3.670 \(\int \frac{(d+e x)^{3/2}}{(f+g x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\)
Optimal. Leaf size=202 \[ -\frac{3 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} (f+g x) (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}-\frac{3 c d \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{(c d f-a e g)^{5/2}} \]
[Out]
(-2*Sqrt[d + e*x])/((c*d*f - a*e*g)*(f + g*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (3*g*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/((c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)) - (3*c*d*Sqrt[g]*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])])/(c*d*f - a*e*g)^(5/2)
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Rubi [A] time = 0.25665, antiderivative size = 202, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used =
{868, 872, 874, 205} \[ -\frac{3 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} (f+g x) (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}-\frac{3 c d \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{(c d f-a e g)^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^(3/2)/((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
[Out]
(-2*Sqrt[d + e*x])/((c*d*f - a*e*g)*(f + g*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (3*g*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/((c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)) - (3*c*d*Sqrt[g]*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])])/(c*d*f - a*e*g)^(5/2)
Rule 868
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))/((p + 1)*(c*e*f + c*d*g - b*e*g)), x]
+ Dist[(e^2*g*(m - n - 2))/((p + 1)*(c*e*f + c*d*g - b*e*g)), 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] && LtQ[p, -1] && RationalQ[n]
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{(d+e x)^{3/2}}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac{2 \sqrt{d+e x}}{(c d f-a e g) (f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{(3 g) \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}{c d f-a e g}\\ &=-\frac{2 \sqrt{d+e x}}{(c d f-a e g) (f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{3 g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt{d+e x} (f+g x)}-\frac{(3 c d g) \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 (c d f-a e g)^2}\\ &=-\frac{2 \sqrt{d+e x}}{(c d f-a e g) (f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{3 g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt{d+e x} (f+g x)}-\frac{\left (3 c d e^2 g\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 )}{(c d f-a e g)^2}\\ &=-\frac{2 \sqrt{d+e x}}{(c d f-a e g) (f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{3 g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt{d+e x} (f+g x)}-\frac{3 c d \sqrt{g} \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 )}{(c d f-a e g)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0357541, size = 73, normalized size = 0.36 \[ -\frac{2 c d \sqrt{d+e x} \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{\sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^(3/2)/((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
[Out]
(-2*c*d*Sqrt[d + e*x]*Hypergeometric2F1[-1/2, 2, 1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*f - a*e*g)^
2*Sqrt[(a*e + c*d*x)*(d + e*x)])
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Maple [A] time = 0.342, size = 225, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ( cdx+ae \right ) \left ( aeg-cdf \right ) ^{2} \left ( gx+f \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) \sqrt{cdx+ae}xcd{g}^{2}+3\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) \sqrt{cdx+ae}cdfg-3\,\sqrt{ \left ( aeg-cdf \right ) g}xcdg-\sqrt{ \left ( aeg-cdf \right ) g}aeg-2\,\sqrt{ \left ( aeg-cdf \right ) g}cdf \right ){\frac{1}{\sqrt{ex+d}}}{\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)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
[Out]
(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(3*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*(c*d*x+a*e)^(1
/2)*x*c*d*g^2+3*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*(c*d*x+a*e)^(1/2)*c*d*f*g-3*((a*e*g-c*d*f
)*g)^(1/2)*x*c*d*g-((a*e*g-c*d*f)*g)^(1/2)*a*e*g-2*((a*e*g-c*d*f)*g)^(1/2)*c*d*f)/(e*x+d)^(1/2)/(c*d*x+a*e)/(a
*e*g-c*d*f)^2/(g*x+f)/((a*e*g-c*d*f)*g)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")
[Out]
integrate((e*x + d)^(3/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^2), x)
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Fricas [B] time = 1.83021, size = 2176, normalized size = 10.77 \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)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")
[Out]
[1/2*(3*(c^2*d^2*e*g*x^3 + a*c*d^2*e*f + (c^2*d^2*e*f + (c^2*d^3 + a*c*d*e^2)*g)*x^2 + (a*c*d^2*e*g + (c^2*d^3
+ a*c*d*e^2)*f)*x)*sqrt(-g/(c*d*f - a*e*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)*(c*d*f - a*e*g)*sqrt(e*x + d)*sqrt(-g/(c*d*f - a*e*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*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d*g*x + 2*c*d*f + a
*e*g)*sqrt(e*x + d))/(a*c^2*d^3*e*f^3 - 2*a^2*c*d^2*e^2*f^2*g + a^3*d*e^3*f*g^2 + (c^3*d^3*e*f^2*g - 2*a*c^2*d
^2*e^2*f*g^2 + a^2*c*d*e^3*g^3)*x^3 + (c^3*d^3*e*f^3 + (c^3*d^4 - a*c^2*d^2*e^2)*f^2*g - (2*a*c^2*d^3*e + a^2*
c*d*e^3)*f*g^2 + (a^2*c*d^2*e^2 + a^3*e^4)*g^3)*x^2 + (a^3*d*e^3*g^3 + (c^3*d^4 + a*c^2*d^2*e^2)*f^3 - (a*c^2*
d^3*e + 2*a^2*c*d*e^3)*f^2*g - (a^2*c*d^2*e^2 - a^3*e^4)*f*g^2)*x), -(3*(c^2*d^2*e*g*x^3 + a*c*d^2*e*f + (c^2*
d^2*e*f + (c^2*d^3 + a*c*d*e^2)*g)*x^2 + (a*c*d^2*e*g + (c^2*d^3 + a*c*d*e^2)*f)*x)*sqrt(g/(c*d*f - a*e*g))*ar
ctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*f - a*e*g)*sqrt(e*x + d)*sqrt(g/(c*d*f - a*e*g))/(c*d*e
*g*x^2 + a*d*e*g + (c*d^2 + a*e^2)*g*x)) + sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d*g*x + 2*c*d*f +
a*e*g)*sqrt(e*x + d))/(a*c^2*d^3*e*f^3 - 2*a^2*c*d^2*e^2*f^2*g + a^3*d*e^3*f*g^2 + (c^3*d^3*e*f^2*g - 2*a*c^2*
d^2*e^2*f*g^2 + a^2*c*d*e^3*g^3)*x^3 + (c^3*d^3*e*f^3 + (c^3*d^4 - a*c^2*d^2*e^2)*f^2*g - (2*a*c^2*d^3*e + a^2
*c*d*e^3)*f*g^2 + (a^2*c*d^2*e^2 + a^3*e^4)*g^3)*x^2 + (a^3*d*e^3*g^3 + (c^3*d^4 + a*c^2*d^2*e^2)*f^3 - (a*c^2
*d^3*e + 2*a^2*c*d*e^3)*f^2*g - (a^2*c*d^2*e^2 - a^3*e^4)*f*g^2)*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((e*x+d)**(3/2)/(g*x+f)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")
[Out]
sage0*x