3.27.43 \(\int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx\) [2643]

Optimal. Leaf size=134 \[ \frac {2 \sqrt {-b c+a d} \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}} \]

[Out]

2*EllipticE(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))*(a*d-b*c)^(1/2)*(b*(d*x+c
)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/b/d^(1/2)/(d*x+c)^(1/2)/(b*(f*x+e)/(-a*f+b*e))^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {115, 114} \begin {gather*} \frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\text {ArcSin}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]),x]

[Out]

(2*Sqrt[-(b*c) + a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/S
qrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqrt[d]*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]
)

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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])

Rule 115

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx &=\frac {\left (\sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x}\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{\sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}\\ &=\frac {2 \sqrt {-b c+a d} \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}\\ \end {align*}

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Mathematica [A]
time = 8.61, size = 154, normalized size = 1.15 \begin {gather*} \frac {2 \sqrt {c+d x} \left (\frac {e+f x}{\sqrt {a+b x}}+\frac {(-b e+a f) \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {a-\frac {b e}{f}}}{\sqrt {a+b x}}\right )|\frac {b c f-a d f}{b d e-a d f}\right )}{b \sqrt {a-\frac {b e}{f}} \sqrt {\frac {b (c+d x)}{d (a+b x)}}}\right )}{d \sqrt {e+f x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]),x]

[Out]

(2*Sqrt[c + d*x]*((e + f*x)/Sqrt[a + b*x] + ((-(b*e) + a*f)*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[ArcSin
[Sqrt[a - (b*e)/f]/Sqrt[a + b*x]], (b*c*f - a*d*f)/(b*d*e - a*d*f)])/(b*Sqrt[a - (b*e)/f]*Sqrt[(b*(c + d*x))/(
d*(a + b*x))])))/(d*Sqrt[e + f*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(557\) vs. \(2(117)=234\).
time = 0.08, size = 558, normalized size = 4.16

method	result	size
elliptic	\(\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right )}\, \left (\frac {2 e \left (-\frac {c}{d}+\frac {e}{f}\right ) \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}+\frac {2 f \left (-\frac {c}{d}+\frac {e}{f}\right ) \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}+\frac {a}{b}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )-\frac {a \EllipticF \left (\sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{b}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}\)	\(498\)
default	\(\frac {2 \left (\EllipticF \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) a c \,f^{2}-\EllipticF \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) a d e f -\EllipticF \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) b c e f +\EllipticF \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) b d \,e^{2}-\EllipticE \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) a c \,f^{2}+\EllipticE \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) a d e f +\EllipticE \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) b c e f -\EllipticE \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) b d \,e^{2}\right ) \sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}\, \sqrt {\frac {\left (b x +a \right ) f}{a f -b e}}\, \sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}\, \sqrt {f x +e}\, \sqrt {b x +a}\, \sqrt {d x +c}}{f d b \left (b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e \right )}\)	\(558\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*(EllipticF((-(f*x+e)*d/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*a*c*f^2-EllipticF((-(f*x+e)*d/(c*f-
d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*a*d*e*f-EllipticF((-(f*x+e)*d/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*
f-b*e))^(1/2))*b*c*e*f+EllipticF((-(f*x+e)*d/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*b*d*e^2-Ellipti
cE((-(f*x+e)*d/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*a*c*f^2+EllipticE((-(f*x+e)*d/(c*f-d*e))^(1/2
),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*a*d*e*f+EllipticE((-(f*x+e)*d/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1
/2))*b*c*e*f-EllipticE((-(f*x+e)*d/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*b*d*e^2)/f*((d*x+c)*f/(c*
f-d*e))^(1/2)*((b*x+a)*f/(a*f-b*e))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)/d/b*(f*x+e)^(1/2)*(b*x+a)^(1/2)*(d*x+c)
^(1/2)/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(f*x + e)/(sqrt(b*x + a)*sqrt(d*x + c)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.33, size = 664, normalized size = 4.96 \begin {gather*} -\frac {2 \, {\left (3 \, \sqrt {b d f} b d f {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right )\right ) - \sqrt {b d f} {\left (2 \, b d e - {\left (b c + a d\right )} f\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right )\right )}}{3 \, b^{2} d^{2} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*sqrt(b*d*f)*b*d*f*weierstrassZeta(4/3*(b^2*d^2*e^2 + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2 - (b^2*c*d + a*
b*d^2)*f*e)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3
- 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*f^2*e - 3*(b^3*c*d^2 + a*b^2*d^3)*f*e^2)/(b^3*d^3*f^3), weierstras
sPInverse(4/3*(b^2*d^2*e^2 + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2 - (b^2*c*d + a*b*d^2)*f*e)/(b^2*d^2*f^2), -4/27
*(2*b^3*d^3*e^3 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3 - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 +
a^2*b*d^3)*f^2*e - 3*(b^3*c*d^2 + a*b^2*d^3)*f*e^2)/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b
*d*f))) - sqrt(b*d*f)*(2*b*d*e - (b*c + a*d)*f)*weierstrassPInverse(4/3*(b^2*d^2*e^2 + (b^2*c^2 - a*b*c*d + a^
2*d^2)*f^2 - (b^2*c*d + a*b*d^2)*f*e)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2
*b*c*d^2 + 2*a^3*d^3)*f^3 - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*f^2*e - 3*(b^3*c*d^2 + a*b^2*d^3)*f*e^2)
/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f)))/(b^2*d^2*f)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e + f x}}{\sqrt {a + b x} \sqrt {c + d x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**(1/2)/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)

[Out]

Integral(sqrt(e + f*x)/(sqrt(a + b*x)*sqrt(c + d*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(f*x + e)/(sqrt(b*x + a)*sqrt(d*x + c)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {e+f\,x}}{\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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