3.26.17 \(\int \frac {1}{\sqrt {a+b x} (e+f x) \sqrt {2 b e-a f+b f x}} \, dx\) [2517]

Optimal. Leaf size=59 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {f} \sqrt {a+b x} \sqrt {2 b e-a f+b f x}}{b e-a f}\right )}{\sqrt {f} (b e-a f)} \]

[Out]

arctan(f^(1/2)*(b*x+a)^(1/2)*(b*f*x-a*f+2*b*e)^(1/2)/(-a*f+b*e))/(-a*f+b*e)/f^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {94, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {f} \sqrt {a+b x} \sqrt {-a f+2 b e+b f x}}{b e-a f}\right )}{\sqrt {f} (b e-a f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x]*(e + f*x)*Sqrt[2*b*e - a*f + b*f*x]),x]

[Out]

ArcTan[(Sqrt[f]*Sqrt[a + b*x]*Sqrt[2*b*e - a*f + b*f*x])/(b*e - a*f)]/(Sqrt[f]*(b*e - a*f))

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b x} (e+f x) \sqrt {2 b e-a f+b f x}} \, dx &=(b f) \text {Subst}\left (\int \frac {1}{b f (b e-a f)^2+b f^2 x^2} \, dx,x,\sqrt {a+b x} \sqrt {2 b e-a f+b f x}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {f} \sqrt {a+b x} \sqrt {2 b e-a f+b f x}}{b e-a f}\right )}{\sqrt {f} (b e-a f)}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 74, normalized size = 1.25 \begin {gather*} \frac {2 \sqrt {\frac {1}{f}} \tan ^{-1}\left (\frac {b e+b f x-\frac {\sqrt {a+b x} \sqrt {2 b e-a f+b f x}}{\sqrt {\frac {1}{f}}}}{b e-a f}\right )}{-b e+a f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x]*(e + f*x)*Sqrt[2*b*e - a*f + b*f*x]),x]

[Out]

(2*Sqrt[f^(-1)]*ArcTan[(b*e + b*f*x - (Sqrt[a + b*x]*Sqrt[2*b*e - a*f + b*f*x])/Sqrt[f^(-1)])/(b*e - a*f)])/(-
(b*e) + a*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(153\) vs. \(2(51)=102\).
time = 0.10, size = 154, normalized size = 2.61

method	result	size
default	\(-\frac {\ln \left (-\frac {2 \left (a^{2} f^{2}-2 a b e f +b^{2} e^{2}-\sqrt {-\frac {\left (a f -b e \right )^{2}}{f}}\, \sqrt {b^{2} f \,x^{2}+2 b^{2} e x -a^{2} f +2 a e b}\, f \right )}{f x +e}\right ) \sqrt {b f x -a f +2 b e}\, \sqrt {b x +a}}{\sqrt {-\frac {\left (a f -b e \right )^{2}}{f}}\, \sqrt {b^{2} f \,x^{2}+2 b^{2} e x -a^{2} f +2 a e b}\, f}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(-2*(a^2*f^2-2*a*b*e*f+b^2*e^2-(-(a*f-b*e)^2/f)^(1/2)*(b^2*f*x^2+2*b^2*e*x-a^2*f+2*a*b*e)^(1/2)*f)/(f*x+e))
*(b*f*x-a*f+2*b*e)^(1/2)*(b*x+a)^(1/2)/(-(a*f-b*e)^2/f)^(1/2)/(b^2*f*x^2+2*b^2*e*x-a^2*f+2*a*b*e)^(1/2)/f

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*f-%e*b>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 0.44, size = 216, normalized size = 3.66 \begin {gather*} \left [\frac {\sqrt {-f} \log \left (-\frac {b^{2} f^{2} x^{2} - 2 \, a^{2} f^{2} - b^{2} e^{2} + 2 \, \sqrt {b f x - a f + 2 \, b e} {\left (a f - b e\right )} \sqrt {b x + a} \sqrt {-f} + 2 \, {\left (b^{2} f x + 2 \, a b f\right )} e}{f^{2} x^{2} + 2 \, f x e + e^{2}}\right )}{2 \, {\left (a f^{2} - b f e\right )}}, \frac {\sqrt {f} \arctan \left (-\frac {\sqrt {b f x - a f + 2 \, b e} {\left (a f - b e\right )} \sqrt {b x + a} \sqrt {f}}{b^{2} f^{2} x^{2} - a^{2} f^{2} + 2 \, {\left (b^{2} f x + a b f\right )} e}\right )}{a f^{2} - b f e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(-f)*log(-(b^2*f^2*x^2 - 2*a^2*f^2 - b^2*e^2 + 2*sqrt(b*f*x - a*f + 2*b*e)*(a*f - b*e)*sqrt(b*x + a)*
sqrt(-f) + 2*(b^2*f*x + 2*a*b*f)*e)/(f^2*x^2 + 2*f*x*e + e^2))/(a*f^2 - b*f*e), sqrt(f)*arctan(-sqrt(b*f*x - a
*f + 2*b*e)*(a*f - b*e)*sqrt(b*x + a)*sqrt(f)/(b^2*f^2*x^2 - a^2*f^2 + 2*(b^2*f*x + a*b*f)*e))/(a*f^2 - b*f*e)
]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b x} \left (e + f x\right ) \sqrt {- a f + 2 b e + b f x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(a + b*x)*(e + f*x)*sqrt(-a*f + 2*b*e + b*f*x)), x)

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Giac [A]
time = 0.59, size = 65, normalized size = 1.10 \begin {gather*} -\frac {2 \, \arctan \left (\frac {{\left (\sqrt {b x + a} \sqrt {f} - \sqrt {{\left (b x + a\right )} f - 2 \, a f + 2 \, b e}\right )}^{2}}{2 \, {\left (a f - b e\right )}}\right )}{{\left (a f - b e\right )} \sqrt {f}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(1/2*(sqrt(b*x + a)*sqrt(f) - sqrt((b*x + a)*f - 2*a*f + 2*b*e))^2/(a*f - b*e))/((a*f - b*e)*sqrt(f))

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Mupad [B]
time = 8.48, size = 998, normalized size = 16.92 \begin {gather*} \frac {2\,\left (\mathrm {atan}\left (\frac {2\,\sqrt {a}\,\sqrt {2\,b\,e-a\,f}\,{\left (f\,\left (b^2\,e^2+a\,f\,\left (a\,f-2\,b\,e\right )\right )\right )}^{5/2}+\frac {b\,e\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,{\left (f\,\left (b^2\,e^2+a\,f\,\left (a\,f-2\,b\,e\right )\right )\right )}^{5/2}}{\sqrt {2\,b\,e-a\,f}-\sqrt {2\,b\,e-a\,f+b\,f\,x}}}{2\,b^6\,e^6\,f^2+2\,a^3\,f^5\,{\left (a\,f-2\,b\,e\right )}^3+6\,a^2\,b^2\,e^2\,f^4\,{\left (a\,f-2\,b\,e\right )}^2+6\,a\,b^4\,e^4\,f^3\,\left (a\,f-2\,b\,e\right )}\right )-\mathrm {atan}\left (\frac {b^2\,e^2\,f^3\,\left (a\,f^2\,\left (a\,f-2\,b\,e\right )+b^2\,e^2\,f\right )\,\left (\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {64}{b\,e\,f^2\,\sqrt {a\,f^2\,\left (a\,f-2\,b\,e\right )+b^2\,e^2\,f}\,\left (2\,a\,f^2\,\left (a\,f-2\,b\,e\right )+2\,b^2\,e^2\,f\right )}+\frac {8\,{\left (b^2\,e^2+a\,f\,\left (a\,f-2\,b\,e\right )\right )}^2}{b^3\,e^3\,f\,{\left (f\,\left (b^2\,e^2+a\,f\,\left (a\,f-2\,b\,e\right )\right )\right )}^{5/2}}-\frac {32\,a\,\left (a\,f-2\,b\,e\right )}{b^2\,e^2\,f^2\,\sqrt {a\,f^2\,\left (a\,f-2\,b\,e\right )+b^2\,e^2\,f}\,\left (4\,b^3\,e^3+4\,a\,b\,e\,f\,\left (a\,f-2\,b\,e\right )\right )}\right )}{\sqrt {2\,b\,e-a\,f}-\sqrt {2\,b\,e-a\,f+b\,f\,x}}-\frac {\left (\frac {f^2\,{\left (b^2\,e^2+a\,f\,\left (a\,f-2\,b\,e\right )\right )}^2\,\left (\frac {4}{b^2\,e^2\,f^2}-\frac {12\,a\,f^2\,\left (a\,f-2\,b\,e\right )+12\,b^2\,e^2\,f}{b^2\,e^2\,f^2\,\left (a\,f^2\,\left (a\,f-2\,b\,e\right )+b^2\,e^2\,f\right )}\right )}{b\,e\,{\left (f\,\left (b^2\,e^2+a\,f\,\left (a\,f-2\,b\,e\right )\right )\right )}^{5/2}}+\frac {32\,a\,\left (a\,f-2\,b\,e\right )}{b^2\,e^2\,f\,\sqrt {a\,f^2\,\left (a\,f-2\,b\,e\right )+b^2\,e^2\,f}\,\left (4\,b^3\,e^3+4\,a\,b\,e\,f\,\left (a\,f-2\,b\,e\right )\right )}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {2\,b\,e-a\,f}-\sqrt {2\,b\,e-a\,f+b\,f\,x}\right )}^3}+\frac {\left (\frac {32\,\sqrt {a}\,\sqrt {2\,b\,e-a\,f}}{b^2\,e^2\,f\,\sqrt {a\,f^2\,\left (a\,f-2\,b\,e\right )+b^2\,e^2\,f}\,\left (2\,a\,f^2\,\left (a\,f-2\,b\,e\right )+2\,b^2\,e^2\,f\right )}+\frac {64\,\sqrt {a}\,\sqrt {2\,b\,e-a\,f}}{b\,e\,f^2\,\sqrt {a\,f^2\,\left (a\,f-2\,b\,e\right )+b^2\,e^2\,f}\,\left (4\,b^3\,e^3+4\,a\,b\,e\,f\,\left (a\,f-2\,b\,e\right )\right )}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {2\,b\,e-a\,f}-\sqrt {2\,b\,e-a\,f+b\,f\,x}\right )}^2}+\frac {32\,\sqrt {a}\,\sqrt {2\,b\,e-a\,f}}{b^2\,e^2\,f^2\,\sqrt {a\,f^2\,\left (a\,f-2\,b\,e\right )+b^2\,e^2\,f}\,\left (2\,a\,f^2\,\left (a\,f-2\,b\,e\right )+2\,b^2\,e^2\,f\right )}\right )}{16}\right )\right )}{\sqrt {a\,f^2\,\left (a\,f-2\,b\,e\right )+b^2\,e^2\,f}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(atan((2*a^(1/2)*(2*b*e - a*f)^(1/2)*(f*(b^2*e^2 + a*f*(a*f - 2*b*e)))^(5/2) + (b*e*((a + b*x)^(1/2) - a^(1
/2))*(f*(b^2*e^2 + a*f*(a*f - 2*b*e)))^(5/2))/((2*b*e - a*f)^(1/2) - (2*b*e - a*f + b*f*x)^(1/2)))/(2*b^6*e^6*
f^2 + 2*a^3*f^5*(a*f - 2*b*e)^3 + 6*a^2*b^2*e^2*f^4*(a*f - 2*b*e)^2 + 6*a*b^4*e^4*f^3*(a*f - 2*b*e))) - atan((
b^2*e^2*f^3*(a*f^2*(a*f - 2*b*e) + b^2*e^2*f)*((((a + b*x)^(1/2) - a^(1/2))*(64/(b*e*f^2*(a*f^2*(a*f - 2*b*e)
+ b^2*e^2*f)^(1/2)*(2*a*f^2*(a*f - 2*b*e) + 2*b^2*e^2*f)) + (8*(b^2*e^2 + a*f*(a*f - 2*b*e))^2)/(b^3*e^3*f*(f*
(b^2*e^2 + a*f*(a*f - 2*b*e)))^(5/2)) - (32*a*(a*f - 2*b*e))/(b^2*e^2*f^2*(a*f^2*(a*f - 2*b*e) + b^2*e^2*f)^(1
/2)*(4*b^3*e^3 + 4*a*b*e*f*(a*f - 2*b*e)))))/((2*b*e - a*f)^(1/2) - (2*b*e - a*f + b*f*x)^(1/2)) - (((f^2*(b^2
*e^2 + a*f*(a*f - 2*b*e))^2*(4/(b^2*e^2*f^2) - (12*a*f^2*(a*f - 2*b*e) + 12*b^2*e^2*f)/(b^2*e^2*f^2*(a*f^2*(a*
f - 2*b*e) + b^2*e^2*f))))/(b*e*(f*(b^2*e^2 + a*f*(a*f - 2*b*e)))^(5/2)) + (32*a*(a*f - 2*b*e))/(b^2*e^2*f*(a*
f^2*(a*f - 2*b*e) + b^2*e^2*f)^(1/2)*(4*b^3*e^3 + 4*a*b*e*f*(a*f - 2*b*e))))*((a + b*x)^(1/2) - a^(1/2))^3)/((
2*b*e - a*f)^(1/2) - (2*b*e - a*f + b*f*x)^(1/2))^3 + (((32*a^(1/2)*(2*b*e - a*f)^(1/2))/(b^2*e^2*f*(a*f^2*(a*
f - 2*b*e) + b^2*e^2*f)^(1/2)*(2*a*f^2*(a*f - 2*b*e) + 2*b^2*e^2*f)) + (64*a^(1/2)*(2*b*e - a*f)^(1/2))/(b*e*f
^2*(a*f^2*(a*f - 2*b*e) + b^2*e^2*f)^(1/2)*(4*b^3*e^3 + 4*a*b*e*f*(a*f - 2*b*e))))*((a + b*x)^(1/2) - a^(1/2))
^2)/((2*b*e - a*f)^(1/2) - (2*b*e - a*f + b*f*x)^(1/2))^2 + (32*a^(1/2)*(2*b*e - a*f)^(1/2))/(b^2*e^2*f^2*(a*f
^2*(a*f - 2*b*e) + b^2*e^2*f)^(1/2)*(2*a*f^2*(a*f - 2*b*e) + 2*b^2*e^2*f))))/16)))/(a*f^2*(a*f - 2*b*e) + b^2*
e^2*f)^(1/2)

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