3.594 \(\int \frac{a+c x^2}{(d+e x)^2 \sqrt{f+g x}} \, dx\)

Optimal. Leaf size=122 \[ -\frac{\sqrt{f+g x} \left (a+\frac{c d^2}{e^2}\right )}{(d+e x) (e f-d g)}+\frac{\left (a e^2 g+c d (4 e f-3 d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} (e f-d g)^{3/2}}+\frac{2 c \sqrt{f+g x}}{e^2 g} \]

[Out]

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

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Rubi [A]  time = 0.203416, antiderivative size = 122, 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, Rules used = {898, 1157, 388, 208} \[ -\frac{\sqrt{f+g x} \left (a+\frac{c d^2}{e^2}\right )}{(d+e x) (e f-d g)}+\frac{\left (a e^2 g+c d (4 e f-3 d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} (e f-d g)^{3/2}}+\frac{2 c \sqrt{f+g x}}{e^2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*Sqrt[f + g*x])/(e^2*g) - ((a + (c*d^2)/e^2)*Sqrt[f + g*x])/((e*f - d*g)*(d + e*x)) + ((a*e^2*g + c*d*(4*e
*f - 3*d*g))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(5/2)*(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 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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)^2 \sqrt{f+g x}} \, 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}}{\left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )^2} \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=-\frac{\left (a+\frac{c d^2}{e^2}\right ) \sqrt{f+g x}}{(e f-d g) (d+e x)}+\frac{\operatorname{Subst}\left (\int \frac{-a+\frac{c d^2}{e^2}-\frac{2 c f^2}{g^2}+\frac{2 c (e f-d g) x^2}{e g^2}}{\frac{-e f+d g}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{e f-d g}\\ &=\frac{2 c \sqrt{f+g x}}{e^2 g}-\frac{\left (a+\frac{c d^2}{e^2}\right ) \sqrt{f+g x}}{(e f-d g) (d+e x)}-\frac{\left (a+\frac{c d (4 e f-3 d g)}{e^2 g}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{-e f+d g}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{e f-d g}\\ &=\frac{2 c \sqrt{f+g x}}{e^2 g}-\frac{\left (a+\frac{c d^2}{e^2}\right ) \sqrt{f+g x}}{(e f-d g) (d+e x)}+\frac{\left (a e^2 g+c d (4 e f-3 d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} (e f-d g)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.32149, size = 171, normalized size = 1.4 \[ \frac{\frac{\left (a e^2+c d^2\right ) \left (\sqrt{e} \sqrt{f+g x} (d g-e f)+g (d+e x) \sqrt{d g-e f} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{d g-e f}}\right )\right )}{(d+e x) (e f-d g)^2}+\frac{4 c d \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e f-d g}}+\frac{2 c \sqrt{e} \sqrt{f+g x}}{g}}{e^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.217, size = 237, normalized size = 1.9 \begin{align*} 2\,{\frac{c\sqrt{gx+f}}{{e}^{2}g}}+{\frac{ag}{ \left ( dg-ef \right ) \left ( egx+dg \right ) }\sqrt{gx+f}}+{\frac{c{d}^{2}g}{{e}^{2} \left ( dg-ef \right ) \left ( egx+dg \right ) }\sqrt{gx+f}}+{\frac{ag}{dg-ef}\arctan \left ({e\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}} \right ){\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}}-3\,{\frac{c{d}^{2}g}{{e}^{2} \left ( dg-ef \right ) \sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }+4\,{\frac{cdf}{ \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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)/(e*x+d)^2/(g*x+f)^(1/2),x)

[Out]

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

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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)^2/(g*x+f)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.86566, size = 1116, normalized size = 9.15 \begin{align*} \left [-\frac{{\left (4 \, c d^{2} e f g -{\left (3 \, c d^{3} - a d e^{2}\right )} g^{2} +{\left (4 \, c d e^{2} f g -{\left (3 \, c d^{2} e - a e^{3}\right )} g^{2}\right )} x\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 d e^{3} f^{2} -{\left (5 \, c d^{2} e^{2} + a e^{4}\right )} f g +{\left (3 \, c d^{3} e + a d e^{3}\right )} g^{2} + 2 \,{\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x\right )} \sqrt{g x + f}}{2 \,{\left (d e^{5} f^{2} g - 2 \, d^{2} e^{4} f g^{2} + d^{3} e^{3} g^{3} +{\left (e^{6} f^{2} g - 2 \, d e^{5} f g^{2} + d^{2} e^{4} g^{3}\right )} x\right )}}, -\frac{{\left (4 \, c d^{2} e f g -{\left (3 \, c d^{3} - a d e^{2}\right )} g^{2} +{\left (4 \, c d e^{2} f g -{\left (3 \, c d^{2} e - a e^{3}\right )} g^{2}\right )} x\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 d e^{3} f^{2} -{\left (5 \, c d^{2} e^{2} + a e^{4}\right )} f g +{\left (3 \, c d^{3} e + a d e^{3}\right )} g^{2} + 2 \,{\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x\right )} \sqrt{g x + f}}{d e^{5} f^{2} g - 2 \, d^{2} e^{4} f g^{2} + d^{3} e^{3} g^{3} +{\left (e^{6} f^{2} g - 2 \, d e^{5} f g^{2} + d^{2} e^{4} g^{3}\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(e*x+d)^2/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)/(e*x+d)**2/(g*x+f)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.14689, size = 200, normalized size = 1.64 \begin{align*} \frac{2 \, \sqrt{g x + f} c e^{\left (-2\right )}}{g} - \frac{{\left (3 \, c d^{2} g - 4 \, c d f e - a g e^{2}\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right )}{{\left (d g e^{2} - f e^{3}\right )} \sqrt{d g e - f e^{2}}} + \frac{\sqrt{g x + f} c d^{2} g + \sqrt{g x + f} a g e^{2}}{{\left (d g e^{2} - f e^{3}\right )}{\left (d g +{\left (g x + f\right )} e - f e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(e*x+d)^2/(g*x+f)^(1/2),x, algorithm="giac")

[Out]

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