3.5.87 \(\int \frac {(a+c x^2)^3}{(d+e x)^{10}} \, dx\) [487]

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

[Out]

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

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Rubi [A]
time = 0.08, 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 = {711} \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]

-1/9*(c*d^2 + a*e^2)^3/(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 711

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*}

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Mathematica [A]
time = 0.04, 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)

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Maple [A]
time = 0.41, size = 218, normalized size = 1.15

method	result	size
risch	\(\frac {-\frac {c^{3} x^{6}}{3 e}-\frac {c^{3} d \,x^{5}}{2 e^{2}}-\frac {c^{2} \left (6 e^{2} a +5 c \,d^{2}\right ) x^{4}}{10 e^{3}}-\frac {c^{2} d \left (6 e^{2} a +5 c \,d^{2}\right ) x^{3}}{15 e^{4}}-\frac {c \left (15 a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x^{2}}{35 e^{5}}-\frac {d c \left (15 a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x}{140 e^{6}}-\frac {140 e^{6} a^{3}+15 e^{4} d^{2} a^{2} c +6 d^{4} e^{2} c^{2} a +5 d^{6} c^{3}}{1260 e^{7}}}{\left (e x +d \right )^{9}}\)	\(199\)
gosper	\(-\frac {420 c^{3} x^{6} e^{6}+630 c^{3} d \,x^{5} e^{5}+756 a \,c^{2} e^{6} x^{4}+630 c^{3} d^{2} e^{4} x^{4}+504 a \,c^{2} d \,e^{5} x^{3}+420 c^{3} d^{3} e^{3} x^{3}+540 a^{2} c \,e^{6} x^{2}+216 a \,c^{2} d^{2} e^{4} x^{2}+180 c^{3} d^{4} e^{2} x^{2}+135 a^{2} c d \,e^{5} x +54 a \,c^{2} d^{3} e^{3} x +45 c^{3} d^{5} e x +140 e^{6} a^{3}+15 e^{4} d^{2} a^{2} c +6 d^{4} e^{2} c^{2} a +5 d^{6} c^{3}}{1260 e^{7} \left (e x +d \right )^{9}}\)	\(205\)
default	\(-\frac {c^{3}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {e^{6} a^{3}+3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{9 e^{7} \left (e x +d \right )^{9}}-\frac {3 c^{2} \left (e^{2} a +5 c \,d^{2}\right )}{5 e^{7} \left (e x +d \right )^{5}}+\frac {3 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{4 e^{7} \left (e x +d \right )^{8}}-\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{7 e^{7} \left (e x +d \right )^{7}}+\frac {3 c^{3} d}{2 e^{7} \left (e x +d \right )^{4}}+\frac {2 c^{2} d \left (3 e^{2} a +5 c \,d^{2}\right )}{3 e^{7} \left (e x +d \right )^{6}}\)	\(218\)
norman	\(\frac {-\frac {c^{3} x^{6}}{3 e}-\frac {c^{3} d \,x^{5}}{2 e^{2}}-\frac {\left (6 e^{4} c^{2} a +5 d^{2} e^{2} c^{3}\right ) x^{4}}{10 e^{5}}-\frac {d \left (6 e^{4} c^{2} a +5 d^{2} e^{2} c^{3}\right ) x^{3}}{15 e^{6}}-\frac {\left (15 e^{6} a^{2} c +6 e^{4} d^{2} c^{2} a +5 d^{4} e^{2} c^{3}\right ) x^{2}}{35 e^{7}}-\frac {d \left (15 e^{6} a^{2} c +6 e^{4} d^{2} c^{2} a +5 d^{4} e^{2} c^{3}\right ) x}{140 e^{8}}-\frac {140 a^{3} e^{8}+15 a^{2} c \,d^{2} e^{6}+6 a \,c^{2} d^{4} e^{4}+5 c^{3} d^{6} e^{2}}{1260 e^{9}}}{\left (e x +d \right )^{9}}\)	\(222\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 270, normalized size = 1.42 \begin {gather*} -\frac {420 \, c^{3} x^{6} e^{6} + 630 \, c^{3} d x^{5} e^{5} + 5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d^{2} e^{4} + 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} + 140 \, a^{3} e^{6} + 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 (x^{9} e^{16} + 9 \, d x^{8} e^{15} + 36 \, d^{2} x^{7} e^{14} + 84 \, d^{3} x^{6} e^{13} + 126 \, d^{4} x^{5} e^{12} + 126 \, d^{5} x^{4} e^{11} + 84 \, d^{6} x^{3} e^{10} + 36 \, d^{7} x^{2} e^{9} + 9 \, d^{8} x e^{8} + 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*x^6*e^6 + 630*c^3*d*x^5*e^5 + 5*c^3*d^6 + 6*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 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 + 140*a^3*e^6 + 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)/(x^9*e^16 + 9*d*x^8*e^15 +
36*d^2*x^7*e^14 + 84*d^3*x^6*e^13 + 126*d^4*x^5*e^12 + 126*d^5*x^4*e^11 + 84*d^6*x^3*e^10 + 36*d^7*x^2*e^9 + 9
*d^8*x*e^8 + d^9*e^7)

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Fricas [A]
time = 3.50, size = 268, normalized size = 1.41 \begin {gather*} -\frac {45 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 4 \, {\left (105 \, c^{3} x^{6} + 189 \, a c^{2} x^{4} + 135 \, a^{2} c x^{2} + 35 \, a^{3}\right )} e^{6} + 9 \, {\left (70 \, c^{3} d x^{5} + 56 \, a c^{2} d x^{3} + 15 \, a^{2} c d x\right )} e^{5} + 3 \, {\left (210 \, c^{3} d^{2} x^{4} + 72 \, a c^{2} d^{2} x^{2} + 5 \, a^{2} c d^{2}\right )} e^{4} + 6 \, {\left (70 \, c^{3} d^{3} x^{3} + 9 \, a c^{2} d^{3} x\right )} e^{3} + 6 \, {\left (30 \, c^{3} d^{4} x^{2} + a c^{2} d^{4}\right )} e^{2}}{1260 \, {\left (x^{9} e^{16} + 9 \, d x^{8} e^{15} + 36 \, d^{2} x^{7} e^{14} + 84 \, d^{3} x^{6} e^{13} + 126 \, d^{4} x^{5} e^{12} + 126 \, d^{5} x^{4} e^{11} + 84 \, d^{6} x^{3} e^{10} + 36 \, d^{7} x^{2} e^{9} + 9 \, d^{8} x e^{8} + 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*(45*c^3*d^5*x*e + 5*c^3*d^6 + 4*(105*c^3*x^6 + 189*a*c^2*x^4 + 135*a^2*c*x^2 + 35*a^3)*e^6 + 9*(70*c^3
*d*x^5 + 56*a*c^2*d*x^3 + 15*a^2*c*d*x)*e^5 + 3*(210*c^3*d^2*x^4 + 72*a*c^2*d^2*x^2 + 5*a^2*c*d^2)*e^4 + 6*(70
*c^3*d^3*x^3 + 9*a*c^2*d^3*x)*e^3 + 6*(30*c^3*d^4*x^2 + a*c^2*d^4)*e^2)/(x^9*e^16 + 9*d*x^8*e^15 + 36*d^2*x^7*
e^14 + 84*d^3*x^6*e^13 + 126*d^4*x^5*e^12 + 126*d^5*x^4*e^11 + 84*d^6*x^3*e^10 + 36*d^7*x^2*e^9 + 9*d^8*x*e^8
+ d^9*e^7)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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Giac [A]
time = 1.55, 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

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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)

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