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

Optimal. Leaf size=91 \[ -\frac{e \sqrt{a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}-\frac{c d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.10308, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{e \sqrt{a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}-\frac{c d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.8885, size = 78, normalized size = 0.86 \[ - \frac{c d \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{\left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{e \sqrt{a + c x^{2}}}{\left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.126371, size = 115, normalized size = 1.26 \[ -\frac{e \sqrt{a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}-\frac{c d \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}+\frac{c d \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 210, normalized size = 2.3 \[ -{\frac{1}{a{e}^{2}+c{d}^{2}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{cd}{e \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271555, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{c x^{2} + a} e -{\left (c d e x + c d^{2}\right )} \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \,{\left (c d^{3} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt{c d^{2} + a e^{2}}}, -\frac{\sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a} e -{\left (c d e x + c d^{2}\right )} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right )}{{\left (c d^{3} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt{-c d^{2} - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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)*e - (c*d*e*x + c*d^2)*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)), -(sq
rt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*e - (c*d*e*x + c*d^2)*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))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 1.34976, size = 439, normalized size = 4.82 \[ -\frac{\sqrt{c d^{2} + a e^{2}} c d e{\rm ln}\left ({\left | -\sqrt{c d^{2} + a e^{2}} c d +{\left (c d^{2} + a e^{2}\right )}{\left (\sqrt{c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}} + \frac{\sqrt{c d^{2} e^{2} + a e^{4}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}{\rm sign}\left (\frac{1}{x e + d}\right )} + \frac{{\left (c^{\frac{3}{2}} d^{2} + \sqrt{c d^{2} + a e^{2}} c d{\rm ln}\left ({\left | c^{\frac{3}{2}} d^{2} - \sqrt{c d^{2} + a e^{2}} c d + a \sqrt{c} e^{2} \right |}\right ) + a \sqrt{c} e^{2}\right )}{\rm sign}\left (\frac{1}{x e + d}\right )}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac{\sqrt{c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}}}{c d^{2}{\rm sign}\left (\frac{1}{x e + d}\right ) + a e^{2}{\rm sign}\left (\frac{1}{x e + d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c*d^2 + a*e^2)*c*d*e*ln(abs(-sqrt(c*d^2 + a*e^2)*c*d + (c*d^2 + a*e^2)*(sq
rt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2) + sqrt(c*d^2*e^2
+ a*e^4)*e^(-1)/(x*e + d))))/((c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)*sign(1/(x*e
+ d))) + (c^(3/2)*d^2 + sqrt(c*d^2 + a*e^2)*c*d*ln(abs(c^(3/2)*d^2 - sqrt(c*d^2
+ a*e^2)*c*d + a*sqrt(c)*e^2)) + a*sqrt(c)*e^2)*sign(1/(x*e + d))/(c^2*d^4 + 2*
a*c*d^2*e^2 + a^2*e^4) - sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x
*e + d)^2)/(c*d^2*sign(1/(x*e + d)) + a*e^2*sign(1/(x*e + d)))