3.1430 \(\int \frac{1}{(a+b x) (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=69 \[ \frac{2}{\sqrt{c+d x} (b c-a d)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{3/2}} \]

[Out]

2/((b*c - a*d)*Sqrt[c + d*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[
b*c - a*d]])/(b*c - a*d)^(3/2)

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Rubi [A]  time = 0.092165, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{2}{\sqrt{c+d x} (b c-a d)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)*(c + d*x)^(3/2)),x]

[Out]

2/((b*c - a*d)*Sqrt[c + d*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[
b*c - a*d]])/(b*c - a*d)^(3/2)

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Rubi in Sympy [A]  time = 13.47, size = 60, normalized size = 0.87 \[ - \frac{2 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{\left (a d - b c\right )^{\frac{3}{2}}} - \frac{2}{\sqrt{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)/(d*x+c)**(3/2),x)

[Out]

-2*sqrt(b)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(a*d - b*c)**(3/2) - 2/(s
qrt(c + d*x)*(a*d - b*c))

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Mathematica [A]  time = 0.0966291, size = 69, normalized size = 1. \[ \frac{2}{\sqrt{c+d x} (b c-a d)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)*(c + d*x)^(3/2)),x]

[Out]

2/((b*c - a*d)*Sqrt[c + d*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[
b*c - a*d]])/(b*c - a*d)^(3/2)

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Maple [A]  time = 0.013, size = 68, normalized size = 1. \[ -2\,{\frac{1}{ \left ( ad-bc \right ) \sqrt{dx+c}}}-2\,{\frac{b}{ \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(d*x+c)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230698, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{d x + c} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \,{\left (b c - a d\right )} \sqrt{d x + c} \sqrt{\frac{b}{b c - a d}}}{b x + a}\right ) - 2}{{\left (b c - a d\right )} \sqrt{d x + c}}, -\frac{2 \,{\left (\sqrt{d x + c} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x + c} b}\right ) - 1\right )}}{{\left (b c - a d\right )} \sqrt{d x + c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)^(3/2)),x, algorithm="fricas")

[Out]

[-(sqrt(d*x + c)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a*d + 2*(b*c - a*d)*sq
rt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) - 2)/((b*c - a*d)*sqrt(d*x + c)), -2
*(sqrt(d*x + c)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(s
qrt(d*x + c)*b)) - 1)/((b*c - a*d)*sqrt(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right ) \left (c + d x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)/(d*x+c)**(3/2),x)

[Out]

Integral(1/((a + b*x)*(c + d*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.219868, size = 93, normalized size = 1.35 \[ \frac{2 \, b \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} + \frac{2}{{\left (b c - a d\right )} \sqrt{d x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)^(3/2)),x, algorithm="giac")

[Out]

2*b*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*(b*c - a*
d)) + 2/((b*c - a*d)*sqrt(d*x + c))