3.350 \(\int \frac{1}{x^3 (d+e x)^2 \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=268 \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2} d^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^4}+\frac{2 e \sqrt{a+c x^2}}{a d^3 x}+\frac{c e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}-\frac{\sqrt{a+c x^2}}{2 a d^2 x^2}+\frac{3 e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^4 \sqrt{a e^2+c d^2}}+\frac{e^4 \sqrt{a+c x^2}}{d^3 (d+e x) \left (a e^2+c d^2\right )} \]

[Out]

-Sqrt[a + c*x^2]/(2*a*d^2*x^2) + (2*e*Sqrt[a + c*x^2])/(a*d^3*x) + (e^4*Sqrt[a +
c*x^2])/(d^3*(c*d^2 + a*e^2)*(d + e*x)) + (c*e^3*ArcTanh[(a*e - c*d*x)/(Sqrt[c*
d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^2*(c*d^2 + a*e^2)^(3/2)) + (3*e^3*ArcTanh[(a*
e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^4*Sqrt[c*d^2 + a*e^2]) + (
c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(3/2)*d^2) - (3*e^2*ArcTanh[Sqrt[a + c*
x^2]/Sqrt[a]])/(Sqrt[a]*d^4)

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Rubi [A]  time = 0.521565, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2} d^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^4}+\frac{2 e \sqrt{a+c x^2}}{a d^3 x}+\frac{c e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}-\frac{\sqrt{a+c x^2}}{2 a d^2 x^2}+\frac{3 e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^4 \sqrt{a e^2+c d^2}}+\frac{e^4 \sqrt{a+c x^2}}{d^3 (d+e x) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + c*x^2]/(2*a*d^2*x^2) + (2*e*Sqrt[a + c*x^2])/(a*d^3*x) + (e^4*Sqrt[a +
c*x^2])/(d^3*(c*d^2 + a*e^2)*(d + e*x)) + (c*e^3*ArcTanh[(a*e - c*d*x)/(Sqrt[c*
d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^2*(c*d^2 + a*e^2)^(3/2)) + (3*e^3*ArcTanh[(a*
e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^4*Sqrt[c*d^2 + a*e^2]) + (
c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(3/2)*d^2) - (3*e^2*ArcTanh[Sqrt[a + c*
x^2]/Sqrt[a]])/(Sqrt[a]*d^4)

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Rubi in Sympy [A]  time = 49.4382, size = 243, normalized size = 0.91 \[ \frac{c e^{3} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d^{2} \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} + \frac{e^{4} \sqrt{a + c x^{2}}}{d^{3} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} + \frac{3 e^{3} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d^{4} \sqrt{a e^{2} + c d^{2}}} - \frac{\sqrt{a + c x^{2}}}{2 a d^{2} x^{2}} + \frac{2 e \sqrt{a + c x^{2}}}{a d^{3} x} - \frac{3 e^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a} d^{4}} + \frac{c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*e**3*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(d**2*(a*e*
*2 + c*d**2)**(3/2)) + e**4*sqrt(a + c*x**2)/(d**3*(d + e*x)*(a*e**2 + c*d**2))
+ 3*e**3*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(d**4*sqr
t(a*e**2 + c*d**2)) - sqrt(a + c*x**2)/(2*a*d**2*x**2) + 2*e*sqrt(a + c*x**2)/(a
*d**3*x) - 3*e**2*atanh(sqrt(a + c*x**2)/sqrt(a))/(sqrt(a)*d**4) + c*atanh(sqrt(
a + c*x**2)/sqrt(a))/(2*a**(3/2)*d**2)

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Mathematica [A]  time = 0.779016, size = 229, normalized size = 0.85 \[ \frac{\frac{\left (c d^2-6 a e^2\right ) \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{a^{3/2}}+\frac{\log (x) \left (6 a e^2-c d^2\right )}{a^{3/2}}+d \sqrt{a+c x^2} \left (\frac{2 e^4}{(d+e x) \left (a e^2+c d^2\right )}-\frac{d-4 e x}{a x^2}\right )+\frac{2 e^3 \left (3 a e^2+4 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{2 e^3 \left (3 a e^2+4 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}}{2 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(d*Sqrt[a + c*x^2]*(-((d - 4*e*x)/(a*x^2)) + (2*e^4)/((c*d^2 + a*e^2)*(d + e*x))
) + ((-(c*d^2) + 6*a*e^2)*Log[x])/a^(3/2) - (2*e^3*(4*c*d^2 + 3*a*e^2)*Log[d + e
*x])/(c*d^2 + a*e^2)^(3/2) + ((c*d^2 - 6*a*e^2)*Log[a + Sqrt[a]*Sqrt[a + c*x^2]]
)/a^(3/2) + (2*e^3*(4*c*d^2 + 3*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqr
t[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/(2*d^4)

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Maple [A]  time = 0.019, size = 452, normalized size = 1.7 \[ -{\frac{1}{2\,a{d}^{2}{x}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{c}{2\,{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-3\,{\frac{{e}^{2}}{{d}^{4}\sqrt{a}}\ln \left ({\frac{2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a}}{x}} \right ) }+3\,{\frac{{e}^{2}}{{d}^{4}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+2\,{\frac{e\sqrt{c{x}^{2}+a}}{a{d}^{3}x}}+{\frac{{e}^{3}}{{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ( x+{\frac{d}{e}} \right ) ^{-1}}+{\frac{{e}^{2}c}{{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.10475, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(2*(c*d^5 + a*d^3*e^2 - 2*(2*c*d^3*e^2 + 3*a*d*e^4)*x^2 - 3*(c*d^4*e + a*d
^2*e^3)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(a) - 2*((4*a*c*d^2*e^4 + 3*a
^2*e^6)*x^3 + (4*a*c*d^3*e^3 + 3*a^2*d*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)
) + ((c^2*d^4*e - 5*a*c*d^2*e^3 - 6*a^2*e^5)*x^3 + (c^2*d^5 - 5*a*c*d^3*e^2 - 6*
a^2*d*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*c*d^6*e + a^2*d^4*e^3)*x^3 + (a*c*d^7 + a^2*d^5*e^2)*x^2)*sqrt
(c*d^2 + a*e^2)*sqrt(a)), -1/4*(2*(c*d^5 + a*d^3*e^2 - 2*(2*c*d^3*e^2 + 3*a*d*e^
4)*x^2 - 3*(c*d^4*e + a*d^2*e^3)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(a)
+ 4*((4*a*c*d^2*e^4 + 3*a^2*e^6)*x^3 + (4*a*c*d^3*e^3 + 3*a^2*d*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)))
+ ((c^2*d^4*e - 5*a*c*d^2*e^3 - 6*a^2*e^5)*x^3 + (c^2*d^5 - 5*a*c*d^3*e^2 - 6*a^
2*d*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*c*d^6*e + a^2*d^4*e^3)*x^3 + (a*c*d^7 + a^2*d^5*e^2)*x^2)*sqrt(
-c*d^2 - a*e^2)*sqrt(a)), -1/2*((c*d^5 + a*d^3*e^2 - 2*(2*c*d^3*e^2 + 3*a*d*e^4)
*x^2 - 3*(c*d^4*e + a*d^2*e^3)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(-a) -
((c^2*d^4*e - 5*a*c*d^2*e^3 - 6*a^2*e^5)*x^3 + (c^2*d^5 - 5*a*c*d^3*e^2 - 6*a^2
*d*e^4)*x^2)*sqrt(c*d^2 + a*e^2)*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - ((4*a*c*d^2*
e^4 + 3*a^2*e^6)*x^3 + (4*a*c*d^3*e^3 + 3*a^2*d*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*c*d^6*e + a^2*d^4*e^3)*x^3 + (a*c*d^7 + a^2*d^5*e^2)*x^2)*sqrt
(c*d^2 + a*e^2)*sqrt(-a)), -1/2*((c*d^5 + a*d^3*e^2 - 2*(2*c*d^3*e^2 + 3*a*d*e^4
)*x^2 - 3*(c*d^4*e + a*d^2*e^3)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(-a)
+ 2*((4*a*c*d^2*e^4 + 3*a^2*e^6)*x^3 + (4*a*c*d^3*e^3 + 3*a^2*d*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)))
- ((c^2*d^4*e - 5*a*c*d^2*e^3 - 6*a^2*e^5)*x^3 + (c^2*d^5 - 5*a*c*d^3*e^2 - 6*a
^2*d*e^4)*x^2)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-a)/sqrt(c*x^2 + a)))/(((a*c*d^6
*e + a^2*d^4*e^3)*x^3 + (a*c*d^7 + a^2*d^5*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. \[ \int \frac{1}{x^{3} \sqrt{a + c x^{2}} \left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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