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

Optimal. Leaf size=90 \[ \frac{d \sqrt{a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}-\frac{a e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.118103, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{d \sqrt{a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}-\frac{a e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.3386, size = 76, normalized size = 0.84 \[ - \frac{a e \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{\left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} + \frac{d \sqrt{a + c x^{2}}}{\left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.122557, size = 114, normalized size = 1.27 \[ \frac{d \sqrt{a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}-\frac{a e \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}+\frac{a e \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.012, size = 340, normalized size = 3.8 \[ -{\frac{1}{{e}^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{d}{e \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ( x+{\frac{d}{e}} \right ) ^{-1}}+{\frac{c{d}^{2}}{{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/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))+d/e/(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)+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))

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.322112, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{c x^{2} + a} d +{\left (a e^{2} x + a d e\right )} \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \,{\left (c d^{3} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt{c d^{2} + a e^{2}}}, \frac{\sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a} d +{\left (a e^{2} x + a d e\right )} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right )}{{\left (c d^{3} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt{-c d^{2} - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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