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

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

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 190, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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 = 0.05, size = 163, normalized size = 0.86 \begin {gather*} -\frac {140 a^3 e^6+15 a^2 c e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+6 a c^2 e^2 \left (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\right )+5 c^3 \left (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\right )}{1260 e^7 (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1260*(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))/(e^7*(d + e*x)^9)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.38, size = 293, normalized size = 1.54 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 191, normalized size = 1.01 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 218, normalized size = 1.15 \begin {gather*} \frac {3 c^{3} d}{2 \left (e x +d \right )^{4} e^{7}}-\frac {c^{3}}{3 \left (e x +d \right )^{3} e^{7}}+\frac {2 \left (3 a \,e^{2}+5 c \,d^{2}\right ) c^{2} d}{3 \left (e x +d \right )^{6} e^{7}}-\frac {3 \left (a \,e^{2}+5 c \,d^{2}\right ) c^{2}}{5 \left (e x +d \right )^{5} e^{7}}+\frac {3 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) c d}{4 \left (e x +d \right )^{8} e^{7}}-\frac {3 \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) c}{7 \left (e x +d \right )^{7} e^{7}}-\frac {e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 a \,c^{2} d^{4} e^{2}+c^{3} d^{6}}{9 \left (e x +d \right )^{9} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.59, size = 293, normalized size = 1.54 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.34, size = 287, normalized size = 1.51 \begin {gather*} -\frac {\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}{1260\,e^7}+\frac {c^3\,x^6}{3\,e}+\frac {c^3\,d\,x^5}{2\,e^2}+\frac {c^2\,x^4\,\left (5\,c\,d^2+6\,a\,e^2\right )}{10\,e^3}+\frac {c\,x^2\,\left (15\,a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{35\,e^5}+\frac {c\,d\,x\,\left (15\,a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{140\,e^6}+\frac {c^2\,d\,x^3\,\left (5\,c\,d^2+6\,a\,e^2\right )}{15\,e^4}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3/(d + e*x)^10,x)

[Out]

-((140*a^3*e^6 + 5*c^3*d^6 + 6*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4)/(1260*e^7) + (c^3*x^6)/(3*e) + (c^3*d*x^5)/(2
*e^2) + (c^2*x^4*(6*a*e^2 + 5*c*d^2))/(10*e^3) + (c*x^2*(15*a^2*e^4 + 5*c^2*d^4 + 6*a*c*d^2*e^2))/(35*e^5) + (
c*d*x*(15*a^2*e^4 + 5*c^2*d^4 + 6*a*c*d^2*e^2))/(140*e^6) + (c^2*d*x^3*(6*a*e^2 + 5*c*d^2))/(15*e^4))/(d^9 + e
^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 84*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e^5*x^5 + 84*d^3*e^6*x^6 +
36*d^2*e^7*x^7 + 9*d^8*e*x)

________________________________________________________________________________________

sympy [A]  time = 37.18, size = 313, normalized size = 1.65 \begin {gather*} \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 {gather*}

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*
(-540*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)

________________________________________________________________________________________