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

Optimal. Leaf size=275 \[ -\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{3/2} d^3}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{5/2} d}+\frac{2 c e x}{a^2 d^2 \sqrt{a+c x^2}}-\frac{3 \sqrt{a+c x^2}}{2 a^2 d x^2}+\frac{e^2}{a d^3 \sqrt{a+c x^2}}+\frac{e}{a d^2 x \sqrt{a+c x^2}}+\frac{e^5 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3 \left (a e^2+c d^2\right )^{3/2}}-\frac{e^3 (a e+c d x)}{a d^3 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{1}{a d x^2 \sqrt{a+c x^2}} \]

[Out]

e^2/(a*d^3*Sqrt[a + c*x^2]) + 1/(a*d*x^2*Sqrt[a + c*x^2]) + e/(a*d^2*x*Sqrt[a +
c*x^2]) + (2*c*e*x)/(a^2*d^2*Sqrt[a + c*x^2]) - (e^3*(a*e + c*d*x))/(a*d^3*(c*d^
2 + a*e^2)*Sqrt[a + c*x^2]) - (3*Sqrt[a + c*x^2])/(2*a^2*d*x^2) + (e^5*ArcTanh[(
a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^3*(c*d^2 + a*e^2)^(3/2))
+ (3*c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(5/2)*d) - (e^2*ArcTanh[Sqrt[a +
c*x^2]/Sqrt[a]])/(a^(3/2)*d^3)

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Rubi [A]  time = 0.539899, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ -\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{3/2} d^3}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{5/2} d}+\frac{2 c e x}{a^2 d^2 \sqrt{a+c x^2}}-\frac{3 \sqrt{a+c x^2}}{2 a^2 d x^2}+\frac{e^2}{a d^3 \sqrt{a+c x^2}}+\frac{e}{a d^2 x \sqrt{a+c x^2}}+\frac{e^5 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3 \left (a e^2+c d^2\right )^{3/2}}-\frac{e^3 (a e+c d x)}{a d^3 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{1}{a d x^2 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

e^2/(a*d^3*Sqrt[a + c*x^2]) + 1/(a*d*x^2*Sqrt[a + c*x^2]) + e/(a*d^2*x*Sqrt[a +
c*x^2]) + (2*c*e*x)/(a^2*d^2*Sqrt[a + c*x^2]) - (e^3*(a*e + c*d*x))/(a*d^3*(c*d^
2 + a*e^2)*Sqrt[a + c*x^2]) - (3*Sqrt[a + c*x^2])/(2*a^2*d*x^2) + (e^5*ArcTanh[(
a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^3*(c*d^2 + a*e^2)^(3/2))
+ (3*c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(5/2)*d) - (e^2*ArcTanh[Sqrt[a +
c*x^2]/Sqrt[a]])/(a^(3/2)*d^3)

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Rubi in Sympy [A]  time = 57.6849, size = 250, normalized size = 0.91 \[ \frac{e^{5} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d^{3} \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} + \frac{1}{a d x^{2} \sqrt{a + c x^{2}}} + \frac{e}{a d^{2} x \sqrt{a + c x^{2}}} - \frac{e^{3} \left (a e + c d x\right )}{a d^{3} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} + \frac{e^{2}}{a d^{3} \sqrt{a + c x^{2}}} + \frac{2 c e x}{a^{2} d^{2} \sqrt{a + c x^{2}}} - \frac{3 \sqrt{a + c x^{2}}}{2 a^{2} d x^{2}} - \frac{e^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} d^{3}} + \frac{3 c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**5*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(d**3*(a*e**2
+ c*d**2)**(3/2)) + 1/(a*d*x**2*sqrt(a + c*x**2)) + e/(a*d**2*x*sqrt(a + c*x**2
)) - e**3*(a*e + c*d*x)/(a*d**3*sqrt(a + c*x**2)*(a*e**2 + c*d**2)) + e**2/(a*d*
*3*sqrt(a + c*x**2)) + 2*c*e*x/(a**2*d**2*sqrt(a + c*x**2)) - 3*sqrt(a + c*x**2)
/(2*a**2*d*x**2) - e**2*atanh(sqrt(a + c*x**2)/sqrt(a))/(a**(3/2)*d**3) + 3*c*at
anh(sqrt(a + c*x**2)/sqrt(a))/(2*a**(5/2)*d)

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Mathematica [A]  time = 0.87437, size = 226, normalized size = 0.82 \[ \frac{1}{2} \left (\frac{\left (3 c d^2-2 a e^2\right ) \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{a^{5/2} d^3}+\frac{\log (x) \left (2 a e^2-3 c d^2\right )}{a^{5/2} d^3}-\frac{\sqrt{a+c x^2} \left (\frac{2 c^2 (d-e x)}{\left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac{2 e}{d^2 x}+\frac{1}{d x^2}\right )}{a^2}+\frac{2 e^5 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{d^3 \left (a e^2+c d^2\right )^{3/2}}-\frac{2 e^5 \log (d+e x)}{d^3 \left (a e^2+c d^2\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-((Sqrt[a + c*x^2]*(1/(d*x^2) - (2*e)/(d^2*x) + (2*c^2*(d - e*x))/((c*d^2 + a*e
^2)*(a + c*x^2))))/a^2) + ((-3*c*d^2 + 2*a*e^2)*Log[x])/(a^(5/2)*d^3) - (2*e^5*L
og[d + e*x])/(d^3*(c*d^2 + a*e^2)^(3/2)) + ((3*c*d^2 - 2*a*e^2)*Log[a + Sqrt[a]*
Sqrt[a + c*x^2]])/(a^(5/2)*d^3) + (2*e^5*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*S
qrt[a + c*x^2]])/(d^3*(c*d^2 + a*e^2)^(3/2)))/2

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Maple [A]  time = 0.019, size = 439, normalized size = 1.6 \[ -{\frac{1}{2\,ad{x}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{3\,c}{2\,d{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{3\,c}{2\,d}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{{e}^{2}}{a{d}^{3}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{{e}^{2}}{{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{{e}^{4}}{{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{e}^{3}cx}{{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{e}^{4}}{{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{e}{a{d}^{2}x}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+2\,{\frac{cex}{{a}^{2}{d}^{2}\sqrt{c{x}^{2}+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*x^2 + a)^(3/2)*(e*x + d)*x^3), x)

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Fricas [A]  time = 0.887489, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(2*(a*c*d^4 + a^2*d^2*e^2 - 2*(2*c^2*d^3*e + a*c*d*e^3)*x^3 + (3*c^2*d^4 +
a*c*d^2*e^2)*x^2 - 2*(a*c*d^3*e + a^2*d*e^3)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2
+ a)*sqrt(a) - 2*(a^2*c*e^5*x^4 + a^3*e^5*x^2)*sqrt(a)*log(((2*a*c*d*e*x - a*c*d
^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2)*sqrt(c*d^2 + a*e^2) - 2*(a*c*d^2*e +
a^2*e^3 - (c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2))
+ ((3*c^3*d^4 + a*c^2*d^2*e^2 - 2*a^2*c*e^4)*x^4 + (3*a*c^2*d^4 + a^2*c*d^2*e^2
- 2*a^3*e^4)*x^2)*sqrt(c*d^2 + a*e^2)*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2
+ a)*a)/x^2))/(((a^2*c^2*d^5 + a^3*c*d^3*e^2)*x^4 + (a^3*c*d^5 + a^4*d^3*e^2)*x
^2)*sqrt(c*d^2 + a*e^2)*sqrt(a)), -1/4*(2*(a*c*d^4 + a^2*d^2*e^2 - 2*(2*c^2*d^3*
e + a*c*d*e^3)*x^3 + (3*c^2*d^4 + a*c*d^2*e^2)*x^2 - 2*(a*c*d^3*e + a^2*d*e^3)*x
)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(a) + 4*(a^2*c*e^5*x^4 + a^3*e^5*x^2)
*sqrt(a)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 +
a))) + ((3*c^3*d^4 + a*c^2*d^2*e^2 - 2*a^2*c*e^4)*x^4 + (3*a*c^2*d^4 + a^2*c*d^
2*e^2 - 2*a^3*e^4)*x^2)*sqrt(-c*d^2 - a*e^2)*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqr
t(c*x^2 + a)*a)/x^2))/(((a^2*c^2*d^5 + a^3*c*d^3*e^2)*x^4 + (a^3*c*d^5 + a^4*d^3
*e^2)*x^2)*sqrt(-c*d^2 - a*e^2)*sqrt(a)), -1/2*((a*c*d^4 + a^2*d^2*e^2 - 2*(2*c^
2*d^3*e + a*c*d*e^3)*x^3 + (3*c^2*d^4 + a*c*d^2*e^2)*x^2 - 2*(a*c*d^3*e + a^2*d*
e^3)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(-a) - ((3*c^3*d^4 + a*c^2*d^2*e
^2 - 2*a^2*c*e^4)*x^4 + (3*a*c^2*d^4 + a^2*c*d^2*e^2 - 2*a^3*e^4)*x^2)*sqrt(c*d^
2 + a*e^2)*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (a^2*c*e^5*x^4 + a^3*e^5*x^2)*sqrt
(-a)*log(((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2)*sqrt(c
*d^2 + a*e^2) - 2*(a*c*d^2*e + a^2*e^3 - (c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*x^2 + a
))/(e^2*x^2 + 2*d*e*x + d^2)))/(((a^2*c^2*d^5 + a^3*c*d^3*e^2)*x^4 + (a^3*c*d^5
+ a^4*d^3*e^2)*x^2)*sqrt(c*d^2 + a*e^2)*sqrt(-a)), -1/2*((a*c*d^4 + a^2*d^2*e^2
- 2*(2*c^2*d^3*e + a*c*d*e^3)*x^3 + (3*c^2*d^4 + a*c*d^2*e^2)*x^2 - 2*(a*c*d^3*e
+ a^2*d*e^3)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(-a) + 2*(a^2*c*e^5*x^
4 + a^3*e^5*x^2)*sqrt(-a)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*
e^2)*sqrt(c*x^2 + a))) - ((3*c^3*d^4 + a*c^2*d^2*e^2 - 2*a^2*c*e^4)*x^4 + (3*a*c
^2*d^4 + a^2*c*d^2*e^2 - 2*a^3*e^4)*x^2)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-a)/sq
rt(c*x^2 + a)))/(((a^2*c^2*d^5 + a^3*c*d^3*e^2)*x^4 + (a^3*c*d^5 + a^4*d^3*e^2)*
x^2)*sqrt(-c*d^2 - a*e^2)*sqrt(-a))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*(a + c*x**2)**(3/2)*(d + e*x)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError