3.483 \(\int \frac {(a+c x^2)^3}{(d+e x)^6} \, dx\)

Optimal. Leaf size=172 \[ -\frac {3 c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac {2 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^2}-\frac {c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac {3 c d \left (a e^2+c d^2\right )^2}{2 e^7 (d+e x)^4}-\frac {\left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac {6 c^3 d \log (d+e x)}{e^7}+\frac {c^3 x}{e^6} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 172, 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} \[ -\frac {3 c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac {2 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^2}-\frac {c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac {3 c d \left (a e^2+c d^2\right )^2}{2 e^7 (d+e x)^4}-\frac {\left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac {6 c^3 d \log (d+e x)}{e^7}+\frac {c^3 x}{e^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3/(d + e*x)^6,x]

[Out]

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

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

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Mathematica [A]  time = 0.08, size = 182, normalized size = 1.06 \[ -\frac {2 a^3 e^6+a^2 c e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )+60 c^3 d (d+e x)^5 \log (d+e x)}{10 e^7 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^3/(d + e*x)^6,x]

[Out]

-1/10*(2*a^3*e^6 + a^2*c*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 6*a*c^2*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10
*d*e^3*x^3 + 5*e^4*x^4) + c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*
d*e^5*x^5 - 10*e^6*x^6) + 60*c^3*d*(d + e*x)^5*Log[d + e*x])/(e^7*(d + e*x)^5)

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fricas [A]  time = 0.92, size = 326, normalized size = 1.90 \[ \frac {10 \, c^{3} e^{6} x^{6} + 50 \, c^{3} d e^{5} x^{5} - 87 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} - 10 \, {\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} - 20 \, {\left (20 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} - 10 \, {\left (60 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} - 5 \, {\left (75 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x - 60 \, {\left (c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{2} e^{4} x^{4} + 10 \, c^{3} d^{3} e^{3} x^{3} + 10 \, c^{3} d^{4} e^{2} x^{2} + 5 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{10 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^6,x, algorithm="fricas")

[Out]

1/10*(10*c^3*e^6*x^6 + 50*c^3*d*e^5*x^5 - 87*c^3*d^6 - 6*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 2*a^3*e^6 - 10*(5*c^3
*d^2*e^4 + 3*a*c^2*e^6)*x^4 - 20*(20*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 - 10*(60*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 +
a^2*c*e^6)*x^2 - 5*(75*c^3*d^5*e + 6*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x - 60*(c^3*d*e^5*x^5 + 5*c^3*d^2*e^4*x^4 +
10*c^3*d^3*e^3*x^3 + 10*c^3*d^4*e^2*x^2 + 5*c^3*d^5*e*x + c^3*d^6)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 1
0*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)

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giac [A]  time = 0.16, size = 188, normalized size = 1.09 \[ -6 \, c^{3} d e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + c^{3} x e^{\left (-6\right )} - \frac {{\left (87 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 30 \, {\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 20 \, {\left (25 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 2 \, a^{3} e^{6} + 10 \, {\left (65 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 5 \, {\left (77 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-7\right )}}{10 \, {\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^6,x, algorithm="giac")

[Out]

-6*c^3*d*e^(-7)*log(abs(x*e + d)) + c^3*x*e^(-6) - 1/10*(87*c^3*d^6 + 6*a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + 30*(5*
c^3*d^2*e^4 + a*c^2*e^6)*x^4 + 20*(25*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 2*a^3*e^6 + 10*(65*c^3*d^4*e^2 + 6*a*
c^2*d^2*e^4 + a^2*c*e^6)*x^2 + 5*(77*c^3*d^5*e + 6*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)*e^(-7)/(x*e + d)^5

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maple [A]  time = 0.05, size = 272, normalized size = 1.58 \[ -\frac {a^{3}}{5 \left (e x +d \right )^{5} e}-\frac {3 a^{2} c \,d^{2}}{5 \left (e x +d \right )^{5} e^{3}}-\frac {3 a \,c^{2} d^{4}}{5 \left (e x +d \right )^{5} e^{5}}-\frac {c^{3} d^{6}}{5 \left (e x +d \right )^{5} e^{7}}+\frac {3 a^{2} c d}{2 \left (e x +d \right )^{4} e^{3}}+\frac {3 a \,c^{2} d^{3}}{\left (e x +d \right )^{4} e^{5}}+\frac {3 c^{3} d^{5}}{2 \left (e x +d \right )^{4} e^{7}}-\frac {a^{2} c}{\left (e x +d \right )^{3} e^{3}}-\frac {6 a \,c^{2} d^{2}}{\left (e x +d \right )^{3} e^{5}}-\frac {5 c^{3} d^{4}}{\left (e x +d \right )^{3} e^{7}}+\frac {6 a \,c^{2} d}{\left (e x +d \right )^{2} e^{5}}+\frac {10 c^{3} d^{3}}{\left (e x +d \right )^{2} e^{7}}-\frac {3 a \,c^{2}}{\left (e x +d \right ) e^{5}}-\frac {15 c^{3} d^{2}}{\left (e x +d \right ) e^{7}}-\frac {6 c^{3} d \ln \left (e x +d \right )}{e^{7}}+\frac {c^{3} x}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3/(e*x+d)^6,x)

[Out]

c^3/e^6*x+6*c^2*d/e^5/(e*x+d)^2*a+10*c^3*d^3/e^7/(e*x+d)^2+3/2*c*d/e^3/(e*x+d)^4*a^2+3*c^2*d^3/e^5/(e*x+d)^4*a
+3/2*c^3*d^5/e^7/(e*x+d)^4-3*c^2/e^5/(e*x+d)*a-15*c^3/e^7/(e*x+d)*d^2-6*c^3*d/e^7*ln(e*x+d)-c/e^3/(e*x+d)^3*a^
2-6*c^2/e^5/(e*x+d)^3*a*d^2-5*c^3/e^7/(e*x+d)^3*d^4-1/5/e/(e*x+d)^5*a^3-3/5/e^3/(e*x+d)^5*d^2*a^2*c-3/5/e^5/(e
*x+d)^5*d^4*a*c^2-1/5/e^7/(e*x+d)^5*c^3*d^6

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maxima [A]  time = 1.52, size = 246, normalized size = 1.43 \[ -\frac {87 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} + 30 \, {\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 20 \, {\left (25 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 10 \, {\left (65 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 5 \, {\left (77 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{10 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} + \frac {c^{3} x}{e^{6}} - \frac {6 \, c^{3} d \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/10*(87*c^3*d^6 + 6*a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + 2*a^3*e^6 + 30*(5*c^3*d^2*e^4 + a*c^2*e^6)*x^4 + 20*(25*
c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 10*(65*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)*x^2 + 5*(77*c^3*d^5*e + 6
*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d
^5*e^7) + c^3*x/e^6 - 6*c^3*d*log(e*x + d)/e^7

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mupad [B]  time = 0.36, size = 247, normalized size = 1.44 \[ \frac {c^3\,x}{e^6}-\frac {x^2\,\left (a^2\,c\,e^5+6\,a\,c^2\,d^2\,e^3+65\,c^3\,d^4\,e\right )+\frac {2\,a^3\,e^6+a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2+87\,c^3\,d^6}{10\,e}+x^4\,\left (15\,c^3\,d^2\,e^3+3\,a\,c^2\,e^5\right )+x\,\left (\frac {a^2\,c\,d\,e^4}{2}+3\,a\,c^2\,d^3\,e^2+\frac {77\,c^3\,d^5}{2}\right )+x^3\,\left (50\,c^3\,d^3\,e^2+6\,a\,c^2\,d\,e^4\right )}{d^5\,e^6+5\,d^4\,e^7\,x+10\,d^3\,e^8\,x^2+10\,d^2\,e^9\,x^3+5\,d\,e^{10}\,x^4+e^{11}\,x^5}-\frac {6\,c^3\,d\,\ln \left (d+e\,x\right )}{e^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3/(d + e*x)^6,x)

[Out]

(c^3*x)/e^6 - (x^2*(a^2*c*e^5 + 65*c^3*d^4*e + 6*a*c^2*d^2*e^3) + (2*a^3*e^6 + 87*c^3*d^6 + 6*a*c^2*d^4*e^2 +
a^2*c*d^2*e^4)/(10*e) + x^4*(3*a*c^2*e^5 + 15*c^3*d^2*e^3) + x*((77*c^3*d^5)/2 + 3*a*c^2*d^3*e^2 + (a^2*c*d*e^
4)/2) + x^3*(50*c^3*d^3*e^2 + 6*a*c^2*d*e^4))/(d^5*e^6 + e^11*x^5 + 5*d^4*e^7*x + 5*d*e^10*x^4 + 10*d^3*e^8*x^
2 + 10*d^2*e^9*x^3) - (6*c^3*d*log(d + e*x))/e^7

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sympy [A]  time = 4.51, size = 264, normalized size = 1.53 \[ - \frac {6 c^{3} d \log {\left (d + e x \right )}}{e^{7}} + \frac {c^{3} x}{e^{6}} + \frac {- 2 a^{3} e^{6} - a^{2} c d^{2} e^{4} - 6 a c^{2} d^{4} e^{2} - 87 c^{3} d^{6} + x^{4} \left (- 30 a c^{2} e^{6} - 150 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 60 a c^{2} d e^{5} - 500 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 10 a^{2} c e^{6} - 60 a c^{2} d^{2} e^{4} - 650 c^{3} d^{4} e^{2}\right ) + x \left (- 5 a^{2} c d e^{5} - 30 a c^{2} d^{3} e^{3} - 385 c^{3} d^{5} e\right )}{10 d^{5} e^{7} + 50 d^{4} e^{8} x + 100 d^{3} e^{9} x^{2} + 100 d^{2} e^{10} x^{3} + 50 d e^{11} x^{4} + 10 e^{12} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**6,x)

[Out]

-6*c**3*d*log(d + e*x)/e**7 + c**3*x/e**6 + (-2*a**3*e**6 - a**2*c*d**2*e**4 - 6*a*c**2*d**4*e**2 - 87*c**3*d*
*6 + x**4*(-30*a*c**2*e**6 - 150*c**3*d**2*e**4) + x**3*(-60*a*c**2*d*e**5 - 500*c**3*d**3*e**3) + x**2*(-10*a
**2*c*e**6 - 60*a*c**2*d**2*e**4 - 650*c**3*d**4*e**2) + x*(-5*a**2*c*d*e**5 - 30*a*c**2*d**3*e**3 - 385*c**3*
d**5*e))/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**1
2*x**5)

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