3.5.5 \(\int \frac {(a+c x^2)^3}{(d+e x)^8} \, dx\)

Optimal. Leaf size=178 \[ -\frac {c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac {c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^4}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac {c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^6}-\frac {\left (a e^2+c d^2\right )^3}{7 e^7 (d+e x)^7}-\frac {c^3}{e^7 (d+e x)}+\frac {3 c^3 d}{e^7 (d+e x)^2} \]

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Rubi [A]  time = 0.12, antiderivative size = 178, 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 {c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac {c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^4}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac {c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^6}-\frac {\left (a e^2+c d^2\right )^3}{7 e^7 (d+e x)^7}-\frac {c^3}{e^7 (d+e x)}+\frac {3 c^3 d}{e^7 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3/(d + e*x)^8,x]

[Out]

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

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

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Mathematica [A]  time = 0.05, size = 161, normalized size = 0.90 \begin {gather*} -\frac {5 a^3 e^6+a^2 c e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a c^2 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )}{35 e^7 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^3/(d + e*x)^8,x]

[Out]

-1/35*(5*a^3*e^6 + a^2*c*e^4*(d^2 + 7*d*e*x + 21*e^2*x^2) + a*c^2*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d
*e^3*x^3 + 35*e^4*x^4) + 5*c^3*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*
x^5 + 7*e^6*x^6))/(e^7*(d + e*x)^7)

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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)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + c*x^2)^3/(d + e*x)^8,x]

[Out]

IntegrateAlgebraic[(a + c*x^2)^3/(d + e*x)^8, x]

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fricas [A]  time = 0.39, size = 263, normalized size = 1.48 \begin {gather*} -\frac {35 \, c^{3} e^{6} x^{6} + 105 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 35 \, {\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 35 \, {\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 21 \, {\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 7 \, {\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{35 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} 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)^8,x, algorithm="fricas")

[Out]

-1/35*(35*c^3*e^6*x^6 + 105*c^3*d*e^5*x^5 + 5*c^3*d^6 + a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + 5*a^3*e^6 + 35*(5*c^3*
d^2*e^4 + a*c^2*e^6)*x^4 + 35*(5*c^3*d^3*e^3 + a*c^2*d*e^5)*x^3 + 21*(5*c^3*d^4*e^2 + a*c^2*d^2*e^4 + a^2*c*e^
6)*x^2 + 7*(5*c^3*d^5*e + a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*
e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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giac [A]  time = 0.17, size = 189, normalized size = 1.06 \begin {gather*} -\frac {{\left (35 \, c^{3} x^{6} e^{6} + 105 \, c^{3} d x^{5} e^{5} + 175 \, c^{3} d^{2} x^{4} e^{4} + 175 \, c^{3} d^{3} x^{3} e^{3} + 105 \, c^{3} d^{4} x^{2} e^{2} + 35 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 35 \, a c^{2} x^{4} e^{6} + 35 \, a c^{2} d x^{3} e^{5} + 21 \, a c^{2} d^{2} x^{2} e^{4} + 7 \, a c^{2} d^{3} x e^{3} + a c^{2} d^{4} e^{2} + 21 \, a^{2} c x^{2} e^{6} + 7 \, a^{2} c d x e^{5} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{35 \, {\left (x e + d\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/35*(35*c^3*x^6*e^6 + 105*c^3*d*x^5*e^5 + 175*c^3*d^2*x^4*e^4 + 175*c^3*d^3*x^3*e^3 + 105*c^3*d^4*x^2*e^2 +
35*c^3*d^5*x*e + 5*c^3*d^6 + 35*a*c^2*x^4*e^6 + 35*a*c^2*d*x^3*e^5 + 21*a*c^2*d^2*x^2*e^4 + 7*a*c^2*d^3*x*e^3
+ a*c^2*d^4*e^2 + 21*a^2*c*x^2*e^6 + 7*a^2*c*d*x*e^5 + a^2*c*d^2*e^4 + 5*a^3*e^6)*e^(-7)/(x*e + d)^7

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maple [A]  time = 0.05, size = 216, normalized size = 1.21 \begin {gather*} \frac {3 c^{3} d}{\left (e x +d \right )^{2} e^{7}}-\frac {c^{3}}{\left (e x +d \right ) e^{7}}+\frac {\left (3 a \,e^{2}+5 c \,d^{2}\right ) c^{2} d}{\left (e x +d \right )^{4} e^{7}}-\frac {\left (a \,e^{2}+5 c \,d^{2}\right ) c^{2}}{\left (e x +d \right )^{3} e^{7}}+\frac {\left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) c d}{\left (e x +d \right )^{6} e^{7}}-\frac {3 \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) c}{5 \left (e x +d \right )^{5} 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}}{7 \left (e x +d \right )^{7} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3/(e*x+d)^8,x)

[Out]

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

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maxima [A]  time = 1.55, size = 263, normalized size = 1.48 \begin {gather*} -\frac {35 \, c^{3} e^{6} x^{6} + 105 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 35 \, {\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 35 \, {\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 21 \, {\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 7 \, {\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{35 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} 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)^8,x, algorithm="maxima")

[Out]

-1/35*(35*c^3*e^6*x^6 + 105*c^3*d*e^5*x^5 + 5*c^3*d^6 + a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + 5*a^3*e^6 + 35*(5*c^3*
d^2*e^4 + a*c^2*e^6)*x^4 + 35*(5*c^3*d^3*e^3 + a*c^2*d*e^5)*x^3 + 21*(5*c^3*d^4*e^2 + a*c^2*d^2*e^4 + a^2*c*e^
6)*x^2 + 7*(5*c^3*d^5*e + a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*
e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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mupad [B]  time = 0.08, size = 254, normalized size = 1.43 \begin {gather*} -\frac {\frac {5\,a^3\,e^6+a^2\,c\,d^2\,e^4+a\,c^2\,d^4\,e^2+5\,c^3\,d^6}{35\,e^7}+\frac {c^3\,x^6}{e}+\frac {3\,c^3\,d\,x^5}{e^2}+\frac {c^2\,x^4\,\left (5\,c\,d^2+a\,e^2\right )}{e^3}+\frac {3\,c\,x^2\,\left (a^2\,e^4+a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{5\,e^5}+\frac {c\,d\,x\,\left (a^2\,e^4+a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{5\,e^6}+\frac {c^2\,d\,x^3\,\left (5\,c\,d^2+a\,e^2\right )}{e^4}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3/(d + e*x)^8,x)

[Out]

-((5*a^3*e^6 + 5*c^3*d^6 + a*c^2*d^4*e^2 + a^2*c*d^2*e^4)/(35*e^7) + (c^3*x^6)/e + (3*c^3*d*x^5)/e^2 + (c^2*x^
4*(a*e^2 + 5*c*d^2))/e^3 + (3*c*x^2*(a^2*e^4 + 5*c^2*d^4 + a*c*d^2*e^2))/(5*e^5) + (c*d*x*(a^2*e^4 + 5*c^2*d^4
+ a*c*d^2*e^2))/(5*e^6) + (c^2*d*x^3*(a*e^2 + 5*c*d^2))/e^4)/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 +
35*d^4*e^3*x^3 + 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)

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sympy [A]  time = 11.35, size = 286, normalized size = 1.61 \begin {gather*} \frac {- 5 a^{3} e^{6} - a^{2} c d^{2} e^{4} - a c^{2} d^{4} e^{2} - 5 c^{3} d^{6} - 105 c^{3} d e^{5} x^{5} - 35 c^{3} e^{6} x^{6} + x^{4} \left (- 35 a c^{2} e^{6} - 175 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 35 a c^{2} d e^{5} - 175 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 21 a^{2} c e^{6} - 21 a c^{2} d^{2} e^{4} - 105 c^{3} d^{4} e^{2}\right ) + x \left (- 7 a^{2} c d e^{5} - 7 a c^{2} d^{3} e^{3} - 35 c^{3} d^{5} e\right )}{35 d^{7} e^{7} + 245 d^{6} e^{8} x + 735 d^{5} e^{9} x^{2} + 1225 d^{4} e^{10} x^{3} + 1225 d^{3} e^{11} x^{4} + 735 d^{2} e^{12} x^{5} + 245 d e^{13} x^{6} + 35 e^{14} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**8,x)

[Out]

(-5*a**3*e**6 - a**2*c*d**2*e**4 - a*c**2*d**4*e**2 - 5*c**3*d**6 - 105*c**3*d*e**5*x**5 - 35*c**3*e**6*x**6 +
x**4*(-35*a*c**2*e**6 - 175*c**3*d**2*e**4) + x**3*(-35*a*c**2*d*e**5 - 175*c**3*d**3*e**3) + x**2*(-21*a**2*
c*e**6 - 21*a*c**2*d**2*e**4 - 105*c**3*d**4*e**2) + x*(-7*a**2*c*d*e**5 - 7*a*c**2*d**3*e**3 - 35*c**3*d**5*e
))/(35*d**7*e**7 + 245*d**6*e**8*x + 735*d**5*e**9*x**2 + 1225*d**4*e**10*x**3 + 1225*d**3*e**11*x**4 + 735*d*
*2*e**12*x**5 + 245*d*e**13*x**6 + 35*e**14*x**7)

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