3.2673 \(\int \frac{\sqrt{e+f x}}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx\)
Optimal. Leaf size=184 \[ \frac{2 \sqrt{f} \sqrt{c+d x} \sqrt{a f-b e} \sqrt{\frac{b (e+f x)}{b e-a f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{a+b x}}{\sqrt{a f-b e}}\right )|\frac{d (b e-a f)}{(b c-a d) f}\right )}{b \sqrt{e+f x} (b c-a d) \sqrt{\frac{b (c+d x)}{b c-a d}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d)} \]
[Out]
(-2*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*Sqrt[a + b*x]) + (2*Sqrt[f]*Sqrt[-(b*e) + a*f]*Sqrt[c + d*x]*Sqr
t[(b*(e + f*x))/(b*e - a*f)]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[a + b*x])/Sqrt[-(b*e) + a*f]], (d*(b*e - a*f))/((b
*c - a*d)*f)])/(b*(b*c - a*d)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x])
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Rubi [A] time = 0.113645, antiderivative size = 184, 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 =
{99, 21, 114, 113} \[ \frac{2 \sqrt{f} \sqrt{c+d x} \sqrt{a f-b e} \sqrt{\frac{b (e+f x)}{b e-a f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{a+b x}}{\sqrt{a f-b e}}\right )|\frac{d (b e-a f)}{(b c-a d) f}\right )}{b \sqrt{e+f x} (b c-a d) \sqrt{\frac{b (c+d x)}{b c-a d}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[e + f*x]/((a + b*x)^(3/2)*Sqrt[c + d*x]),x]
[Out]
(-2*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*Sqrt[a + b*x]) + (2*Sqrt[f]*Sqrt[-(b*e) + a*f]*Sqrt[c + d*x]*Sqr
t[(b*(e + f*x))/(b*e - a*f)]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[a + b*x])/Sqrt[-(b*e) + a*f]], (d*(b*e - a*f))/((b
*c - a*d)*f)])/(b*(b*c - a*d)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x])
Rule 99
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])
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{\sqrt{e+f x}}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx &=-\frac{2 \sqrt{c+d x} \sqrt{e+f x}}{(b c-a d) \sqrt{a+b x}}+\frac{2 \int \frac{\frac{c f}{2}+\frac{d f x}{2}}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{b c-a d}\\ &=-\frac{2 \sqrt{c+d x} \sqrt{e+f x}}{(b c-a d) \sqrt{a+b x}}+\frac{f \int \frac{\sqrt{c+d x}}{\sqrt{a+b x} \sqrt{e+f x}} \, dx}{b c-a d}\\ &=-\frac{2 \sqrt{c+d x} \sqrt{e+f x}}{(b c-a d) \sqrt{a+b x}}+\frac{\left (f \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}\right ) \int \frac{\sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}}}{\sqrt{a+b x} \sqrt{\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}}} \, dx}{(b c-a d) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x}}\\ &=-\frac{2 \sqrt{c+d x} \sqrt{e+f x}}{(b c-a d) \sqrt{a+b x}}+\frac{2 \sqrt{f} \sqrt{-b e+a f} \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{a+b x}}{\sqrt{-b e+a f}}\right )|\frac{d (b e-a f)}{(b c-a d) f}\right )}{b (b c-a d) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x}}\\ \end{align*}
Mathematica [A] time = 0.799666, size = 126, normalized size = 0.68 \[ -\frac{2 \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{d (a+b x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{a-\frac{b c}{d}}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )}{b \sqrt{c+d x} \sqrt{a-\frac{b c}{d}} \sqrt{\frac{b (e+f x)}{f (a+b x)}}} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[e + f*x]/((a + b*x)^(3/2)*Sqrt[c + d*x]),x]
[Out]
(-2*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[e + f*x]*EllipticE[ArcSin[Sqrt[a - (b*c)/d]/Sqrt[a + b*x]], (b*d*e
- a*d*f)/(b*c*f - a*d*f)])/(b*Sqrt[a - (b*c)/d]*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(f*(a + b*x))])
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Maple [B] time = 0.054, size = 1022, normalized size = 5.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x+e)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(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*b*c*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*d^2*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^2*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))*b^2*c*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))*a^2*d^2*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*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*d^2*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*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)-x^2*b^2*d^2*f-x*b^2*c*d*f-x*b^2*d^2*e-b^2*c*d*e)*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(f*x+e)^(1/2)
/d/b^2/(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{\sqrt{f x + e}}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(f*x + e)/((b*x + a)^(3/2)*sqrt(d*x + c)), 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 x^{3} + a^{2} c +{\left (b^{2} c + 2 \, a b d\right )} x^{2} +{\left (2 \, a b c + a^{2} d\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="fricas")
[Out]
integral(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)/(b^2*d*x^3 + a^2*c + (b^2*c + 2*a*b*d)*x^2 + (2*a*b*c + a^2
*d)*x), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e + f x}}{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)**(1/2)/(b*x+a)**(3/2)/(d*x+c)**(1/2),x)
[Out]
Integral(sqrt(e + f*x)/((a + b*x)**(3/2)*sqrt(c + d*x)), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{f x + e}}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(f*x + e)/((b*x + a)^(3/2)*sqrt(d*x + c)), x)