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

Optimal. Leaf size=116 \[ -\frac{2 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} \sqrt{e f-d g}}+\frac{2 \sqrt{f+g x} (b e g-c (d g+e f))}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2} \]

[Out]

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

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Rubi [A]  time = 0.168351, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {897, 1153, 208} \[ -\frac{2 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} \sqrt{e f-d g}}+\frac{2 \sqrt{f+g x} (b e g-c (d g+e f))}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 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+b x+c x^2}{(d+e x) \sqrt{f+g x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\frac{c f^2-b f g+a g^2}{g^2}-\frac{(2 c f-b g) x^2}{g^2}+\frac{c x^4}{g^2}}{\frac{-e f+d g}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{b e g-c (e f+d g)}{e^2 g}+\frac{c x^2}{e g}+\frac{c d^2-b d e+a e^2}{e^2 \left (d-\frac{e f}{g}+\frac{e x^2}{g}\right )}\right ) \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=\frac{2 (b e g-c (e f+d g)) \sqrt{f+g x}}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2}+\frac{\left (2 \left (c d^2-b d e+a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{d-\frac{e f}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{e^2 g}\\ &=\frac{2 (b e g-c (e f+d g)) \sqrt{f+g x}}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2}-\frac{2 \left (c d^2-b d e+a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} \sqrt{e f-d g}}\\ \end{align*}

Mathematica [A]  time = 0.199735, size = 118, normalized size = 1.02 \[ \frac{2 \left (-\frac{g^2 \left (c d^2-e (b d-a e)\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} \sqrt{e f-d g}}+\frac{\sqrt{f+g x} (b e g-c (d g+e f))}{e^2}+\frac{c (f+g x)^{3/2}}{3 e}\right )}{g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.217, size = 189, normalized size = 1.6 \begin{align*}{\frac{2\,c}{3\,e{g}^{2}} \left ( gx+f \right ) ^{{\frac{3}{2}}}}+2\,{\frac{b\sqrt{gx+f}}{eg}}-2\,{\frac{\sqrt{gx+f}cd}{g{e}^{2}}}-2\,{\frac{\sqrt{gx+f}cf}{e{g}^{2}}}+2\,{\frac{a}{\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }-2\,{\frac{bd}{e\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}}{{e}^{2}\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+b*x+a)/(e*x+d)/(g*x+f)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.85776, size = 714, normalized size = 6.16 \begin{align*} \left [\frac{3 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{e^{2} f - d e g} g^{2} \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^{2} +{\left (c d e^{2} - 3 \, b e^{3}\right )} f g - 3 \,{\left (c d^{2} e - b d e^{2}\right )} g^{2} -{\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt{g x + f}}{3 \,{\left (e^{4} f g^{2} - d e^{3} g^{3}\right )}}, \frac{2 \,{\left (3 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-e^{2} f + d e g} g^{2} \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^{2} +{\left (c d e^{2} - 3 \, b e^{3}\right )} f g - 3 \,{\left (c d^{2} e - b d e^{2}\right )} g^{2} -{\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt{g x + f}\right )}}{3 \,{\left (e^{4} f g^{2} - d e^{3} g^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 31.4552, size = 112, normalized size = 0.97 \begin{align*} \frac{2 c \left (f + g x\right )^{\frac{3}{2}}}{3 e g^{2}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right ) \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{e}{d g - e f}} \sqrt{f + g x}} \right )}}{e^{2} \sqrt{\frac{e}{d g - e f}} \left (d g - e f\right )} + \frac{2 \sqrt{f + g x} \left (b e g - c d g - c e f\right )}{e^{2} g^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12425, size = 173, normalized size = 1.49 \begin{align*} \frac{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right ) e^{\left (-2\right )}}{\sqrt{d g e - f e^{2}}} - \frac{2 \,{\left (3 \, \sqrt{g x + f} c d g^{5} e -{\left (g x + f\right )}^{\frac{3}{2}} c g^{4} e^{2} + 3 \, \sqrt{g x + f} c f g^{4} e^{2} - 3 \, \sqrt{g x + f} b g^{5} e^{2}\right )} e^{\left (-3\right )}}{3 \, g^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(c*d^2 - b*d*e + a*e^2)*arctan(sqrt(g*x + f)*e/sqrt(d*g*e - f*e^2))*e^(-2)/sqrt(d*g*e - f*e^2) - 2/3*(3*sqrt
(g*x + f)*c*d*g^5*e - (g*x + f)^(3/2)*c*g^4*e^2 + 3*sqrt(g*x + f)*c*f*g^4*e^2 - 3*sqrt(g*x + f)*b*g^5*e^2)*e^(
-3)/g^6