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

Optimal. Leaf size=145 \[ -\frac{3 c d e \sqrt{a+c x^2}}{2 (d+e x) \left (a e^2+c d^2\right )^2}-\frac{e \sqrt{a+c x^2}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{c \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.222421, antiderivative size = 145, 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 \[ -\frac{3 c d e \sqrt{a+c x^2}}{2 (d+e x) \left (a e^2+c d^2\right )^2}-\frac{e \sqrt{a+c x^2}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{c \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.9569, size = 128, normalized size = 0.88 \[ - \frac{3 c d e \sqrt{a + c x^{2}}}{2 \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{2}} + \frac{c \left (a e^{2} - 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{\frac{5}{2}}} - \frac{e \sqrt{a + c x^{2}}}{2 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.218076, size = 161, normalized size = 1.11 \[ \frac{-e \sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left (a e^2+c d (4 d+3 e x)\right )-c (d+e x)^2 \left (2 c d^2-a e^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+c (d+e x)^2 \left (2 c d^2-a e^2\right ) \log (d+e x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-(e*Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(a*e^2 + c*d*(4*d + 3*e*x))) + c*(2*c*d
^2 - a*e^2)*(d + e*x)^2*Log[d + e*x] - c*(2*c*d^2 - a*e^2)*(d + e*x)^2*Log[a*e -
c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(2*(c*d^2 + a*e^2)^(5/2)*(d + e*x
)^2)

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Maple [B]  time = 0.017, size = 426, normalized size = 2.9 \[ -{\frac{1}{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-2}}-{\frac{3\,cd}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{3\,{c}^{2}{d}^{2}}{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{c}{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/e/(a*e^2+c*d^2)/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(
1/2)-3/2*c*d/(a*e^2+c*d^2)^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)-3/2/e*c^2*d^2/(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))+1/2/e*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*(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)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.405997, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, c d e^{2} x + 4 \, c d^{2} e + a e^{3}\right )} \sqrt{c d^{2} + a e^{2}} \sqrt{c x^{2} + a} +{\left (2 \, c^{2} d^{4} - a c d^{2} e^{2} +{\left (2 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 2 \,{\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} - 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{4 \,{\left (c^{2} d^{6} + 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} +{\left (c^{2} d^{4} e^{2} + 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e + 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )} \sqrt{c d^{2} + a e^{2}}}, -\frac{{\left (3 \, c d e^{2} x + 4 \, c d^{2} e + a e^{3}\right )} \sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a} -{\left (2 \, c^{2} d^{4} - a c d^{2} e^{2} +{\left (2 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 2 \,{\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right )}{2 \,{\left (c^{2} d^{6} + 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} +{\left (c^{2} d^{4} e^{2} + 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e + 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )} \sqrt{-c d^{2} - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(2*(3*c*d*e^2*x + 4*c*d^2*e + a*e^3)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) +
(2*c^2*d^4 - a*c*d^2*e^2 + (2*c^2*d^2*e^2 - a*c*e^4)*x^2 + 2*(2*c^2*d^3*e - a*c
*d*e^3)*x)*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^6 + 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (
c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e + 2*a*c*d^3*e^3 + a^2*
d*e^5)*x)*sqrt(c*d^2 + a*e^2)), -1/2*((3*c*d*e^2*x + 4*c*d^2*e + a*e^3)*sqrt(-c*
d^2 - a*e^2)*sqrt(c*x^2 + a) - (2*c^2*d^4 - a*c*d^2*e^2 + (2*c^2*d^2*e^2 - a*c*e
^4)*x^2 + 2*(2*c^2*d^3*e - a*c*d*e^3)*x)*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^6 + 2*a*c*d^4*e^2 + a^2*d^2*e^4 +
(c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e + 2*a*c*d^3*e^3 + a^
2*d*e^5)*x)*sqrt(-c*d^2 - a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.244883, size = 466, normalized size = 3.21 \[ -c{\left (\frac{{\left (2 \, c d^{2} - a e^{2}\right )} \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{-c d^{2} - a e^{2}}} + \frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} c d^{2} e + 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} c^{\frac{3}{2}} d^{3} - 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a c d^{2} e - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a \sqrt{c} d e^{2} -{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a e^{3} + 3 \, a^{2} \sqrt{c} d e^{2} -{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{2} e^{3}}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} e + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} \sqrt{c} d - a e\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*((2*c*d^2 - a*e^2)*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(
-c*d^2 - a*e^2))/((c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c*d^2 - a*e^2)) + (2
*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c*d^2*e + 6*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(
3/2)*d^3 - 10*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c*d^2*e - 3*(sqrt(c)*x - sqrt(c*x^
2 + a))^2*a*sqrt(c)*d*e^2 - (sqrt(c)*x - sqrt(c*x^2 + a))^3*a*e^3 + 3*a^2*sqrt(c
)*d*e^2 - (sqrt(c)*x - sqrt(c*x^2 + a))*a^2*e^3)/((c^2*d^4 + 2*a*c*d^2*e^2 + a^2
*e^4)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(
c)*d - a*e)^2))