3.2857 \(\int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx\)
Optimal. Leaf size=204 \[ \frac{2 \sqrt{f} \sqrt{a+b x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|-\frac{b (d e-c f)}{(b c-a d) f}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) \sqrt{-\frac{d (a+b x)}{b c-a d}}}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d) (b e-a f)} \]
[Out]
(-2*b*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) + (2*Sqrt[f]*Sqrt[-(d*e) + c*f]*Sqr
t[a + b*x]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], -((b*
(d*e - c*f))/((b*c - a*d)*f))])/((b*c - a*d)*(b*e - a*f)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Sqrt[e + f*x])
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Rubi [A] time = 0.136025, antiderivative size = 204, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used =
{104, 21, 114, 113} \[ \frac{2 \sqrt{f} \sqrt{a+b x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|-\frac{b (d e-c f)}{(b c-a d) f}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) \sqrt{-\frac{d (a+b x)}{b c-a d}}}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
[Out]
(-2*b*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) + (2*Sqrt[f]*Sqrt[-(d*e) + c*f]*Sqr
t[a + b*x]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], -((b*
(d*e - c*f))/((b*c - a*d)*f))])/((b*c - a*d)*(b*e - a*f)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Sqrt[e + f*x])
Rule 104
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]
Rule 21
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])
Rule 114
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) && !LtQ[-((b*c - a*d)/d), 0]
Rule 113
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !LtQ[-((b*c - a*d)/d),
0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)
/b, 0])
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx &=-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{(b c-a d) (b e-a f) \sqrt{a+b x}}-\frac{2 \int \frac{-\frac{1}{2} a d f-\frac{1}{2} b d f x}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{(b c-a d) (b e-a f)}\\ &=-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{(b c-a d) (b e-a f) \sqrt{a+b x}}+\frac{(d f) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{(b c-a d) (b e-a f)}\\ &=-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{(b c-a d) (b e-a f) \sqrt{a+b x}}+\frac{\left (d f \sqrt{a+b x} \sqrt{\frac{d (e+f x)}{d e-c f}}\right ) \int \frac{\sqrt{\frac{a d}{-b c+a d}+\frac{b d x}{-b c+a d}}}{\sqrt{c+d x} \sqrt{\frac{d e}{d e-c f}+\frac{d f x}{d e-c f}}} \, dx}{(b c-a d) (b e-a f) \sqrt{\frac{d (a+b x)}{-b c+a d}} \sqrt{e+f x}}\\ &=-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{(b c-a d) (b e-a f) \sqrt{a+b x}}+\frac{2 \sqrt{f} \sqrt{-d e+c f} \sqrt{a+b x} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{-d e+c f}}\right )|-\frac{b (d e-c f)}{(b c-a d) f}\right )}{(b c-a d) (b e-a f) \sqrt{-\frac{d (a+b x)}{b c-a d}} \sqrt{e+f x}}\\ \end{align*}
Mathematica [C] time = 1.09885, size = 201, normalized size = 0.99 \[ \frac{2 b \sqrt{c+d x} \sqrt{e+f x} \left (-1-\frac{i \sqrt{\frac{d (a+b x)}{b (c+d x)}} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{d (a+b x)}{b c-a d}}\right )|\frac{b c f-a d f}{b d e-a d f}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{d (a+b x)}{b c-a d}}\right ),\frac{b c f-a d f}{b d e-a d f}\right )\right )}{\sqrt{\frac{b (e+f x)}{b e-a f}}}\right )}{\sqrt{a+b x} (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
[Out]
(2*b*Sqrt[c + d*x]*Sqrt[e + f*x]*(-1 - (I*Sqrt[(d*(a + b*x))/(b*(c + d*x))]*(EllipticE[I*ArcSinh[Sqrt[(d*(a +
b*x))/(b*c - a*d)]], (b*c*f - a*d*f)/(b*d*e - a*d*f)] - EllipticF[I*ArcSinh[Sqrt[(d*(a + b*x))/(b*c - a*d)]],
(b*c*f - a*d*f)/(b*d*e - a*d*f)]))/Sqrt[(b*(e + f*x))/(b*e - a*f)]))/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x])
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Maple [B] time = 0.042, size = 1011, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)
[Out]
2*(EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*d*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*
(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d
/(a*f-b*e))^(1/2))*a*b*c*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/
2)-EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b*d*e*(d*(b*x+a)/(a*d-b*c))^(1/2)*
(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d
/(a*f-b*e))^(1/2))*b^2*c*e*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/
2)-EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*d*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*
(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d
/(a*f-b*e))^(1/2))*a*b*c*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/
2)+EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b*d*e*(d*(b*x+a)/(a*d-b*c))^(1/2)*
(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d
/(a*f-b*e))^(1/2))*b^2*c*e*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/
2)-x^2*b^2*d*f-x*b^2*c*f-x*b^2*d*e-b^2*c*e)*(f*x+e)^(1/2)*(d*x+c)^(1/2)*(b*x+a)^(1/2)/b/(a*f-b*e)/(a*d-b*c)/(b
*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e}}{b^{2} d f x^{4} + a^{2} c e +{\left (b^{2} d e +{\left (b^{2} c + 2 \, a b d\right )} f\right )} x^{3} +{\left ({\left (b^{2} c + 2 \, a b d\right )} e +{\left (2 \, a b c + a^{2} d\right )} f\right )} x^{2} +{\left (a^{2} c f +{\left (2 \, a b c + a^{2} d\right )} e\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="fricas")
[Out]
integral(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)/(b^2*d*f*x^4 + a^2*c*e + (b^2*d*e + (b^2*c + 2*a*b*d)*f)*x^
3 + ((b^2*c + 2*a*b*d)*e + (2*a*b*c + a^2*d)*f)*x^2 + (a^2*c*f + (2*a*b*c + a^2*d)*e)*x), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \sqrt{e + f x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)
[Out]
Integral(1/((a + b*x)**(3/2)*sqrt(c + d*x)*sqrt(e + f*x)), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")
[Out]
integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)), x)