 + 245*d**6*e**8*x + 735*d**5*e**9*x**2 + 1225*d**4*e**10*x**3 + 1225*d**3*e**11*x**4 + 735*d**2*e
**12*x**5 + 245*d*e**13*x**6 + 35*e**14*x**7)

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Giac [A]  time = 1.22163, size = 255, normalized size = 1.43 \begin{align*} -\frac{{\left (35 \, c^{3} x^{6} e^{6} + 105 \, c^{3} d x^{5} e^{5} + 175 \, c^{3} d^{2} x^{4} e^{4} + 175 \, c^{3} d^{3} x^{3} e^{3} + 105 \, c^{3} d^{4} x^{2} e^{2} + 35 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 35 \, a c^{2} x^{4} e^{6} + 35 \, a c^{2} d x^{3} e^{5} + 21 \, a c^{2} d^{2} x^{2} e^{4} + 7 \, a c^{2} d^{3} x e^{3} + a c^{2} d^{4} e^{2} + 21 \, a^{2} c x^{2} e^{6} + 7 \, a^{2} c d x e^{5} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{35 \,{\left (x e + d\right )}^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/35*(35*c^3*x^6*e^6 + 105*c^3*d*x^5*e^5 + 175*c^3*d^2*x^4*e^4 + 175*c^3*d^3*x^3*e^3 + 105*c^3*d^4*x^2*e^2 +
35*c^3*d^5*x*e + 5*c^3*d^6 + 35*a*c^2*x^4*e^6 + 35*a*c^2*d*x^3*e^5 + 21*a*c^2*d^2*x^2*e^4 + 7*a*c^2*d^3*x*e^3
+ a*c^2*d^4*e^2 + 21*a^2*c*x^2*e^6 + 7*a^2*c*d*x*e^5 + a^2*c*d^2*e^4 + 5*a^3*e^6)*e^(-7)/(x*e + d)^7