3.16.51 \(\int \frac {1}{\sqrt {\frac {-b+b c}{d}+b x} \sqrt {c+d x}} \, dx\) [1551]

Optimal. Leaf size=43 \[ \frac {2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {-\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]

[Out]

2*arcsinh(d^(1/2)*(-b*(1-c)/d+b*x)^(1/2)/b^(1/2))/b^(1/2)/d^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {65, 221} \begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {b x-\frac {b (1-c)}{d}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[(-b + b*c)/d + b*x]*Sqrt[c + d*x]),x]

[Out]

(2*ArcSinh[(Sqrt[d]*Sqrt[-((b*(1 - c))/d) + b*x])/Sqrt[b]])/(Sqrt[b]*Sqrt[d])

Rule 65

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 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\frac {-b+b c}{d}+b x} \sqrt {c+d x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {-b+b c}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {\frac {-b+b c}{d}+b x}\right )}{b}\\ &=\frac {2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {-\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 51, normalized size = 1.19 \begin {gather*} \frac {2 \sqrt {-1+c+d x} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {-1+c+d x}}\right )}{d \sqrt {\frac {b (-1+c+d x)}{d}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[(-b + b*c)/d + b*x]*Sqrt[c + d*x]),x]

[Out]

(2*Sqrt[-1 + c + d*x]*ArcTanh[Sqrt[c + d*x]/Sqrt[-1 + c + d*x]])/(d*Sqrt[(b*(-1 + c + d*x))/d])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs. \(2(33)=66\).
time = 0.19, size = 100, normalized size = 2.33

method	result	size
default	\(\frac {\sqrt {\left (b x +\frac {b \left (c -1\right )}{d}\right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {b \left (c -1\right )}{2}+\frac {b c}{2}+b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (b \left (c -1\right )+b c \right ) x +\frac {b \left (c -1\right ) c}{d}}\right )}{\sqrt {b x +\frac {b \left (c -1\right )}{d}}\, \sqrt {d x +c}\, \sqrt {b d}}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x+b*(c-1)/d)*(d*x+c))^(1/2)/(b*x+b*(c-1)/d)^(1/2)/(d*x+c)^(1/2)*ln((1/2*b*(c-1)+1/2*b*c+b*d*x)/(b*d)^(1/2)
+(b*d*x^2+(b*(c-1)+b*c)*x+b*(c-1)/d*c)^(1/2))/(b*d)^(1/2)

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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/((b*c-b)/d+b*x)^(1/2)/(d*x+c)^(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(2*c-1>0)', see `assume?` for m
ore details)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (30) = 60\).
time = 1.18, size = 175, normalized size = 4.07 \begin {gather*} \left [\frac {\sqrt {b d} \log \left (8 \, b d^{2} x^{2} + 8 \, b c^{2} + 8 \, {\left (2 \, b c - b\right )} d x + 4 \, \sqrt {b d} {\left (2 \, d x + 2 \, c - 1\right )} \sqrt {d x + c} \sqrt {\frac {b d x + b c - b}{d}} - 8 \, b c + b\right )}{2 \, b d}, -\frac {\sqrt {-b d} \arctan \left (\frac {\sqrt {-b d} {\left (2 \, d x + 2 \, c - 1\right )} \sqrt {d x + c} \sqrt {\frac {b d x + b c - b}{d}}}{2 \, {\left (b d^{2} x^{2} + b c^{2} + {\left (2 \, b c - b\right )} d x - b c\right )}}\right )}{b d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(b*d)*log(8*b*d^2*x^2 + 8*b*c^2 + 8*(2*b*c - b)*d*x + 4*sqrt(b*d)*(2*d*x + 2*c - 1)*sqrt(d*x + c)*sqr
t((b*d*x + b*c - b)/d) - 8*b*c + b)/(b*d), -sqrt(-b*d)*arctan(1/2*sqrt(-b*d)*(2*d*x + 2*c - 1)*sqrt(d*x + c)*s
qrt((b*d*x + b*c - b)/d)/(b*d^2*x^2 + b*c^2 + (2*b*c - b)*d*x - b*c))/(b*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((b*c-b)/d+b*x)**(1/2)/(d*x+c)**(1/2),x)

[Out]

Integral(1/(sqrt(b*(c/d + x - 1/d))*sqrt(c + d*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (30) = 60\).
time = 0.99, size = 66, normalized size = 1.53 \begin {gather*} -\frac {2 \, {\left | b \right |} \log \left (-\sqrt {b d^{2} x + b c d - b d} \sqrt {b d} + \sqrt {b^{2} d^{2} + {\left (b d^{2} x + b c d - b d\right )} b d}\right )}{\sqrt {b d} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*abs(b)*log(-sqrt(b*d^2*x + b*c*d - b*d)*sqrt(b*d) + sqrt(b^2*d^2 + (b*d^2*x + b*c*d - b*d)*b*d))/(sqrt(b*d)
*b)

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Mupad [B]
time = 0.50, size = 66, normalized size = 1.53 \begin {gather*} \frac {4\,\mathrm {atan}\left (-\frac {d\,\left (\sqrt {b\,x-\frac {b-b\,c}{d}}-\sqrt {-\frac {b-b\,c}{d}}\right )}{\sqrt {-b\,d}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )}{\sqrt {-b\,d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x - (b - b*c)/d)^(1/2)*(c + d*x)^(1/2)),x)

[Out]

(4*atan(-(d*((b*x - (b - b*c)/d)^(1/2) - (-(b - b*c)/d)^(1/2)))/((-b*d)^(1/2)*((c + d*x)^(1/2) - c^(1/2)))))/(
-b*d)^(1/2)

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