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

Optimal. Leaf size=214 \[ -\frac{\sqrt{f+g x} \left (a e^2+c d^2\right )}{2 e (d+e x)^2 (e f-d g)^2}-\frac{\left (15 a e^2 g^2+c \left (-d^2 g^2+8 d e f g+8 e^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{4 e^{3/2} (e f-d g)^{7/2}}+\frac{\sqrt{f+g x} \left (7 a e^2 g+c d (8 e f-d g)\right )}{4 e (d+e x) (e f-d g)^3}+\frac{2 \left (a g^2+c f^2\right )}{\sqrt{f+g x} (e f-d g)^3} \]

[Out]

(2*(c*f^2 + a*g^2))/((e*f - d*g)^3*Sqrt[f + g*x]) - ((c*d^2 + a*e^2)*Sqrt[f + g*
x])/(2*e*(e*f - d*g)^2*(d + e*x)^2) + ((7*a*e^2*g + c*d*(8*e*f - d*g))*Sqrt[f +
g*x])/(4*e*(e*f - d*g)^3*(d + e*x)) - ((15*a*e^2*g^2 + c*(8*e^2*f^2 + 8*d*e*f*g
- d^2*g^2))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(4*e^(3/2)*(e*f -
d*g)^(7/2))

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Rubi [A]  time = 1.00825, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{f+g x} \left (a e^2+c d^2\right )}{2 e (d+e x)^2 (e f-d g)^2}-\frac{\left (15 a e^2 g^2+c \left (-d^2 g^2+8 d e f g+8 e^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{4 e^{3/2} (e f-d g)^{7/2}}+\frac{\sqrt{f+g x} \left (7 a e^2 g+c d (8 e f-d g)\right )}{4 e (d+e x) (e f-d g)^3}+\frac{2 \left (a g^2+c f^2\right )}{\sqrt{f+g x} (e f-d g)^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c*f^2 + a*g^2))/((e*f - d*g)^3*Sqrt[f + g*x]) - ((c*d^2 + a*e^2)*Sqrt[f + g*
x])/(2*e*(e*f - d*g)^2*(d + e*x)^2) + ((7*a*e^2*g + c*d*(8*e*f - d*g))*Sqrt[f +
g*x])/(4*e*(e*f - d*g)^3*(d + e*x)) - ((15*a*e^2*g^2 + c*(8*e^2*f^2 + 8*d*e*f*g
- d^2*g^2))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(4*e^(3/2)*(e*f -
d*g)^(7/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 1.37627, size = 190, normalized size = 0.89 \[ \frac{1}{4} \left (\frac{\sqrt{f+g x} \left (\frac{2 \left (a e^2+c d^2\right ) (d g-e f)}{e (d+e x)^2}+\frac{7 a e^2 g+c d (8 e f-d g)}{e (d+e x)}+\frac{8 \left (a g^2+c f^2\right )}{f+g x}\right )}{(e f-d g)^3}-\frac{\left (15 a e^2 g^2+c \left (-d^2 g^2+8 d e f g+8 e^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{3/2} (e f-d g)^{7/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[f + g*x]*((2*(c*d^2 + a*e^2)*(-(e*f) + d*g))/(e*(d + e*x)^2) + (7*a*e^2*g
+ c*d*(8*e*f - d*g))/(e*(d + e*x)) + (8*(c*f^2 + a*g^2))/(f + g*x)))/(e*f - d*g
)^3 - ((15*a*e^2*g^2 + c*(8*e^2*f^2 + 8*d*e*f*g - d^2*g^2))*ArcTanh[(Sqrt[e]*Sqr
t[f + g*x])/Sqrt[e*f - d*g]])/(e^(3/2)*(e*f - d*g)^(7/2)))/4

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Maple [B]  time = 0.033, size = 546, normalized size = 2.6 \[ -2\,{\frac{a{g}^{2}}{ \left ( dg-ef \right ) ^{3}\sqrt{gx+f}}}-2\,{\frac{c{f}^{2}}{ \left ( dg-ef \right ) ^{3}\sqrt{gx+f}}}-{\frac{7\,a{e}^{2}{g}^{2}}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}} \left ( gx+f \right ) ^{{\frac{3}{2}}}}+{\frac{c{d}^{2}{g}^{2}}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}} \left ( gx+f \right ) ^{{\frac{3}{2}}}}-2\,{\frac{ \left ( gx+f \right ) ^{3/2}cdefg}{ \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}}}-{\frac{9\,{g}^{3}ead}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}}\sqrt{gx+f}}+{\frac{9\,a{e}^{2}{g}^{2}f}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}}\sqrt{gx+f}}-{\frac{c{d}^{3}{g}^{3}}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}e}\sqrt{gx+f}}-{\frac{7\,c{d}^{2}{g}^{2}f}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}}\sqrt{gx+f}}+2\,{\frac{eg\sqrt{gx+f}cd{f}^{2}}{ \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}}}-{\frac{15\,ae{g}^{2}}{4\, \left ( dg-ef \right ) ^{3}}\arctan \left ({e\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}} \right ){\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}}+{\frac{c{d}^{2}{g}^{2}}{4\, \left ( dg-ef \right ) ^{3}e}\arctan \left ({e\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}} \right ){\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}}-2\,{\frac{cdfg}{ \left ( dg-ef \right ) ^{3}\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }-2\,{\frac{ce{f}^{2}}{ \left ( dg-ef \right ) ^{3}\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(d*g-e*f)^3/(g*x+f)^(1/2)*a*g^2-2/(d*g-e*f)^3/(g*x+f)^(1/2)*c*f^2-7/4/(d*g-e*
f)^3/(e*g*x+d*g)^2*(g*x+f)^(3/2)*a*e^2*g^2+1/4/(d*g-e*f)^3/(e*g*x+d*g)^2*(g*x+f)
^(3/2)*c*d^2*g^2-2/(d*g-e*f)^3/(e*g*x+d*g)^2*(g*x+f)^(3/2)*c*d*e*f*g-9/4/(d*g-e*
f)^3/(e*g*x+d*g)^2*g^3*e*(g*x+f)^(1/2)*a*d+9/4/(d*g-e*f)^3/(e*g*x+d*g)^2*g^2*e^2
*(g*x+f)^(1/2)*a*f-1/4/(d*g-e*f)^3/(e*g*x+d*g)^2*g^3/e*(g*x+f)^(1/2)*c*d^3-7/4/(
d*g-e*f)^3/(e*g*x+d*g)^2*g^2*(g*x+f)^(1/2)*c*d^2*f+2/(d*g-e*f)^3/(e*g*x+d*g)^2*g
*e*(g*x+f)^(1/2)*c*d*f^2-15/4/(d*g-e*f)^3*e/((d*g-e*f)*e)^(1/2)*arctan((g*x+f)^(
1/2)*e/((d*g-e*f)*e)^(1/2))*a*g^2+1/4/(d*g-e*f)^3/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*g^2-2/(d*g-e*f)^3/((d*g-e*f)*e)^(1/2)*
arctan((g*x+f)^(1/2)*e/((d*g-e*f)*e)^(1/2))*c*d*f*g-2/(d*g-e*f)^3*e/((d*g-e*f)*e
)^(1/2)*arctan((g*x+f)^(1/2)*e/((d*g-e*f)*e)^(1/2))*c*f^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*(g*x + f)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.311973, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*((8*c*d^2*e^2*f^2 + 8*c*d^3*e*f*g - (c*d^4 - 15*a*d^2*e^2)*g^2 + (8*c*e^4*
f^2 + 8*c*d*e^3*f*g - (c*d^2*e^2 - 15*a*e^4)*g^2)*x^2 + 2*(8*c*d*e^3*f^2 + 8*c*d
^2*e^2*f*g - (c*d^3*e - 15*a*d*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*(8*a*
d^2*e*g^2 + 2*(7*c*d^2*e - a*e^3)*f^2 + (c*d^3 + 9*a*d*e^2)*f*g + (8*c*e^3*f^2 +
8*c*d*e^2*f*g - (c*d^2*e - 15*a*e^3)*g^2)*x^2 + (24*c*d*e^2*f^2 + 5*(c*d^2*e +
a*e^3)*f*g + (c*d^3 + 25*a*d*e^2)*g^2)*x)*sqrt(e^2*f - d*e*g))/((d^2*e^4*f^3 - 3
*d^3*e^3*f^2*g + 3*d^4*e^2*f*g^2 - d^5*e*g^3 + (e^6*f^3 - 3*d*e^5*f^2*g + 3*d^2*
e^4*f*g^2 - d^3*e^3*g^3)*x^2 + 2*(d*e^5*f^3 - 3*d^2*e^4*f^2*g + 3*d^3*e^3*f*g^2
- d^4*e^2*g^3)*x)*sqrt(e^2*f - d*e*g)*sqrt(g*x + f)), -1/4*((8*c*d^2*e^2*f^2 + 8
*c*d^3*e*f*g - (c*d^4 - 15*a*d^2*e^2)*g^2 + (8*c*e^4*f^2 + 8*c*d*e^3*f*g - (c*d^
2*e^2 - 15*a*e^4)*g^2)*x^2 + 2*(8*c*d*e^3*f^2 + 8*c*d^2*e^2*f*g - (c*d^3*e - 15*
a*d*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))) - (8*a*d^2*e*g^2 + 2*(7*c*d^2*e - a*e^3)*f^2 + (c*d^3 + 9*a*d*e^2)*f*g
+ (8*c*e^3*f^2 + 8*c*d*e^2*f*g - (c*d^2*e - 15*a*e^3)*g^2)*x^2 + (24*c*d*e^2*f^2
+ 5*(c*d^2*e + a*e^3)*f*g + (c*d^3 + 25*a*d*e^2)*g^2)*x)*sqrt(-e^2*f + d*e*g))/
((d^2*e^4*f^3 - 3*d^3*e^3*f^2*g + 3*d^4*e^2*f*g^2 - d^5*e*g^3 + (e^6*f^3 - 3*d*e
^5*f^2*g + 3*d^2*e^4*f*g^2 - d^3*e^3*g^3)*x^2 + 2*(d*e^5*f^3 - 3*d^2*e^4*f^2*g +
3*d^3*e^3*f*g^2 - d^4*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)**3/(g*x+f)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282456, size = 487, normalized size = 2.28 \[ \frac{{\left (c d^{2} g^{2} - 8 \, c d f g e - 8 \, c f^{2} e^{2} - 15 \, a g^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right )}{4 \,{\left (d^{3} g^{3} e - 3 \, d^{2} f g^{2} e^{2} + 3 \, d f^{2} g e^{3} - f^{3} e^{4}\right )} \sqrt{d g e - f e^{2}}} - \frac{2 \,{\left (c f^{2} + a g^{2}\right )}}{{\left (d^{3} g^{3} - 3 \, d^{2} f g^{2} e + 3 \, d f^{2} g e^{2} - f^{3} e^{3}\right )} \sqrt{g x + f}} - \frac{\sqrt{g x + f} c d^{3} g^{3} -{\left (g x + f\right )}^{\frac{3}{2}} c d^{2} g^{2} e + 7 \, \sqrt{g x + f} c d^{2} f g^{2} e + 8 \,{\left (g x + f\right )}^{\frac{3}{2}} c d f g e^{2} - 8 \, \sqrt{g x + f} c d f^{2} g e^{2} + 9 \, \sqrt{g x + f} a d g^{3} e^{2} + 7 \,{\left (g x + f\right )}^{\frac{3}{2}} a g^{2} e^{3} - 9 \, \sqrt{g x + f} a f g^{2} e^{3}}{4 \,{\left (d^{3} g^{3} e - 3 \, d^{2} f g^{2} e^{2} + 3 \, d f^{2} g e^{3} - f^{3} e^{4}\right )}{\left (d g +{\left (g x + f\right )} e - f e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(c*d^2*g^2 - 8*c*d*f*g*e - 8*c*f^2*e^2 - 15*a*g^2*e^2)*arctan(sqrt(g*x + f)*
e/sqrt(d*g*e - f*e^2))/((d^3*g^3*e - 3*d^2*f*g^2*e^2 + 3*d*f^2*g*e^3 - f^3*e^4)*
sqrt(d*g*e - f*e^2)) - 2*(c*f^2 + a*g^2)/((d^3*g^3 - 3*d^2*f*g^2*e + 3*d*f^2*g*e
^2 - f^3*e^3)*sqrt(g*x + f)) - 1/4*(sqrt(g*x + f)*c*d^3*g^3 - (g*x + f)^(3/2)*c*
d^2*g^2*e + 7*sqrt(g*x + f)*c*d^2*f*g^2*e + 8*(g*x + f)^(3/2)*c*d*f*g*e^2 - 8*sq
rt(g*x + f)*c*d*f^2*g*e^2 + 9*sqrt(g*x + f)*a*d*g^3*e^2 + 7*(g*x + f)^(3/2)*a*g^
2*e^3 - 9*sqrt(g*x + f)*a*f*g^2*e^3)/((d^3*g^3*e - 3*d^2*f*g^2*e^2 + 3*d*f^2*g*e
^3 - f^3*e^4)*(d*g + (g*x + f)*e - f*e)^2)