3.664 \(\int \frac{\sqrt{d+e x}}{(f+g x)^4 \sqrt{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=280 \[ \frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 \sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{5 c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{8 \sqrt{g} (c d f-a e g)^{7/2}}+\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^3 (c d f-a e g)} \]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) + (5*c*d*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^2) + (5*c^2*d^2*Sqrt[a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) + (5*c^3*d^3*ArcTan[(Sqrt[g]*Sqrt
[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])])/(8*Sqrt[g]*(c*d*f - a*e*g)^(7/2
))

________________________________________________________________________________________

Rubi [A]  time = 0.423973, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {872, 874, 205} \[ \frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 \sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{5 c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{8 \sqrt{g} (c d f-a e g)^{7/2}}+\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^3 (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) + (5*c*d*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^2) + (5*c^2*d^2*Sqrt[a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) + (5*c^3*d^3*ArcTan[(Sqrt[g]*Sqrt
[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])])/(8*Sqrt[g]*(c*d*f - a*e*g)^(7/2
))

Rule 872

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

Rule 874

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[
2*e^2, Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; Fre
eQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x}}{(f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt{d+e x} (f+g x)^3}+\frac{(5 c d) \int \frac{\sqrt{d+e x}}{(f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 (c d f-a e g)}\\ &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt{d+e x} (f+g x)^3}+\frac{5 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^2}+\frac{\left (5 c^2 d^2\right ) \int \frac{\sqrt{d+e x}}{(f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 (c d f-a e g)^2}\\ &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt{d+e x} (f+g x)^3}+\frac{5 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^2}+\frac{5 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 (c d f-a e g)^3 \sqrt{d+e x} (f+g x)}+\frac{\left (5 c^3 d^3\right ) \int \frac{\sqrt{d+e x}}{(f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 (c d f-a e g)^3}\\ &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt{d+e x} (f+g x)^3}+\frac{5 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^2}+\frac{5 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 (c d f-a e g)^3 \sqrt{d+e x} (f+g x)}+\frac{\left (5 c^3 d^3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}}\right )}{8 (c d f-a e g)^3}\\ &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt{d+e x} (f+g x)^3}+\frac{5 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^2}+\frac{5 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 (c d f-a e g)^3 \sqrt{d+e x} (f+g x)}+\frac{5 c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{c d f-a e g} \sqrt{d+e x}}\right )}{8 \sqrt{g} (c d f-a e g)^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0456276, size = 77, normalized size = 0.28 \[ \frac{2 c^3 d^3 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{\sqrt{d+e x} (c d f-a e g)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(2*c^3*d^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[1/2, 4, 3/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])
/((c*d*f - a*e*g)^4*Sqrt[d + e*x])

________________________________________________________________________________________

Maple [A]  time = 0.378, size = 450, normalized size = 1.6 \begin{align*}{\frac{1}{24\, \left ( aeg-cdf \right ) ^{3} \left ( gx+f \right ) ^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{3}{c}^{3}{d}^{3}{g}^{3}+45\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}{c}^{3}{d}^{3}f{g}^{2}+45\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{3}{d}^{3}{f}^{2}g+15\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){c}^{3}{d}^{3}{f}^{3}-15\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{x}^{2}{c}^{2}{d}^{2}{g}^{2}+10\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}xacde{g}^{2}-40\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}x{c}^{2}{d}^{2}fg-8\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{2}{e}^{2}{g}^{2}+26\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}acdefg-33\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{c}^{2}{d}^{2}{f}^{2} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

1/24*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(15*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x^3*c^3*
d^3*g^3+45*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x^2*c^3*d^3*f*g^2+45*arctanh((c*d*x+a*e)^(1/2)
*g/((a*e*g-c*d*f)*g)^(1/2))*x*c^3*d^3*f^2*g+15*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*f^
3-15*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x^2*c^2*d^2*g^2+10*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x*
a*c*d*e*g^2-40*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x*c^2*d^2*f*g-8*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(
1/2)*a^2*e^2*g^2+26*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*f*g-33*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*
e)^(1/2)*c^2*d^2*f^2)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/(a*e*g-c*d*f)^3/(g*x+f)^3/((a*e*g-c*d*f)*g)^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^4), x)

________________________________________________________________________________________

Fricas [B]  time = 2.00879, size = 4030, normalized size = 14.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

[1/48*(15*(c^3*d^3*e*g^3*x^4 + c^3*d^4*f^3 + (3*c^3*d^3*e*f*g^2 + c^3*d^4*g^3)*x^3 + 3*(c^3*d^3*e*f^2*g + c^3*
d^4*f*g^2)*x^2 + (c^3*d^3*e*f^3 + 3*c^3*d^4*f^2*g)*x)*sqrt(-c*d*f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2
*a*d*e*g - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*f*g + a
*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f + d*g)*x)) + 2*(33*c^3*d^3*f^3*g - 59*a*c^2*d^2*e*f^2*g^2 + 34*a^
2*c*d*e^2*f*g^3 - 8*a^3*e^3*g^4 + 15*(c^3*d^3*f*g^3 - a*c^2*d^2*e*g^4)*x^2 + 10*(4*c^3*d^3*f^2*g^2 - 5*a*c^2*d
^2*e*f*g^3 + a^2*c*d*e^2*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^5*f^7*g - 4
*a*c^3*d^4*e*f^6*g^2 + 6*a^2*c^2*d^3*e^2*f^5*g^3 - 4*a^3*c*d^2*e^3*f^4*g^4 + a^4*d*e^4*f^3*g^5 + (c^4*d^4*e*f^
4*g^4 - 4*a*c^3*d^3*e^2*f^3*g^5 + 6*a^2*c^2*d^2*e^3*f^2*g^6 - 4*a^3*c*d*e^4*f*g^7 + a^4*e^5*g^8)*x^4 + (3*c^4*
d^4*e*f^5*g^3 + a^4*d*e^4*g^8 + (c^4*d^5 - 12*a*c^3*d^3*e^2)*f^4*g^4 - 2*(2*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f
^3*g^5 + 6*(a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^2*g^6 - (4*a^3*c*d^2*e^3 - 3*a^4*e^5)*f*g^7)*x^3 + 3*(c^4*d^4*e
*f^6*g^2 + a^4*d*e^4*f*g^7 + (c^4*d^5 - 4*a*c^3*d^3*e^2)*f^5*g^3 - 2*(2*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^4*g
^4 + 2*(3*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^3*g^5 - (4*a^3*c*d^2*e^3 - a^4*e^5)*f^2*g^6)*x^2 + (c^4*d^4*e*f^7
*g + 3*a^4*d*e^4*f^2*g^6 + (3*c^4*d^5 - 4*a*c^3*d^3*e^2)*f^6*g^2 - 6*(2*a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^5*g^3
+ 2*(9*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^4*g^4 - (12*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g^5)*x), -1/24*(15*(c^3*d^
3*e*g^3*x^4 + c^3*d^4*f^3 + (3*c^3*d^3*e*f*g^2 + c^3*d^4*g^3)*x^3 + 3*(c^3*d^3*e*f^2*g + c^3*d^4*f*g^2)*x^2 +
(c^3*d^3*e*f^3 + 3*c^3*d^4*f^2*g)*x)*sqrt(c*d*f*g - a*e*g^2)*arctan(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
)*sqrt(c*d*f*g - a*e*g^2)*sqrt(e*x + d)/(c*d*e*g*x^2 + a*d*e*g + (c*d^2 + a*e^2)*g*x)) - (33*c^3*d^3*f^3*g - 5
9*a*c^2*d^2*e*f^2*g^2 + 34*a^2*c*d*e^2*f*g^3 - 8*a^3*e^3*g^4 + 15*(c^3*d^3*f*g^3 - a*c^2*d^2*e*g^4)*x^2 + 10*(
4*c^3*d^3*f^2*g^2 - 5*a*c^2*d^2*e*f*g^3 + a^2*c*d*e^2*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt
(e*x + d))/(c^4*d^5*f^7*g - 4*a*c^3*d^4*e*f^6*g^2 + 6*a^2*c^2*d^3*e^2*f^5*g^3 - 4*a^3*c*d^2*e^3*f^4*g^4 + a^4*
d*e^4*f^3*g^5 + (c^4*d^4*e*f^4*g^4 - 4*a*c^3*d^3*e^2*f^3*g^5 + 6*a^2*c^2*d^2*e^3*f^2*g^6 - 4*a^3*c*d*e^4*f*g^7
+ a^4*e^5*g^8)*x^4 + (3*c^4*d^4*e*f^5*g^3 + a^4*d*e^4*g^8 + (c^4*d^5 - 12*a*c^3*d^3*e^2)*f^4*g^4 - 2*(2*a*c^3
*d^4*e - 9*a^2*c^2*d^2*e^3)*f^3*g^5 + 6*(a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^2*g^6 - (4*a^3*c*d^2*e^3 - 3*a^4*e
^5)*f*g^7)*x^3 + 3*(c^4*d^4*e*f^6*g^2 + a^4*d*e^4*f*g^7 + (c^4*d^5 - 4*a*c^3*d^3*e^2)*f^5*g^3 - 2*(2*a*c^3*d^4
*e - 3*a^2*c^2*d^2*e^3)*f^4*g^4 + 2*(3*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^3*g^5 - (4*a^3*c*d^2*e^3 - a^4*e^5)*
f^2*g^6)*x^2 + (c^4*d^4*e*f^7*g + 3*a^4*d*e^4*f^2*g^6 + (3*c^4*d^5 - 4*a*c^3*d^3*e^2)*f^6*g^2 - 6*(2*a*c^3*d^4
*e - a^2*c^2*d^2*e^3)*f^5*g^3 + 2*(9*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^4*g^4 - (12*a^3*c*d^2*e^3 - a^4*e^5)*f
^3*g^5)*x)]

________________________________________________________________________________________

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((e*x+d)**(1/2)/(g*x+f)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [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((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

Timed out