3.493 \(\int \frac{1}{(d+e x)^3 \left (a+c x^2\right )} \, dx\)

Optimal. Leaf size=176 \[ \frac{c^{3/2} d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^2+c d^2\right )^3}-\frac{c e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}-\frac{2 c d e}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac{e}{2 (d+e x)^2 \left (a e^2+c d^2\right )}+\frac{c e \left (3 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^3} \]

[Out]

-e/(2*(c*d^2 + a*e^2)*(d + e*x)^2) - (2*c*d*e)/((c*d^2 + a*e^2)^2*(d + e*x)) + (
c^(3/2)*d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*(c*d^2 + a*e^2
)^3) + (c*e*(3*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^3 - (c*e*(3*c*d^2 -
a*e^2)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^3)

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Rubi [A]  time = 0.406056, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{c^{3/2} d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^2+c d^2\right )^3}-\frac{c e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}-\frac{2 c d e}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac{e}{2 (d+e x)^2 \left (a e^2+c d^2\right )}+\frac{c e \left (3 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(a + c*x^2)),x]

[Out]

-e/(2*(c*d^2 + a*e^2)*(d + e*x)^2) - (2*c*d*e)/((c*d^2 + a*e^2)^2*(d + e*x)) + (
c^(3/2)*d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*(c*d^2 + a*e^2
)^3) + (c*e*(3*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^3 - (c*e*(3*c*d^2 -
a*e^2)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^3)

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Rubi in Sympy [A]  time = 54.7813, size = 163, normalized size = 0.93 \[ - \frac{2 c d e}{\left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{2}} + \frac{c e \left (a e^{2} - 3 c d^{2}\right ) \log{\left (a + c x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{3}} - \frac{c e \left (a e^{2} - 3 c d^{2}\right ) \log{\left (d + e x \right )}}{\left (a e^{2} + c d^{2}\right )^{3}} - \frac{e}{2 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )} - \frac{c^{\frac{3}{2}} d \left (3 a e^{2} - c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{\sqrt{a} \left (a e^{2} + c d^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**3/(c*x**2+a),x)

[Out]

-2*c*d*e/((d + e*x)*(a*e**2 + c*d**2)**2) + c*e*(a*e**2 - 3*c*d**2)*log(a + c*x*
*2)/(2*(a*e**2 + c*d**2)**3) - c*e*(a*e**2 - 3*c*d**2)*log(d + e*x)/(a*e**2 + c*
d**2)**3 - e/(2*(d + e*x)**2*(a*e**2 + c*d**2)) - c**(3/2)*d*(3*a*e**2 - c*d**2)
*atan(sqrt(c)*x/sqrt(a))/(sqrt(a)*(a*e**2 + c*d**2)**3)

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Mathematica [A]  time = 0.569542, size = 140, normalized size = 0.8 \[ \frac{\frac{2 c^{3/2} d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a}}+e \left (c \left (a e^2-3 c d^2\right ) \log \left (a+c x^2\right )-\frac{\left (a e^2+c d^2\right ) \left (a e^2+c d (5 d+4 e x)\right )}{(d+e x)^2}+2 c \left (3 c d^2-a e^2\right ) \log (d+e x)\right )}{2 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(a + c*x^2)),x]

[Out]

((2*c^(3/2)*d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[a] + e*(-(((c*
d^2 + a*e^2)*(a*e^2 + c*d*(5*d + 4*e*x)))/(d + e*x)^2) + 2*c*(3*c*d^2 - a*e^2)*L
og[d + e*x] + c*(-3*c*d^2 + a*e^2)*Log[a + c*x^2]))/(2*(c*d^2 + a*e^2)^3)

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Maple [A]  time = 0.013, size = 233, normalized size = 1.3 \[ -{\frac{e}{ \left ( 2\,a{e}^{2}+2\,c{d}^{2} \right ) \left ( ex+d \right ) ^{2}}}-2\,{\frac{cde}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( ex+d \right ) }}-{\frac{c{e}^{3}\ln \left ( ex+d \right ) a}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}+3\,{\frac{e{c}^{2}\ln \left ( ex+d \right ){d}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}+{\frac{c\ln \left ( c{x}^{2}+a \right ){e}^{3}a}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}-{\frac{3\,{c}^{2}\ln \left ( c{x}^{2}+a \right ){d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}-3\,{\frac{d{e}^{2}a{c}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{{c}^{3}{d}^{3}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(c*x^2+a),x)

[Out]

-1/2*e/(a*e^2+c*d^2)/(e*x+d)^2-2*c*d*e/(a*e^2+c*d^2)^2/(e*x+d)-e^3*c/(a*e^2+c*d^
2)^3*ln(e*x+d)*a+3*e*c^2/(a*e^2+c*d^2)^3*ln(e*x+d)*d^2+1/2*c/(a*e^2+c*d^2)^3*ln(
c*x^2+a)*e^3*a-3/2*c^2/(a*e^2+c*d^2)^3*ln(c*x^2+a)*d^2*e-3*c^2/(a*e^2+c*d^2)^3/(
a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*a*d*e^2+c^3/(a*e^2+c*d^2)^3/(a*c)^(1/2)*arcta
n(c*x/(a*c)^(1/2))*d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.24518, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)*(e*x + d)^3),x, algorithm="fricas")

[Out]

[-1/2*(5*c^2*d^4*e + 6*a*c*d^2*e^3 + a^2*e^5 + (c^2*d^5 - 3*a*c*d^3*e^2 + (c^2*d
^3*e^2 - 3*a*c*d*e^4)*x^2 + 2*(c^2*d^4*e - 3*a*c*d^2*e^3)*x)*sqrt(-c/a)*log((c*x
^2 - 2*a*x*sqrt(-c/a) - a)/(c*x^2 + a)) + 4*(c^2*d^3*e^2 + a*c*d*e^4)*x + (3*c^2
*d^4*e - a*c*d^2*e^3 + (3*c^2*d^2*e^3 - a*c*e^5)*x^2 + 2*(3*c^2*d^3*e^2 - a*c*d*
e^4)*x)*log(c*x^2 + a) - 2*(3*c^2*d^4*e - a*c*d^2*e^3 + (3*c^2*d^2*e^3 - a*c*e^5
)*x^2 + 2*(3*c^2*d^3*e^2 - a*c*d*e^4)*x)*log(e*x + d))/(c^3*d^8 + 3*a*c^2*d^6*e^
2 + 3*a^2*c*d^4*e^4 + a^3*d^2*e^6 + (c^3*d^6*e^2 + 3*a*c^2*d^4*e^4 + 3*a^2*c*d^2
*e^6 + a^3*e^8)*x^2 + 2*(c^3*d^7*e + 3*a*c^2*d^5*e^3 + 3*a^2*c*d^3*e^5 + a^3*d*e
^7)*x), -1/2*(5*c^2*d^4*e + 6*a*c*d^2*e^3 + a^2*e^5 - 2*(c^2*d^5 - 3*a*c*d^3*e^2
+ (c^2*d^3*e^2 - 3*a*c*d*e^4)*x^2 + 2*(c^2*d^4*e - 3*a*c*d^2*e^3)*x)*sqrt(c/a)*
arctan(c*x/(a*sqrt(c/a))) + 4*(c^2*d^3*e^2 + a*c*d*e^4)*x + (3*c^2*d^4*e - a*c*d
^2*e^3 + (3*c^2*d^2*e^3 - a*c*e^5)*x^2 + 2*(3*c^2*d^3*e^2 - a*c*d*e^4)*x)*log(c*
x^2 + a) - 2*(3*c^2*d^4*e - a*c*d^2*e^3 + (3*c^2*d^2*e^3 - a*c*e^5)*x^2 + 2*(3*c
^2*d^3*e^2 - a*c*d*e^4)*x)*log(e*x + d))/(c^3*d^8 + 3*a*c^2*d^6*e^2 + 3*a^2*c*d^
4*e^4 + a^3*d^2*e^6 + (c^3*d^6*e^2 + 3*a*c^2*d^4*e^4 + 3*a^2*c*d^2*e^6 + a^3*e^8
)*x^2 + 2*(c^3*d^7*e + 3*a*c^2*d^5*e^3 + 3*a^2*c*d^3*e^5 + a^3*d*e^7)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**3/(c*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214169, size = 363, normalized size = 2.06 \[ -\frac{{\left (3 \, c^{2} d^{2} e - a c e^{3}\right )}{\rm ln}\left (c x^{2} + a\right )}{2 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac{{\left (3 \, c^{2} d^{2} e^{2} - a c e^{4}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}} + \frac{{\left (c^{3} d^{3} - 3 \, a c^{2} d e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt{a c}} - \frac{5 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} + a^{2} e^{5} + 4 \,{\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x}{2 \,{\left (c d^{2} + a e^{2}\right )}^{3}{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)*(e*x + d)^3),x, algorithm="giac")

[Out]

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