3.487 \(\int \frac{(a+c x^2)^3}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=190 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac{2 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^6}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7 (d+e x)^7}+\frac{3 c d \left (a e^2+c d^2\right )^2}{4 e^7 (d+e x)^8}-\frac{\left (a e^2+c d^2\right )^3}{9 e^7 (d+e x)^9}-\frac{c^3}{3 e^7 (d+e x)^3}+\frac{3 c^3 d}{2 e^7 (d+e x)^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.12076, antiderivative size = 190, 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} \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac{2 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^6}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7 (d+e x)^7}+\frac{3 c d \left (a e^2+c d^2\right )^2}{4 e^7 (d+e x)^8}-\frac{\left (a e^2+c d^2\right )^3}{9 e^7 (d+e x)^9}-\frac{c^3}{3 e^7 (d+e x)^3}+\frac{3 c^3 d}{2 e^7 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3/(d + e*x)^10,x]

[Out]

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

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)^{10}} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^{10}}-\frac{6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^9}+\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)^8}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)^7}+\frac{3 c^2 \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^6}-\frac{6 c^3 d}{e^6 (d+e x)^5}+\frac{c^3}{e^6 (d+e x)^4}\right ) \, dx\\ &=-\frac{\left (c d^2+a e^2\right )^3}{9 e^7 (d+e x)^9}+\frac{3 c d \left (c d^2+a e^2\right )^2}{4 e^7 (d+e x)^8}-\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{7 e^7 (d+e x)^7}+\frac{2 c^2 d \left (5 c d^2+3 a e^2\right )}{3 e^7 (d+e x)^6}-\frac{3 c^2 \left (5 c d^2+a e^2\right )}{5 e^7 (d+e x)^5}+\frac{3 c^3 d}{2 e^7 (d+e x)^4}-\frac{c^3}{3 e^7 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0537566, size = 163, normalized size = 0.86 \[ -\frac{15 a^2 c e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+140 a^3 e^6+6 a c^2 e^2 \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )+5 c^3 \left (36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+9 d^5 e x+d^6+126 d e^5 x^5+84 e^6 x^6\right )}{1260 e^7 (d+e x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^3/(d + e*x)^10,x]

[Out]

-(140*a^3*e^6 + 15*a^2*c*e^4*(d^2 + 9*d*e*x + 36*e^2*x^2) + 6*a*c^2*e^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84
*d*e^3*x^3 + 126*e^4*x^4) + 5*c^3*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d
*e^5*x^5 + 84*e^6*x^6))/(1260*e^7*(d + e*x)^9)

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 218, normalized size = 1.2 \begin{align*}{\frac{3\,{c}^{3}d}{2\,{e}^{7} \left ( ex+d \right ) ^{4}}}+{\frac{2\,{c}^{2}d \left ( 3\,a{e}^{2}+5\,c{d}^{2} \right ) }{3\,{e}^{7} \left ( ex+d \right ) ^{6}}}+{\frac{3\,cd \left ({a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{8}}}-{\frac{{c}^{3}}{3\,{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}}{9\,{e}^{7} \left ( ex+d \right ) ^{9}}}-{\frac{3\,c \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}+5\,{c}^{2}{d}^{4} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{3\,{c}^{2} \left ( a{e}^{2}+5\,c{d}^{2} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3/(e*x+d)^10,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.25438, size = 396, normalized size = 2.08 \begin{align*} -\frac{420 \, c^{3} e^{6} x^{6} + 630 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6} + 126 \,{\left (5 \, c^{3} d^{2} e^{4} + 6 \, a c^{2} e^{6}\right )} x^{4} + 84 \,{\left (5 \, c^{3} d^{3} e^{3} + 6 \, a c^{2} d e^{5}\right )} x^{3} + 36 \,{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + 15 \, a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5}\right )} x}{1260 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} 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)^10,x, algorithm="maxima")

[Out]

-1/1260*(420*c^3*e^6*x^6 + 630*c^3*d*e^5*x^5 + 5*c^3*d^6 + 6*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 140*a^3*e^6 +
126*(5*c^3*d^2*e^4 + 6*a*c^2*e^6)*x^4 + 84*(5*c^3*d^3*e^3 + 6*a*c^2*d*e^5)*x^3 + 36*(5*c^3*d^4*e^2 + 6*a*c^2*d
^2*e^4 + 15*a^2*c*e^6)*x^2 + 9*(5*c^3*d^5*e + 6*a*c^2*d^3*e^3 + 15*a^2*c*d*e^5)*x)/(e^16*x^9 + 9*d*e^15*x^8 +
36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9
*d^8*e^8*x + d^9*e^7)

________________________________________________________________________________________

Fricas [A]  time = 2.20905, size = 625, normalized size = 3.29 \begin{align*} -\frac{420 \, c^{3} e^{6} x^{6} + 630 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6} + 126 \,{\left (5 \, c^{3} d^{2} e^{4} + 6 \, a c^{2} e^{6}\right )} x^{4} + 84 \,{\left (5 \, c^{3} d^{3} e^{3} + 6 \, a c^{2} d e^{5}\right )} x^{3} + 36 \,{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + 15 \, a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5}\right )} x}{1260 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} 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)^10,x, algorithm="fricas")

[Out]

-1/1260*(420*c^3*e^6*x^6 + 630*c^3*d*e^5*x^5 + 5*c^3*d^6 + 6*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 140*a^3*e^6 +
126*(5*c^3*d^2*e^4 + 6*a*c^2*e^6)*x^4 + 84*(5*c^3*d^3*e^3 + 6*a*c^2*d*e^5)*x^3 + 36*(5*c^3*d^4*e^2 + 6*a*c^2*d
^2*e^4 + 15*a^2*c*e^6)*x^2 + 9*(5*c^3*d^5*e + 6*a*c^2*d^3*e^3 + 15*a^2*c*d*e^5)*x)/(e^16*x^9 + 9*d*e^15*x^8 +
36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9
*d^8*e^8*x + d^9*e^7)

________________________________________________________________________________________

Sympy [A]  time = 44.4539, size = 308, normalized size = 1.62 \begin{align*} - \frac{140 a^{3} e^{6} + 15 a^{2} c d^{2} e^{4} + 6 a c^{2} d^{4} e^{2} + 5 c^{3} d^{6} + 630 c^{3} d e^{5} x^{5} + 420 c^{3} e^{6} x^{6} + x^{4} \left (756 a c^{2} e^{6} + 630 c^{3} d^{2} e^{4}\right ) + x^{3} \left (504 a c^{2} d e^{5} + 420 c^{3} d^{3} e^{3}\right ) + x^{2} \left (540 a^{2} c e^{6} + 216 a c^{2} d^{2} e^{4} + 180 c^{3} d^{4} e^{2}\right ) + x \left (135 a^{2} c d e^{5} + 54 a c^{2} d^{3} e^{3} + 45 c^{3} d^{5} e\right )}{1260 d^{9} e^{7} + 11340 d^{8} e^{8} x + 45360 d^{7} e^{9} x^{2} + 105840 d^{6} e^{10} x^{3} + 158760 d^{5} e^{11} x^{4} + 158760 d^{4} e^{12} x^{5} + 105840 d^{3} e^{13} x^{6} + 45360 d^{2} e^{14} x^{7} + 11340 d e^{15} x^{8} + 1260 e^{16} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**10,x)

[Out]

-(140*a**3*e**6 + 15*a**2*c*d**2*e**4 + 6*a*c**2*d**4*e**2 + 5*c**3*d**6 + 630*c**3*d*e**5*x**5 + 420*c**3*e**
6*x**6 + x**4*(756*a*c**2*e**6 + 630*c**3*d**2*e**4) + x**3*(504*a*c**2*d*e**5 + 420*c**3*d**3*e**3) + x**2*(5
40*a**2*c*e**6 + 216*a*c**2*d**2*e**4 + 180*c**3*d**4*e**2) + x*(135*a**2*c*d*e**5 + 54*a*c**2*d**3*e**3 + 45*
c**3*d**5*e))/(1260*d**9*e**7 + 11340*d**8*e**8*x + 45360*d**7*e**9*x**2 + 105840*d**6*e**10*x**3 + 158760*d**
5*e**11*x**4 + 158760*d**4*e**12*x**5 + 105840*d**3*e**13*x**6 + 45360*d**2*e**14*x**7 + 11340*d*e**15*x**8 +
1260*e**16*x**9)

________________________________________________________________________________________

Giac [A]  time = 1.33224, size = 258, normalized size = 1.36 \begin{align*} -\frac{{\left (420 \, c^{3} x^{6} e^{6} + 630 \, c^{3} d x^{5} e^{5} + 630 \, c^{3} d^{2} x^{4} e^{4} + 420 \, c^{3} d^{3} x^{3} e^{3} + 180 \, c^{3} d^{4} x^{2} e^{2} + 45 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 756 \, a c^{2} x^{4} e^{6} + 504 \, a c^{2} d x^{3} e^{5} + 216 \, a c^{2} d^{2} x^{2} e^{4} + 54 \, a c^{2} d^{3} x e^{3} + 6 \, a c^{2} d^{4} e^{2} + 540 \, a^{2} c x^{2} e^{6} + 135 \, a^{2} c d x e^{5} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{1260 \,{\left (x e + d\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/1260*(420*c^3*x^6*e^6 + 630*c^3*d*x^5*e^5 + 630*c^3*d^2*x^4*e^4 + 420*c^3*d^3*x^3*e^3 + 180*c^3*d^4*x^2*e^2
+ 45*c^3*d^5*x*e + 5*c^3*d^6 + 756*a*c^2*x^4*e^6 + 504*a*c^2*d*x^3*e^5 + 216*a*c^2*d^2*x^2*e^4 + 54*a*c^2*d^3
*x*e^3 + 6*a*c^2*d^4*e^2 + 540*a^2*c*x^2*e^6 + 135*a^2*c*d*x*e^5 + 15*a^2*c*d^2*e^4 + 140*a^3*e^6)*e^(-7)/(x*e
+ d)^9