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

Optimal. Leaf size=77 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} c^{3/2}}-\frac{2 d \sqrt{a+b x}}{c \sqrt{c+d x} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.140739, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} c^{3/2}}-\frac{2 d \sqrt{a+b x}}{c \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.4704, size = 68, normalized size = 0.88 \[ \frac{2 d \sqrt{a + b x}}{c \sqrt{c + d x} \left (a d - b c\right )} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d*sqrt(a + b*x)/(c*sqrt(c + d*x)*(a*d - b*c)) - 2*atanh(sqrt(c)*sqrt(a + b*x)/
(sqrt(a)*sqrt(c + d*x)))/(sqrt(a)*c**(3/2))

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Mathematica [A]  time = 0.210665, size = 104, normalized size = 1.35 \[ -\frac{\log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a} c^{3/2}}-\frac{2 d \sqrt{a+b x}}{c \sqrt{c+d x} (b c-a d)}+\frac{\log (x)}{\sqrt{a} c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.039, size = 243, normalized size = 3.2 \[{\frac{1}{c \left ( ad-bc \right ) } \left ( -\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xa{d}^{2}+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xbcd-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) acd+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) b{c}^{2}+2\,d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac} \right ) \sqrt{bx+a}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(3/2)*x), x)

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Fricas [A]  time = 0.290874, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} d -{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{2 \,{\left (b c^{3} - a c^{2} d +{\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt{a c}}, -\frac{2 \, \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} d +{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{{\left (b c^{3} - a c^{2} d +{\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(4*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c)*d - (b*c^2 - a*c*d + (b*c*d - a*d
^2)*x)*log(-(4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c) -
(8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqr
t(a*c))/x^2))/((b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x)*sqrt(a*c)), -(2*sqrt(-a
*c)*sqrt(b*x + a)*sqrt(d*x + c)*d + (b*c^2 - a*c*d + (b*c*d - a*d^2)*x)*arctan(1
/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)))/((b*c^
3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x)*sqrt(-a*c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.231039, size = 189, normalized size = 2.45 \[ -\frac{2 \, \sqrt{b x + a} b^{2} d}{{\left (b c^{2}{\left | b \right |} - a c d{\left | b \right |}\right )} \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} - \frac{2 \, \sqrt{b d} b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} c{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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