3.601 \(\int \frac{a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{\sqrt{f+g x} \left (a e^2+c d^2\right )}{e (d+e x) (e f-d g)^2}+\frac{\left (3 a e^2 g+c d (4 e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}}-\frac{2 \left (a g^2+c f^2\right )}{g \sqrt{f+g x} (e f-d g)^2} \]

[Out]

(-2*(c*f^2 + a*g^2))/(g*(e*f - d*g)^2*Sqrt[f + g*x]) - ((c*d^2 + a*e^2)*Sqrt[f +
g*x])/(e*(e*f - d*g)^2*(d + e*x)) + ((3*a*e^2*g + c*d*(4*e*f - d*g))*ArcTanh[(S
qrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(3/2)*(e*f - d*g)^(5/2))

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Rubi [A]  time = 0.610172, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\sqrt{f+g x} \left (a e^2+c d^2\right )}{e (d+e x) (e f-d g)^2}+\frac{\left (3 a e^2 g+c d (4 e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}}-\frac{2 \left (a g^2+c f^2\right )}{g \sqrt{f+g x} (e f-d g)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(c*f^2 + a*g^2))/(g*(e*f - d*g)^2*Sqrt[f + g*x]) - ((c*d^2 + a*e^2)*Sqrt[f +
g*x])/(e*(e*f - d*g)^2*(d + e*x)) + ((3*a*e^2*g + c*d*(4*e*f - d*g))*ArcTanh[(S
qrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(3/2)*(e*f - d*g)^(5/2))

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Rubi in Sympy [A]  time = 174.434, size = 141, normalized size = 0.98 \[ - \frac{2 \left (a g^{2} + c f^{2}\right )}{g \sqrt{f + g x} \left (d g - e f\right )^{2}} - \frac{g \sqrt{f + g x} \left (a e^{2} + c d^{2}\right )}{e \left (d g - e f\right )^{2} \left (d g - e f + e \left (f + g x\right )\right )} - \frac{\left (3 a e^{2} g - c d^{2} g + 4 c d e f\right ) \operatorname{atan}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{d g - e f}} \right )}}{e^{\frac{3}{2}} \left (d g - e f\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a*g**2 + c*f**2)/(g*sqrt(f + g*x)*(d*g - e*f)**2) - g*sqrt(f + g*x)*(a*e**2
+ c*d**2)/(e*(d*g - e*f)**2*(d*g - e*f + e*(f + g*x))) - (3*a*e**2*g - c*d**2*g
+ 4*c*d*e*f)*atan(sqrt(e)*sqrt(f + g*x)/sqrt(d*g - e*f))/(e**(3/2)*(d*g - e*f)**
(5/2))

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Mathematica [A]  time = 0.572277, size = 135, normalized size = 0.94 \[ \frac{\left (3 a e^2 g+c d (4 e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}}-\frac{\sqrt{f+g x} \left (\frac{a e^2+c d^2}{d e+e^2 x}+\frac{2 \left (a g^2+c f^2\right )}{g (f+g x)}\right )}{(e f-d g)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.03, size = 269, normalized size = 1.9 \[ -2\,{\frac{ag}{ \left ( dg-ef \right ) ^{2}\sqrt{gx+f}}}-2\,{\frac{c{f}^{2}}{g \left ( dg-ef \right ) ^{2}\sqrt{gx+f}}}-{\frac{aeg}{ \left ( dg-ef \right ) ^{2} \left ( egx+dg \right ) }\sqrt{gx+f}}-{\frac{c{d}^{2}g}{ \left ( dg-ef \right ) ^{2}e \left ( egx+dg \right ) }\sqrt{gx+f}}-3\,{\frac{aeg}{ \left ( dg-ef \right ) ^{2}\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }+{\frac{c{d}^{2}g}{ \left ( dg-ef \right ) ^{2}e}\arctan \left ({e\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}} \right ){\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}}-4\,{\frac{cdf}{ \left ( dg-ef \right ) ^{2}\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296011, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (4 \, c d^{2} e f g -{\left (c d^{3} - 3 \, a d e^{2}\right )} g^{2} +{\left (4 \, c d e^{2} f g -{\left (c d^{2} e - 3 \, a e^{3}\right )} g^{2}\right )} x\right )} \sqrt{g x + f} \log \left (\frac{\sqrt{e^{2} f - d e g}{\left (e g x + 2 \, e f - d g\right )} + 2 \,{\left (e^{2} f - d e g\right )} \sqrt{g x + f}}{e x + d}\right ) - 2 \,{\left (2 \, c d e f^{2} + 2 \, a d e g^{2} +{\left (c d^{2} + a e^{2}\right )} f g +{\left (2 \, c e^{2} f^{2} +{\left (c d^{2} + 3 \, a e^{2}\right )} g^{2}\right )} x\right )} \sqrt{e^{2} f - d e g}}{2 \,{\left (d e^{3} f^{2} g - 2 \, d^{2} e^{2} f g^{2} + d^{3} e g^{3} +{\left (e^{4} f^{2} g - 2 \, d e^{3} f g^{2} + d^{2} e^{2} g^{3}\right )} x\right )} \sqrt{e^{2} f - d e g} \sqrt{g x + f}}, \frac{{\left (4 \, c d^{2} e f g -{\left (c d^{3} - 3 \, a d e^{2}\right )} g^{2} +{\left (4 \, c d e^{2} f g -{\left (c d^{2} e - 3 \, a e^{3}\right )} g^{2}\right )} x\right )} \sqrt{g x + f} \arctan \left (-\frac{e f - d g}{\sqrt{-e^{2} f + d e g} \sqrt{g x + f}}\right ) -{\left (2 \, c d e f^{2} + 2 \, a d e g^{2} +{\left (c d^{2} + a e^{2}\right )} f g +{\left (2 \, c e^{2} f^{2} +{\left (c d^{2} + 3 \, a e^{2}\right )} g^{2}\right )} x\right )} \sqrt{-e^{2} f + d e g}}{{\left (d e^{3} f^{2} g - 2 \, d^{2} e^{2} f g^{2} + d^{3} e g^{3} +{\left (e^{4} f^{2} g - 2 \, d e^{3} f g^{2} + d^{2} e^{2} g^{3}\right )} x\right )} \sqrt{-e^{2} f + d e g} \sqrt{g x + f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*((4*c*d^2*e*f*g - (c*d^3 - 3*a*d*e^2)*g^2 + (4*c*d*e^2*f*g - (c*d^2*e - 3*a
*e^3)*g^2)*x)*sqrt(g*x + f)*log((sqrt(e^2*f - d*e*g)*(e*g*x + 2*e*f - d*g) + 2*(
e^2*f - d*e*g)*sqrt(g*x + f))/(e*x + d)) - 2*(2*c*d*e*f^2 + 2*a*d*e*g^2 + (c*d^2
+ a*e^2)*f*g + (2*c*e^2*f^2 + (c*d^2 + 3*a*e^2)*g^2)*x)*sqrt(e^2*f - d*e*g))/((
d*e^3*f^2*g - 2*d^2*e^2*f*g^2 + d^3*e*g^3 + (e^4*f^2*g - 2*d*e^3*f*g^2 + d^2*e^2
*g^3)*x)*sqrt(e^2*f - d*e*g)*sqrt(g*x + f)), ((4*c*d^2*e*f*g - (c*d^3 - 3*a*d*e^
2)*g^2 + (4*c*d*e^2*f*g - (c*d^2*e - 3*a*e^3)*g^2)*x)*sqrt(g*x + f)*arctan(-(e*f
- d*g)/(sqrt(-e^2*f + d*e*g)*sqrt(g*x + f))) - (2*c*d*e*f^2 + 2*a*d*e*g^2 + (c*
d^2 + a*e^2)*f*g + (2*c*e^2*f^2 + (c*d^2 + 3*a*e^2)*g^2)*x)*sqrt(-e^2*f + d*e*g)
)/((d*e^3*f^2*g - 2*d^2*e^2*f*g^2 + d^3*e*g^3 + (e^4*f^2*g - 2*d*e^3*f*g^2 + d^2
*e^2*g^3)*x)*sqrt(-e^2*f + d*e*g)*sqrt(g*x + f))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27459, size = 304, normalized size = 2.11 \[ \frac{{\left (c d^{2} g - 4 \, c d f e - 3 \, a g e^{2}\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right )}{{\left (d^{2} g^{2} e - 2 \, d f g e^{2} + f^{2} e^{3}\right )} \sqrt{d g e - f e^{2}}} - \frac{{\left (g x + f\right )} c d^{2} g^{2} + 2 \, c d f^{2} g e + 2 \, a d g^{3} e + 2 \,{\left (g x + f\right )} c f^{2} e^{2} - 2 \, c f^{3} e^{2} + 3 \,{\left (g x + f\right )} a g^{2} e^{2} - 2 \, a f g^{2} e^{2}}{{\left (d^{2} g^{3} e - 2 \, d f g^{2} e^{2} + f^{2} g e^{3}\right )}{\left (\sqrt{g x + f} d g +{\left (g x + f\right )}^{\frac{3}{2}} e - \sqrt{g x + f} f e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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