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

Optimal. Leaf size=151 \[ \frac{e \sqrt{a+c x^2} \left (c d^2-2 a e^2\right )}{a (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )}-\frac{3 c d e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{5/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.268005, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{e \sqrt{a+c x^2} \left (c d^2-2 a e^2\right )}{a (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )}-\frac{3 c d e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 36.5609, size = 133, normalized size = 0.88 \[ - \frac{3 c d e^{2} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{\left (a e^{2} + c d^{2}\right )^{\frac{5}{2}}} - \frac{e \sqrt{a + c x^{2}} \left (2 a e^{2} - c d^{2}\right )}{a \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{2}} + \frac{a e + c d x}{a \sqrt{a + c x^{2}} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.431725, size = 167, normalized size = 1.11 \[ \frac{-a^2 e^3+a c e \left (2 d^2+d e x-2 e^2 x^2\right )+c^2 d^2 x (d+e x)}{a \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )^2}-\frac{3 c d e^2 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{5/2}}+\frac{3 c d e^2 \log (d+e x)}{\left (a e^2+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.016, size = 400, normalized size = 2.7 \[ -{\frac{1}{a{e}^{2}+c{d}^{2}} \left ({\frac{d}{e}}+x \right ) ^{-1}{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+3\,{\frac{ced}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+3\,{\frac{{c}^{2}{d}^{2}x}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-3\,{\frac{ced}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-2\,{\frac{cx}{ \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

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)^(3/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.341825, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{2} e - 2 \, a c e^{3}\right )} x^{2} +{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c d^{2} + a e^{2}} \sqrt{c x^{2} + a} + 3 \,{\left (a c^{2} d e^{3} x^{3} + a c^{2} d^{2} e^{2} x^{2} + a^{2} c d e^{3} x + a^{2} c d^{2} e^{2}\right )} \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \,{\left (a^{2} c^{2} d^{5} + 2 \, a^{3} c d^{3} e^{2} + a^{4} d e^{4} +{\left (a c^{3} d^{4} e + 2 \, a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{3} +{\left (a c^{3} d^{5} + 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} x^{2} +{\left (a^{2} c^{2} d^{4} e + 2 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} x\right )} \sqrt{c d^{2} + a e^{2}}}, \frac{{\left (2 \, a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{2} e - 2 \, a c e^{3}\right )} x^{2} +{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a} + 3 \,{\left (a c^{2} d e^{3} x^{3} + a c^{2} d^{2} e^{2} x^{2} + a^{2} c d e^{3} x + a^{2} c d^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right )}{{\left (a^{2} c^{2} d^{5} + 2 \, a^{3} c d^{3} e^{2} + a^{4} d e^{4} +{\left (a c^{3} d^{4} e + 2 \, a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{3} +{\left (a c^{3} d^{5} + 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} x^{2} +{\left (a^{2} c^{2} d^{4} e + 2 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} x\right )} \sqrt{-c d^{2} - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*(2*a*c*d^2*e - a^2*e^3 + (c^2*d^2*e - 2*a*c*e^3)*x^2 + (c^2*d^3 + a*c*d*
e^2)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) + 3*(a*c^2*d*e^3*x^3 + a*c^2*d^2*e^2
*x^2 + a^2*c*d*e^3*x + a^2*c*d^2*e^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)))/((a^2*c^2*d^5 +
2*a^3*c*d^3*e^2 + a^4*d*e^4 + (a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3
+ (a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e + 2*a^3*c*d
^2*e^3 + a^4*e^5)*x)*sqrt(c*d^2 + a*e^2)), ((2*a*c*d^2*e - a^2*e^3 + (c^2*d^2*e
- 2*a*c*e^3)*x^2 + (c^2*d^3 + a*c*d*e^2)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)
+ 3*(a*c^2*d*e^3*x^3 + a*c^2*d^2*e^2*x^2 + a^2*c*d*e^3*x + a^2*c*d^2*e^2)*arcta
n(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((a^2*c
^2*d^5 + 2*a^3*c*d^3*e^2 + a^4*d*e^4 + (a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3 + a^3*c*
e^5)*x^3 + (a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e +
2*a^3*c*d^2*e^3 + a^4*e^5)*x)*sqrt(-c*d^2 - a*e^2))]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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