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

Optimal. Leaf size=112 \[ -\frac{2 \left (a e^2+c d^2\right )}{e^2 \sqrt{d+e x} (e f-d g)}-\frac{2 \left (a g^2+c f^2\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right )}{g^{3/2} (e f-d g)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^2 g} \]

[Out]

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

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Rubi [A]  time = 0.456118, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 \left (a e^2+c d^2\right )}{e^2 \sqrt{d+e x} (e f-d g)}-\frac{2 \left (a g^2+c f^2\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right )}{g^{3/2} (e f-d g)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^2 g} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 \left (a g^{2} + c f^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{d + e x}}{\sqrt{d g - e f}} \right )}}{g^{\frac{3}{2}} \left (d g - e f\right )^{\frac{3}{2}}} + \frac{2 \left (a e^{2} + c d^{2}\right )}{e^{2} \sqrt{d + e x} \left (d g - e f\right )} + \frac{2 \int ^{\sqrt{d + e x}} c\, dx}{e^{2} g} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a*g**2 + c*f**2)*atanh(sqrt(g)*sqrt(d + e*x)/sqrt(d*g - e*f))/(g**(3/2)*(d*g
- e*f)**(3/2)) + 2*(a*e**2 + c*d**2)/(e**2*sqrt(d + e*x)*(d*g - e*f)) + 2*Integ
ral(c, (x, sqrt(d + e*x)))/(e**2*g)

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Mathematica [A]  time = 0.333416, size = 108, normalized size = 0.96 \[ \frac{2 \sqrt{d+e x} \left (\frac{a e^2+c d^2}{(d+e x) (d g-e f)}+\frac{c}{g}\right )}{e^2}-\frac{2 \left (a g^2+c f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{d g-e f}}\right )}{g^{3/2} (d g-e f)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 114, normalized size = 1. \[ 2\,{\frac{1}{{e}^{2}} \left ({\frac{c\sqrt{ex+d}}{g}}-{\frac{{e}^{2} \left ( a{g}^{2}+c{f}^{2} \right ) }{ \left ( dg-ef \right ) g\sqrt{ \left ( dg-ef \right ) g}}{\it Artanh} \left ({\frac{g\sqrt{ex+d}}{\sqrt{ \left ( dg-ef \right ) g}}} \right ) }-{\frac{-a{e}^{2}-c{d}^{2}}{ \left ( dg-ef \right ) \sqrt{ex+d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/e^2*(c/g*(e*x+d)^(1/2)-e^2*(a*g^2+c*f^2)/(d*g-e*f)/g/((d*g-e*f)*g)^(1/2)*arcta
nh(g*(e*x+d)^(1/2)/((d*g-e*f)*g)^(1/2))-(-a*e^2-c*d^2)/(d*g-e*f)/(e*x+d)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290642, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (c e^{2} f^{2} + a e^{2} g^{2}\right )} \sqrt{e x + d} \log \left (\frac{\sqrt{-e f g + d g^{2}}{\left (e g x - e f + 2 \, d g\right )} + 2 \,{\left (e f g - d g^{2}\right )} \sqrt{e x + d}}{g x + f}\right ) - 2 \,{\left (c d e f -{\left (2 \, c d^{2} + a e^{2}\right )} g +{\left (c e^{2} f - c d e g\right )} x\right )} \sqrt{-e f g + d g^{2}}}{{\left (e^{3} f g - d e^{2} g^{2}\right )} \sqrt{-e f g + d g^{2}} \sqrt{e x + d}}, -\frac{2 \,{\left ({\left (c e^{2} f^{2} + a e^{2} g^{2}\right )} \sqrt{e x + d} \arctan \left (-\frac{e f - d g}{\sqrt{e f g - d g^{2}} \sqrt{e x + d}}\right ) -{\left (c d e f -{\left (2 \, c d^{2} + a e^{2}\right )} g +{\left (c e^{2} f - c d e g\right )} x\right )} \sqrt{e f g - d g^{2}}\right )}}{{\left (e^{3} f g - d e^{2} g^{2}\right )} \sqrt{e f g - d g^{2}} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-((c*e^2*f^2 + a*e^2*g^2)*sqrt(e*x + d)*log((sqrt(-e*f*g + d*g^2)*(e*g*x - e*f
+ 2*d*g) + 2*(e*f*g - d*g^2)*sqrt(e*x + d))/(g*x + f)) - 2*(c*d*e*f - (2*c*d^2 +
a*e^2)*g + (c*e^2*f - c*d*e*g)*x)*sqrt(-e*f*g + d*g^2))/((e^3*f*g - d*e^2*g^2)*
sqrt(-e*f*g + d*g^2)*sqrt(e*x + d)), -2*((c*e^2*f^2 + a*e^2*g^2)*sqrt(e*x + d)*a
rctan(-(e*f - d*g)/(sqrt(e*f*g - d*g^2)*sqrt(e*x + d))) - (c*d*e*f - (2*c*d^2 +
a*e^2)*g + (c*e^2*f - c*d*e*g)*x)*sqrt(e*f*g - d*g^2))/((e^3*f*g - d*e^2*g^2)*sq
rt(e*f*g - d*g^2)*sqrt(e*x + d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.288194, size = 157, normalized size = 1.4 \[ \frac{2 \, \sqrt{x e + d} c e^{\left (-2\right )}}{g} + \frac{2 \,{\left (c f^{2} + a g^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} g}{\sqrt{-d g^{2} + f g e}}\right )}{{\left (d g^{2} - f g e\right )} \sqrt{-d g^{2} + f g e}} + \frac{2 \,{\left (c d^{2} + a e^{2}\right )}}{{\left (d g e^{2} - f e^{3}\right )} \sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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