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

Optimal. Leaf size=140 \[ -\frac{\sqrt{f+g x} \left (a+\frac{d (c d-b e)}{e^2}\right )}{(d+e x) (e f-d g)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) (c d (4 e f-3 d g)-e (-a e g-b d g+2 b e f))}{e^{5/2} (e f-d g)^{3/2}}+\frac{2 c \sqrt{f+g x}}{e^2 g} \]

[Out]

(2*c*Sqrt[f + g*x])/(e^2*g) - ((a + (d*(c*d - b*e))/e^2)*Sqrt[f + g*x])/((e*f - d*g)*(d + e*x)) + ((c*d*(4*e*f
- 3*d*g) - e*(2*b*e*f - b*d*g - a*e*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 = 0.290021, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {897, 1157, 388, 208} \[ -\frac{\sqrt{f+g x} \left (a+\frac{d (c d-b e)}{e^2}\right )}{(d+e x) (e f-d g)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) (c d (4 e f-3 d g)-e (-a e g-b d g+2 b e f))}{e^{5/2} (e f-d g)^{3/2}}+\frac{2 c \sqrt{f+g x}}{e^2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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+b x+c x^2}{(d+e x)^2 \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}}{\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{d (c d-b e)}{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{b d}{e}-\frac{2 c f^2}{g^2}+\frac{2 b f}{g}+\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{d (c d-b e)}{e^2}\right ) \sqrt{f+g x}}{(e f-d g) (d+e x)}-\frac{(c d (4 e f-3 d g)-e (2 b e f-b d g-a e g)) \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^2 g (e f-d g)}\\ &=\frac{2 c \sqrt{f+g x}}{e^2 g}-\frac{\left (a+\frac{d (c d-b e)}{e^2}\right ) \sqrt{f+g x}}{(e f-d g) (d+e x)}+\frac{(c d (4 e f-3 d g)-e (2 b e f-b d g-a e g)) \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 = 0.647903, size = 150, normalized size = 1.07 \[ \frac{\sqrt{f+g x} \left (e g (b d-a e)+c \left (-3 d^2 g+2 d e (f-g x)+2 e^2 f x\right )\right )}{e^2 g (d+e x) (e f-d g)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) (e (-a e g-b d g+2 b e f)+c d (3 d g-4 e f))}{e^{5/2} (e f-d g)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[f + g*x]*(e*(b*d - a*e)*g + c*(-3*d^2*g + 2*e^2*f*x + 2*d*e*(f - g*x))))/(e^2*g*(e*f - d*g)*(d + e*x)) -
((c*d*(-4*e*f + 3*d*g) + e*(2*b*e*f - b*d*g - a*e*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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Maple [B]  time = 0.263, size = 371, normalized size = 2.7 \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{bdg}{ \left ( dg-ef \right ) e \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}}}}+{\frac{bdg}{ \left ( dg-ef \right ) 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{bf}{ \left ( dg-ef \right ) \sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }-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+b*x+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/(d*g-e*f)*(g*x+f)^(1/2)/(e*g*x+d*g)*b*d+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+g/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))*b*d-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))*b*f-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))*d*c*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+b*x+a)/(e*x+d)^2/(g*x+f)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.88276, size = 1300, normalized size = 9.29 \begin{align*} \left [-\frac{\sqrt{e^{2} f - d e g}{\left (2 \,{\left (2 \, c d^{2} e - b d e^{2}\right )} f g -{\left (3 \, c d^{3} - b d^{2} e - a d e^{2}\right )} g^{2} +{\left (2 \,{\left (2 \, c d e^{2} - b e^{3}\right )} f g -{\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} g^{2}\right )} x\right )} \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} - b d e^{3} + a e^{4}\right )} f g +{\left (3 \, c d^{3} e - b d^{2} e^{2} + 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{\sqrt{-e^{2} f + d e g}{\left (2 \,{\left (2 \, c d^{2} e - b d e^{2}\right )} f g -{\left (3 \, c d^{3} - b d^{2} e - a d e^{2}\right )} g^{2} +{\left (2 \,{\left (2 \, c d e^{2} - b e^{3}\right )} f g -{\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} g^{2}\right )} x\right )} \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} - b d e^{3} + a e^{4}\right )} f g +{\left (3 \, c d^{3} e - b d^{2} e^{2} + 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+b*x+a)/(e*x+d)^2/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.1144, size = 236, normalized size = 1.69 \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 - b d g e + 2 \, b f e^{2} - 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} b d g e + \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+b*x+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 - b*d*g*e + 2*b*f*e^2 - a*g*e^2)*arctan(sqrt(g*x + f)*e/sq
rt(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)*b*d*g*e +
sqrt(g*x + f)*a*g*e^2)/((d*g*e^2 - f*e^3)*(d*g + (g*x + f)*e - f*e))