3.5.86 \(\int \frac {(a+c x^2)^3}{(d+e x)^9} \, dx\) [486]

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

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 188, 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 )}{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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3/(d + e*x)^9,x]

[Out]

-1/8*(c*d^2 + a*e^2)^3/(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 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)^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*}

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Mathematica [A]
time = 0.04, size = 163, normalized size = 0.87 \begin {gather*} -\frac {35 a^3 e^6+5 a^2 c e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+3 a c^2 e^2 \left (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\right )+5 c^3 \left (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\right )}{280 e^7 (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/280*(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))/(e^7*(d + e*x)^8)

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Maple [A]
time = 0.43, size = 218, normalized size = 1.16

method	result	size
risch	\(\frac {-\frac {c^{3} x^{6}}{2 e}-\frac {c^{3} d \,x^{5}}{e^{2}}-\frac {c^{2} \left (3 e^{2} a +5 c \,d^{2}\right ) x^{4}}{4 e^{3}}-\frac {d \,c^{2} \left (3 e^{2} a +5 c \,d^{2}\right ) x^{3}}{5 e^{4}}-\frac {c \left (5 a^{2} e^{4}+3 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x^{2}}{10 e^{5}}-\frac {d c \left (5 a^{2} e^{4}+3 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x}{35 e^{6}}-\frac {35 e^{6} a^{3}+5 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a +5 d^{6} c^{3}}{280 e^{7}}}{\left (e x +d \right )^{8}}\)	\(199\)
gosper	\(-\frac {140 c^{3} x^{6} e^{6}+280 c^{3} d \,x^{5} e^{5}+210 a \,c^{2} e^{6} x^{4}+350 c^{3} d^{2} e^{4} x^{4}+168 a \,c^{2} d \,e^{5} x^{3}+280 c^{3} d^{3} e^{3} x^{3}+140 a^{2} c \,e^{6} x^{2}+84 a \,c^{2} d^{2} e^{4} x^{2}+140 c^{3} d^{4} e^{2} x^{2}+40 a^{2} c d \,e^{5} x +24 a \,c^{2} d^{3} e^{3} x +40 c^{3} d^{5} e x +35 e^{6} a^{3}+5 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a +5 d^{6} c^{3}}{280 e^{7} \left (e x +d \right )^{8}}\)	\(205\)
norman	\(\frac {-\frac {c^{3} x^{6}}{2 e}-\frac {c^{3} d \,x^{5}}{e^{2}}-\frac {\left (3 e^{3} c^{2} a +5 d^{2} e \,c^{3}\right ) x^{4}}{4 e^{4}}-\frac {d \left (3 e^{3} c^{2} a +5 d^{2} e \,c^{3}\right ) x^{3}}{5 e^{5}}-\frac {\left (5 e^{5} a^{2} c +3 d^{2} e^{3} c^{2} a +5 d^{4} e \,c^{3}\right ) x^{2}}{10 e^{6}}-\frac {d \left (5 e^{5} a^{2} c +3 d^{2} e^{3} c^{2} a +5 d^{4} e \,c^{3}\right ) x}{35 e^{7}}-\frac {35 a^{3} e^{7}+5 a^{2} c \,d^{2} e^{5}+3 a \,c^{2} d^{4} e^{3}+5 c^{3} d^{6} e}{280 e^{8}}}{\left (e x +d \right )^{8}}\)	\(212\)
default	\(\frac {2 c^{3} d}{e^{7} \left (e x +d \right )^{3}}+\frac {4 c^{2} d \left (3 e^{2} a +5 c \,d^{2}\right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {e^{6} a^{3}+3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{8 e^{7} \left (e x +d \right )^{8}}-\frac {c^{3}}{2 e^{7} \left (e x +d \right )^{2}}+\frac {6 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {3 c^{2} \left (e^{2} a +5 c \,d^{2}\right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{2 e^{7} \left (e x +d \right )^{6}}\)	\(218\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3/(e*x+d)^9,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 260, normalized size = 1.38 \begin {gather*} -\frac {140 \, c^{3} x^{6} e^{6} + 280 \, c^{3} d x^{5} e^{5} + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 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} + 35 \, a^{3} e^{6} + 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 (x^{8} e^{15} + 8 \, d x^{7} e^{14} + 28 \, d^{2} x^{6} e^{13} + 56 \, d^{3} x^{5} e^{12} + 70 \, d^{4} x^{4} e^{11} + 56 \, d^{5} x^{3} e^{10} + 28 \, d^{6} x^{2} e^{9} + 8 \, d^{7} x e^{8} + d^{8} 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)^9,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 3.03, size = 255, normalized size = 1.36 \begin {gather*} -\frac {40 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 35 \, {\left (4 \, c^{3} x^{6} + 6 \, a c^{2} x^{4} + 4 \, a^{2} c x^{2} + a^{3}\right )} e^{6} + 8 \, {\left (35 \, c^{3} d x^{5} + 21 \, a c^{2} d x^{3} + 5 \, a^{2} c d x\right )} e^{5} + {\left (350 \, c^{3} d^{2} x^{4} + 84 \, a c^{2} d^{2} x^{2} + 5 \, a^{2} c d^{2}\right )} e^{4} + 8 \, {\left (35 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{3} x\right )} e^{3} + {\left (140 \, c^{3} d^{4} x^{2} + 3 \, a c^{2} d^{4}\right )} e^{2}}{280 \, {\left (x^{8} e^{15} + 8 \, d x^{7} e^{14} + 28 \, d^{2} x^{6} e^{13} + 56 \, d^{3} x^{5} e^{12} + 70 \, d^{4} x^{4} e^{11} + 56 \, d^{5} x^{3} e^{10} + 28 \, d^{6} x^{2} e^{9} + 8 \, d^{7} x e^{8} + d^{8} 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)^9,x, algorithm="fricas")

[Out]

-1/280*(40*c^3*d^5*x*e + 5*c^3*d^6 + 35*(4*c^3*x^6 + 6*a*c^2*x^4 + 4*a^2*c*x^2 + a^3)*e^6 + 8*(35*c^3*d*x^5 +
21*a*c^2*d*x^3 + 5*a^2*c*d*x)*e^5 + (350*c^3*d^2*x^4 + 84*a*c^2*d^2*x^2 + 5*a^2*c*d^2)*e^4 + 8*(35*c^3*d^3*x^3
+ 3*a*c^2*d^3*x)*e^3 + (140*c^3*d^4*x^2 + 3*a*c^2*d^4)*e^2)/(x^8*e^15 + 8*d*x^7*e^14 + 28*d^2*x^6*e^13 + 56*d
^3*x^5*e^12 + 70*d^4*x^4*e^11 + 56*d^5*x^3*e^10 + 28*d^6*x^2*e^9 + 8*d^7*x*e^8 + d^8*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)**9,x)

[Out]

Timed out

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Giac [A]
time = 1.49, size = 191, normalized size = 1.02 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 0.33, size = 275, normalized size = 1.46 \begin {gather*} -\frac {\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\,e^7}+\frac {c^3\,x^6}{2\,e}+\frac {c^3\,d\,x^5}{e^2}+\frac {c^2\,x^4\,\left (5\,c\,d^2+3\,a\,e^2\right )}{4\,e^3}+\frac {c\,x^2\,\left (5\,a^2\,e^4+3\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{10\,e^5}+\frac {c\,d\,x\,\left (5\,a^2\,e^4+3\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{35\,e^6}+\frac {c^2\,d\,x^3\,\left (5\,c\,d^2+3\,a\,e^2\right )}{5\,e^4}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3/(d + e*x)^9,x)

[Out]

-((35*a^3*e^6 + 5*c^3*d^6 + 3*a*c^2*d^4*e^2 + 5*a^2*c*d^2*e^4)/(280*e^7) + (c^3*x^6)/(2*e) + (c^3*d*x^5)/e^2 +
(c^2*x^4*(3*a*e^2 + 5*c*d^2))/(4*e^3) + (c*x^2*(5*a^2*e^4 + 5*c^2*d^4 + 3*a*c*d^2*e^2))/(10*e^5) + (c*d*x*(5*
a^2*e^4 + 5*c^2*d^4 + 3*a*c*d^2*e^2))/(35*e^6) + (c^2*d*x^3*(3*a*e^2 + 5*c*d^2))/(5*e^4))/(d^8 + e^8*x^8 + 8*d
*e^7*x^7 + 28*d^6*e^2*x^2 + 56*d^5*e^3*x^3 + 70*d^4*e^4*x^4 + 56*d^3*e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*e*x)

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