3.706 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx\)

Optimal. Leaf size=235 \[ -\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{g^3 \sqrt{d+e x}}+\frac{5 c d (c d f-a e g)^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{7/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.377965, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {862, 864, 874, 205} \[ -\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{g^3 \sqrt{d+e x}}+\frac{5 c d (c d f-a e g)^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{7/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^2),x]

[Out]

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

Rule 862

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

Rule 864

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

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{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac{(5 c d) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx}{2 g}\\ &=\frac{5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}-\frac{(5 c d (c d f-a e g)) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)} \, dx}{2 g^2}\\ &=-\frac{5 c d (c d f-a e g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^3 \sqrt{d+e x}}+\frac{5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac{\left (5 c d (c d f-a e g)^2\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}{2 g^3}\\ &=-\frac{5 c d (c d f-a e g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^3 \sqrt{d+e x}}+\frac{5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac{\left (5 c d e^2 (c d f-a e g)^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 )}{g^3}\\ &=-\frac{5 c d (c d f-a e g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^3 \sqrt{d+e x}}+\frac{5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac{5 c d (c d f-a e g)^{3/2} \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 )}{g^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0759045, size = 75, normalized size = 0.32 \[ \frac{2 c d ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{7 (d+e x)^{7/2} (c d f-a e g)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^2),x]

[Out]

(2*c*d*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(
7*(c*d*f - a*e*g)^2*(d + e*x)^(7/2))

________________________________________________________________________________________

Maple [B]  time = 0.336, size = 523, normalized size = 2.2 \begin{align*} -{\frac{1}{3\,{g}^{3} \left ( gx+f \right ) }\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{a}^{2}cd{e}^{2}{g}^{3}-30\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) xa{c}^{2}{d}^{2}ef{g}^{2}+15\,{\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 ){a}^{2}cd{e}^{2}f{g}^{2}-30\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) a{c}^{2}{d}^{2}e{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}-2\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{x}^{2}{c}^{2}{d}^{2}{g}^{2}-14\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}xacde{g}^{2}+10\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}x{c}^{2}{d}^{2}fg+3\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{2}{e}^{2}{g}^{2}-20\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}acdefg+15\,\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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^2,x)

[Out]

-1/3*(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*a^2*c*
d*e^2*g^3-30*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x*a*c^2*d^2*e*f*g^2+15*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))*a^2*c*
d*e^2*f*g^2-30*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*a*c^2*d^2*e*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^3-2*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x^2*c^2*d^2*g^2-14*((a
*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x*a*c*d*e*g^2+10*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x*c^2*d^2*f*
g+3*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^2*g^2-20*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e
*f*g+15*((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)/g^3/(g*x+f)/((a
*e*g-c*d*f)*g)^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^2,x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^2), x)

________________________________________________________________________________________

Fricas [A]  time = 2.30612, size = 1423, normalized size = 6.06 \begin{align*} \left [-\frac{15 \,{\left (c^{2} d^{3} f^{2} - a c d^{2} e f g +{\left (c^{2} d^{2} e f g - a c d e^{2} g^{2}\right )} x^{2} +{\left (c^{2} d^{2} e f^{2} - a c d^{2} e g^{2} +{\left (c^{2} d^{3} - a c d e^{2}\right )} f g\right )} x\right )} \sqrt{-\frac{c d f - a e g}{g}} \log \left (-\frac{c d e g x^{2} - c d^{2} f + 2 \, a d e g - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} g \sqrt{-\frac{c d f - a e g}{g}} -{\left (c d e f -{\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f +{\left (e f + d g\right )} x}\right ) - 2 \,{\left (2 \, c^{2} d^{2} g^{2} x^{2} - 15 \, c^{2} d^{2} f^{2} + 20 \, a c d e f g - 3 \, a^{2} e^{2} g^{2} - 2 \,{\left (5 \, c^{2} d^{2} f g - 7 \, a c d e g^{2}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{6 \,{\left (e g^{4} x^{2} + d f g^{3} +{\left (e f g^{3} + d g^{4}\right )} x\right )}}, -\frac{15 \,{\left (c^{2} d^{3} f^{2} - a c d^{2} e f g +{\left (c^{2} d^{2} e f g - a c d e^{2} g^{2}\right )} x^{2} +{\left (c^{2} d^{2} e f^{2} - a c d^{2} e g^{2} +{\left (c^{2} d^{3} - a c d e^{2}\right )} f g\right )} x\right )} \sqrt{\frac{c d f - a e g}{g}} \arctan \left (\frac{\sqrt{e x + d} \sqrt{\frac{c d f - a e g}{g}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\right ) -{\left (2 \, c^{2} d^{2} g^{2} x^{2} - 15 \, c^{2} d^{2} f^{2} + 20 \, a c d e f g - 3 \, a^{2} e^{2} g^{2} - 2 \,{\left (5 \, c^{2} d^{2} f g - 7 \, a c d e g^{2}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{3 \,{\left (e g^{4} x^{2} + d f g^{3} +{\left (e f g^{3} + d g^{4}\right )} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^2,x, algorithm="fricas")

[Out]

[-1/6*(15*(c^2*d^3*f^2 - a*c*d^2*e*f*g + (c^2*d^2*e*f*g - a*c*d*e^2*g^2)*x^2 + (c^2*d^2*e*f^2 - a*c*d^2*e*g^2
+ (c^2*d^3 - a*c*d*e^2)*f*g)*x)*sqrt(-(c*d*f - a*e*g)/g)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - 2*sqrt(c*d*
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*g*sqrt(-(c*d*f - a*e*g)/g) - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*
x)/(e*g*x^2 + d*f + (e*f + d*g)*x)) - 2*(2*c^2*d^2*g^2*x^2 - 15*c^2*d^2*f^2 + 20*a*c*d*e*f*g - 3*a^2*e^2*g^2 -
2*(5*c^2*d^2*f*g - 7*a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(e*g^4*x^2 +
d*f*g^3 + (e*f*g^3 + d*g^4)*x), -1/3*(15*(c^2*d^3*f^2 - a*c*d^2*e*f*g + (c^2*d^2*e*f*g - a*c*d*e^2*g^2)*x^2 +
(c^2*d^2*e*f^2 - a*c*d^2*e*g^2 + (c^2*d^3 - a*c*d*e^2)*f*g)*x)*sqrt((c*d*f - a*e*g)/g)*arctan(sqrt(e*x + d)*sq
rt((c*d*f - a*e*g)/g)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)) - (2*c^2*d^2*g^2*x^2 - 15*c^2*d^2*f^2 + 20*
a*c*d*e*f*g - 3*a^2*e^2*g^2 - 2*(5*c^2*d^2*f*g - 7*a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)
*sqrt(e*x + d))/(e*g^4*x^2 + d*f*g^3 + (e*f*g^3 + d*g^4)*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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2)/(g*x+f)**2,x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^2,x, algorithm="giac")

[Out]

Exception raised: AttributeError