3.5.6 \(\int \frac {(a+c x^2)^3}{(d+e x)^9} \, dx\)

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

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Rubi [A]  time = 0.13, 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 = {697} \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]

-(c*d^2 + a*e^2)^3/(8*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 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)^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 = 0.06, 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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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)^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 282, normalized size = 1.50 \begin {gather*} -\frac {140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} + 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} + 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 (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + 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*(140*c^3*e^6*x^6 + 280*c^3*d*e^5*x^5 + 5*c^3*d^6 + 3*a*c^2*d^4*e^2 + 5*a^2*c*d^2*e^4 + 35*a^3*e^6 + 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 + 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)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2
*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

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giac [A]  time = 0.22, 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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maple [A]  time = 0.05, size = 218, normalized size = 1.16 \begin {gather*} \frac {2 c^{3} d}{\left (e x +d \right )^{3} e^{7}}-\frac {c^{3}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {4 \left (3 a \,e^{2}+5 c \,d^{2}\right ) c^{2} d}{5 \left (e x +d \right )^{5} e^{7}}-\frac {3 \left (a \,e^{2}+5 c \,d^{2}\right ) c^{2}}{4 \left (e x +d \right )^{4} e^{7}}+\frac {6 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) c d}{7 \left (e x +d \right )^{7} e^{7}}-\frac {\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) c}{2 \left (e x +d \right )^{6} 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}}{8 \left (e x +d \right )^{8} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.52, size = 282, normalized size = 1.50 \begin {gather*} -\frac {140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} + 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} + 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 (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + 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*e^6*x^6 + 280*c^3*d*e^5*x^5 + 5*c^3*d^6 + 3*a*c^2*d^4*e^2 + 5*a^2*c*d^2*e^4 + 35*a^3*e^6 + 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 + 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)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2
*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

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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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sympy [A]  time = 19.81, size = 301, normalized size = 1.60 \begin {gather*} \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 c^{3} d e^{5} x^{5} - 140 c^{3} e^{6} x^{6} + x^{4} \left (- 210 a c^{2} e^{6} - 350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 168 a c^{2} d e^{5} - 280 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 140 a^{2} c e^{6} - 84 a c^{2} d^{2} e^{4} - 140 c^{3} d^{4} e^{2}\right ) + x \left (- 40 a^{2} c d e^{5} - 24 a c^{2} d^{3} e^{3} - 40 c^{3} d^{5} e\right )}{280 d^{8} e^{7} + 2240 d^{7} e^{8} x + 7840 d^{6} e^{9} x^{2} + 15680 d^{5} e^{10} x^{3} + 19600 d^{4} e^{11} x^{4} + 15680 d^{3} e^{12} x^{5} + 7840 d^{2} e^{13} x^{6} + 2240 d e^{14} x^{7} + 280 e^{15} x^{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)

[Out]

(-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*c**3*d*e**5*x**5 - 140*c**3*e**6*
x**6 + x**4*(-210*a*c**2*e**6 - 350*c**3*d**2*e**4) + x**3*(-168*a*c**2*d*e**5 - 280*c**3*d**3*e**3) + x**2*(-
140*a**2*c*e**6 - 84*a*c**2*d**2*e**4 - 140*c**3*d**4*e**2) + x*(-40*a**2*c*d*e**5 - 24*a*c**2*d**3*e**3 - 40*
c**3*d**5*e))/(280*d**8*e**7 + 2240*d**7*e**8*x + 7840*d**6*e**9*x**2 + 15680*d**5*e**10*x**3 + 19600*d**4*e**
11*x**4 + 15680*d**3*e**12*x**5 + 7840*d**2*e**13*x**6 + 2240*d*e**14*x**7 + 280*e**15*x**8)

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