3.2300 \(\int \frac{\sqrt{c+d x}}{\sqrt{a+b x} (e+f x)} \, dx\)
Optimal. Leaf size=119 \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} f}-\frac{2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{d e-c f}}{\sqrt{c+d x} \sqrt{b e-a f}}\right )}{f \sqrt{b e-a f}} \]
[Out]
(2*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*f) - (2*Sqrt[d*e - c*f]*ArcTanh[
(Sqrt[d*e - c*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*Sqrt[c + d*x])])/(f*Sqrt[b*e - a*f])
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Rubi [A] time = 0.072831, antiderivative size = 119, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used =
{105, 63, 217, 206, 93, 208} \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} f}-\frac{2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{d e-c f}}{\sqrt{c+d x} \sqrt{b e-a f}}\right )}{f \sqrt{b e-a f}} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[c + d*x]/(Sqrt[a + b*x]*(e + f*x)),x]
[Out]
(2*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*f) - (2*Sqrt[d*e - c*f]*ArcTanh[
(Sqrt[d*e - c*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*Sqrt[c + d*x])])/(f*Sqrt[b*e - a*f])
Rule 105
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] || !SumSimplerQ[n, -1])))
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 93
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x}}{\sqrt{a+b x} (e+f x)} \, dx &=\frac{d \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{f}-\frac{(d e-c f) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} (e+f x)} \, dx}{f}\\ &=\frac{(2 d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{b f}-\frac{(2 (d e-c f)) \operatorname{Subst}\left (\int \frac{1}{b e-a f-(d e-c f) x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{f}\\ &=-\frac{2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d e-c f} \sqrt{a+b x}}{\sqrt{b e-a f} \sqrt{c+d x}}\right )}{f \sqrt{b e-a f}}+\frac{(2 d) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{b f}\\ &=\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} f}-\frac{2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d e-c f} \sqrt{a+b x}}{\sqrt{b e-a f} \sqrt{c+d x}}\right )}{f \sqrt{b e-a f}}\\ \end{align*}
Mathematica [A] time = 0.526597, size = 151, normalized size = 1.27 \[ \frac{2 \left (\frac{\sqrt{c f-d e} \tan ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{c f-d e}}{\sqrt{c+d x} \sqrt{b e-a f}}\right )}{\sqrt{b e-a f}}+\frac{\sqrt{d} \sqrt{c+d x} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{f} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[c + d*x]/(Sqrt[a + b*x]*(e + f*x)),x]
[Out]
(2*((Sqrt[d]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x
))/(b*c - a*d)]) + (Sqrt[-(d*e) + c*f]*ArcTan[(Sqrt[-(d*e) + c*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*Sqrt[c + d*x
])])/Sqrt[b*e - a*f]))/f
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Maple [B] time = 0.023, size = 300, normalized size = 2.5 \begin{align*}{\frac{1}{{f}^{2}} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) df\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}-\ln \left ({\frac{1}{fx+e} \left ( adfx+bcfx-2\,bdex+2\,\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }f+2\,acf-ade-bce \right ) } \right ) cf\sqrt{bd}+\ln \left ({\frac{1}{fx+e} \left ( adfx+bcfx-2\,bdex+2\,\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }f+2\,acf-ade-bce \right ) } \right ) de\sqrt{bd} \right ) \sqrt{bx+a}\sqrt{dx+c}{\frac{1}{\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^(1/2)/(f*x+e)/(b*x+a)^(1/2),x)
[Out]
(ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*d*f*((c*f-d*e)*(a*f-b*e)/f^2)^(1/
2)-ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*f+2*a*c*f-a*d*e-b*c
*e)/(f*x+e))*c*f*(b*d)^(1/2)+ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)*f+2*a*c*f-a*d*e-b*c*e)/(f*x+e))*d*e*(b*d)^(1/2))*(b*x+a)^(1/2)*(d*x+c)^(1/2)/((c*f-d*e)*(a*f-b*e)/f^2)^
(1/2)/(b*d)^(1/2)/f^2/((b*x+a)*(d*x+c))^(1/2)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(1/2)/(f*x+e)/(b*x+a)^(1/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 36.9625, size = 2912, normalized size = 24.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(1/2)/(f*x+e)/(b*x+a)^(1/2),x, algorithm="fricas")
[Out]
[1/2*(sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a
)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) + sqrt((d*e - c*f)/(b*e - a*f))*log((8*a^2*c^2*f^2 + (b^2
*c^2 + 6*a*b*c*d + a^2*d^2)*e^2 - 8*(a*b*c^2 + a^2*c*d)*e*f + (8*b^2*d^2*e^2 - 8*(b^2*c*d + a*b*d^2)*e*f + (b^
2*c^2 + 6*a*b*c*d + a^2*d^2)*f^2)*x^2 - 4*(2*a^2*c*f^2 + (b^2*c + a*b*d)*e^2 - (3*a*b*c + a^2*d)*e*f + (2*b^2*
d*e^2 - (b^2*c + 3*a*b*d)*e*f + (a*b*c + a^2*d)*f^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt((d*e - c*f)/(b*e - a*
f)) + 2*(4*(b^2*c*d + a*b*d^2)*e^2 - (3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*e*f + 4*(a*b*c^2 + a^2*c*d)*f^2)*x)/
(f^2*x^2 + 2*e*f*x + e^2)))/f, 1/2*(2*sqrt(-(d*e - c*f)/(b*e - a*f))*arctan(-1/2*(2*a*c*f - (b*c + a*d)*e - (2
*b*d*e - (b*c + a*d)*f)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-(d*e - c*f)/(b*e - a*f))/(a*c*d*e - a*c^2*f + (b*
d^2*e - b*c*d*f)*x^2 + ((b*c*d + a*d^2)*e - (b*c^2 + a*c*d)*f)*x)) + sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6
*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^
2)*x))/f, -1/2*(2*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b)/(b*d^2*x^
2 + a*c*d + (b*c*d + a*d^2)*x)) - sqrt((d*e - c*f)/(b*e - a*f))*log((8*a^2*c^2*f^2 + (b^2*c^2 + 6*a*b*c*d + a^
2*d^2)*e^2 - 8*(a*b*c^2 + a^2*c*d)*e*f + (8*b^2*d^2*e^2 - 8*(b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 + 6*a*b*c*d + a
^2*d^2)*f^2)*x^2 - 4*(2*a^2*c*f^2 + (b^2*c + a*b*d)*e^2 - (3*a*b*c + a^2*d)*e*f + (2*b^2*d*e^2 - (b^2*c + 3*a*
b*d)*e*f + (a*b*c + a^2*d)*f^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt((d*e - c*f)/(b*e - a*f)) + 2*(4*(b^2*c*d +
a*b*d^2)*e^2 - (3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*e*f + 4*(a*b*c^2 + a^2*c*d)*f^2)*x)/(f^2*x^2 + 2*e*f*x +
e^2)))/f, (sqrt(-(d*e - c*f)/(b*e - a*f))*arctan(-1/2*(2*a*c*f - (b*c + a*d)*e - (2*b*d*e - (b*c + a*d)*f)*x)*
sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-(d*e - c*f)/(b*e - a*f))/(a*c*d*e - a*c^2*f + (b*d^2*e - b*c*d*f)*x^2 + ((b*
c*d + a*d^2)*e - (b*c^2 + a*c*d)*f)*x)) - sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x +
c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)))/f]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x}}{\sqrt{a + b x} \left (e + f x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**(1/2)/(f*x+e)/(b*x+a)**(1/2),x)
[Out]
Integral(sqrt(c + d*x)/(sqrt(a + b*x)*(e + f*x)), x)
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Giac [B] time = 1.7944, size = 321, normalized size = 2.7 \begin{align*} -\frac{{\left (\frac{\sqrt{b d} \log \left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{f} + \frac{2 \,{\left (\sqrt{b d} b^{2} c f - \sqrt{b d} b^{2} d e\right )} \arctan \left (-\frac{b^{2} c f + a b d f - 2 \, b^{2} d e -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} f}{2 \, \sqrt{-a b c d f^{2} + b^{2} c d f e + a b d^{2} f e - b^{2} d^{2} e^{2}} b}\right )}{\sqrt{-a b c d f^{2} + b^{2} c d f e + a b d^{2} f e - b^{2} d^{2} e^{2}} b f}\right )}{\left | b \right |}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(1/2)/(f*x+e)/(b*x+a)^(1/2),x, algorithm="giac")
[Out]
-(sqrt(b*d)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/f + 2*(sqrt(b*d)*b^2*c*f -
sqrt(b*d)*b^2*d*e)*arctan(-1/2*(b^2*c*f + a*b*d*f - 2*b^2*d*e - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^2*f)/(sqrt(-a*b*c*d*f^2 + b^2*c*d*f*e + a*b*d^2*f*e - b^2*d^2*e^2)*b))/(sqrt(-a*b*c*d*f^2 +
b^2*c*d*f*e + a*b*d^2*f*e - b^2*d^2*e^2)*b*f))*abs(b)/b^2