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

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

[Out]

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

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Rubi [A]  time = 0.173682, 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, Rules used = {898, 1261, 208} \[ -\frac{2 \left (a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{3/2} (e f-d g)^{3/2}}+\frac{2 \left (a g^2+c f^2\right )}{g^2 \sqrt{f+g x} (e f-d g)}+\frac{2 c \sqrt{f+g x}}{e g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 898

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 + a*e^2)/e^2 - (2*c
*d*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]

Rule 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\frac{c f^2+a g^2}{g^2}-\frac{2 c f x^2}{g^2}+\frac{c x^4}{g^2}}{x^2 \left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )} \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{c}{e g}+\frac{c f^2+a g^2}{g (-e f+d g) x^2}-\frac{\left (c d^2+a e^2\right ) g}{e (e f-d g) \left (e f-d g-e x^2\right )}\right ) \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=\frac{2 \left (c f^2+a g^2\right )}{g^2 (e f-d g) \sqrt{f+g x}}+\frac{2 c \sqrt{f+g x}}{e g^2}-\frac{\left (2 \left (c d^2+a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{e f-d g-e x^2} \, dx,x,\sqrt{f+g x}\right )}{e (e f-d g)}\\ &=\frac{2 \left (c f^2+a g^2\right )}{g^2 (e f-d g) \sqrt{f+g x}}+\frac{2 c \sqrt{f+g x}}{e g^2}-\frac{2 \left (c d^2+a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{3/2} (e f-d g)^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.0714895, size = 90, normalized size = 0.8 \[ -\frac{2 \left (g^2 \left (a e^2+c d^2\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right )+c (e f-d g) (d g+2 e f+e g x)\right )}{e^2 g^2 \sqrt{f+g x} (d g-e f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(c*(e*f - d*g)*(2*e*f + d*g + e*g*x) + (c*d^2 + a*e^2)*g^2*Hypergeometric2F1[-1/2, 1, 1/2, (e*(f + g*x))/(
e*f - d*g)]))/(e^2*g^2*(-(e*f) + d*g)*Sqrt[f + g*x])

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Maple [A]  time = 0.214, size = 165, normalized size = 1.5 \begin{align*} 2\,{\frac{c\sqrt{gx+f}}{e{g}^{2}}}-2\,{\frac{ae}{ \left ( dg-ef \right ) \sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }-2\,{\frac{c{d}^{2}}{ \left ( dg-ef \right ) e\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }-2\,{\frac{a}{ \left ( dg-ef \right ) \sqrt{gx+f}}}-2\,{\frac{c{f}^{2}}{{g}^{2} \left ( dg-ef \right ) \sqrt{gx+f}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.86707, size = 1015, normalized size = 9.06 \begin{align*} \left [-\frac{{\left ({\left (c d^{2} + a e^{2}\right )} g^{3} x +{\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt{e^{2} f - d e g} \log \left (\frac{e g x + 2 \, e f - d g + 2 \, \sqrt{e^{2} f - d e g} \sqrt{g x + f}}{e x + d}\right ) - 2 \,{\left (2 \, c e^{3} f^{3} - 3 \, c d e^{2} f^{2} g - a d e^{2} g^{3} +{\left (c d^{2} e + a e^{3}\right )} f g^{2} +{\left (c e^{3} f^{2} g - 2 \, c d e^{2} f g^{2} + c d^{2} e g^{3}\right )} x\right )} \sqrt{g x + f}}{e^{4} f^{3} g^{2} - 2 \, d e^{3} f^{2} g^{3} + d^{2} e^{2} f g^{4} +{\left (e^{4} f^{2} g^{3} - 2 \, d e^{3} f g^{4} + d^{2} e^{2} g^{5}\right )} x}, \frac{2 \,{\left ({\left ({\left (c d^{2} + a e^{2}\right )} g^{3} x +{\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt{-e^{2} f + d e g} \arctan \left (\frac{\sqrt{-e^{2} f + d e g} \sqrt{g x + f}}{e g x + e f}\right ) +{\left (2 \, c e^{3} f^{3} - 3 \, c d e^{2} f^{2} g - a d e^{2} g^{3} +{\left (c d^{2} e + a e^{3}\right )} f g^{2} +{\left (c e^{3} f^{2} g - 2 \, c d e^{2} f g^{2} + c d^{2} e g^{3}\right )} x\right )} \sqrt{g x + f}\right )}}{e^{4} f^{3} g^{2} - 2 \, d e^{3} f^{2} g^{3} + d^{2} e^{2} f g^{4} +{\left (e^{4} f^{2} g^{3} - 2 \, d e^{3} f g^{4} + d^{2} e^{2} g^{5}\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(((c*d^2 + a*e^2)*g^3*x + (c*d^2 + a*e^2)*f*g^2)*sqrt(e^2*f - d*e*g)*log((e*g*x + 2*e*f - d*g + 2*sqrt(e^2*f
- d*e*g)*sqrt(g*x + f))/(e*x + d)) - 2*(2*c*e^3*f^3 - 3*c*d*e^2*f^2*g - a*d*e^2*g^3 + (c*d^2*e + a*e^3)*f*g^2
+ (c*e^3*f^2*g - 2*c*d*e^2*f*g^2 + c*d^2*e*g^3)*x)*sqrt(g*x + f))/(e^4*f^3*g^2 - 2*d*e^3*f^2*g^3 + d^2*e^2*f*
g^4 + (e^4*f^2*g^3 - 2*d*e^3*f*g^4 + d^2*e^2*g^5)*x), 2*(((c*d^2 + a*e^2)*g^3*x + (c*d^2 + a*e^2)*f*g^2)*sqrt(
-e^2*f + d*e*g)*arctan(sqrt(-e^2*f + d*e*g)*sqrt(g*x + f)/(e*g*x + e*f)) + (2*c*e^3*f^3 - 3*c*d*e^2*f^2*g - a*
d*e^2*g^3 + (c*d^2*e + a*e^3)*f*g^2 + (c*e^3*f^2*g - 2*c*d*e^2*f*g^2 + c*d^2*e*g^3)*x)*sqrt(g*x + f))/(e^4*f^3
*g^2 - 2*d*e^3*f^2*g^3 + d^2*e^2*f*g^4 + (e^4*f^2*g^3 - 2*d*e^3*f*g^4 + d^2*e^2*g^5)*x)]

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Sympy [A]  time = 20.3886, size = 104, normalized size = 0.93 \begin{align*} \frac{2 c \sqrt{f + g x}}{e g^{2}} - \frac{2 \left (a g^{2} + c f^{2}\right )}{g^{2} \sqrt{f + g x} \left (d g - e f\right )} - \frac{2 \left (a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{f + g x}}{\sqrt{\frac{d g - e f}{e}}} \right )}}{e^{2} \sqrt{\frac{d g - e f}{e}} \left (d g - e f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c*sqrt(f + g*x)/(e*g**2) - 2*(a*g**2 + c*f**2)/(g**2*sqrt(f + g*x)*(d*g - e*f)) - 2*(a*e**2 + c*d**2)*atan(s
qrt(f + g*x)/sqrt((d*g - e*f)/e))/(e**2*sqrt((d*g - e*f)/e)*(d*g - e*f))

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Giac [A]  time = 1.16887, size = 136, normalized size = 1.21 \begin{align*} -\frac{2 \,{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right )}{{\left (d g e - f e^{2}\right )}^{\frac{3}{2}}} + \frac{2 \, \sqrt{g x + f} c e^{\left (-1\right )}}{g^{2}} - \frac{2 \,{\left (c f^{2} + a g^{2}\right )}}{{\left (d g^{3} - f g^{2} e\right )} \sqrt{g x + f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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