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

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

[Out]

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

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Rubi [A]  time = 0.103733, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} d^{3/2}}-\frac{2 c \sqrt{a+b x}}{d \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.80418, size = 68, normalized size = 0.88 \[ \frac{2 c \sqrt{a + b x}}{d \sqrt{c + d x} \left (a d - b c\right )} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{b} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.113126, size = 89, normalized size = 1.16 \[ \frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{b} d^{3/2}}+\frac{2 c \sqrt{a+b x}}{d \sqrt{c+d x} (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.032, size = 251, normalized size = 3.3 \[{\frac{1}{d \left ( ad-bc \right ) }\sqrt{bx+a} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) xa{d}^{2}-\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) xbcd+\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) acd-\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) b{c}^{2}+2\,c\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312545, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} c -{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{2 \,{\left (b c^{2} d - a c d^{2} +{\left (b c d^{2} - a d^{3}\right )} x\right )} \sqrt{b d}}, -\frac{2 \, \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} c -{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{{\left (b c^{2} d - a c d^{2} +{\left (b c d^{2} - a d^{3}\right )} x\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.246948, size = 146, normalized size = 1.9 \[ -\frac{2 \,{\left (\frac{\sqrt{b x + a} b^{3} c{\left | b \right |}}{{\left (b^{3} c d - a b^{2} d^{2}\right )} \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{{\left | b \right |}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d}\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(sqrt(b*x + a)*b^3*c*abs(b)/((b^3*c*d - a*b^2*d^2)*sqrt(b^2*c + (b*x + a)*b*d
- a*b*d)) + abs(b)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d
- a*b*d)))/(sqrt(b*d)*d))/b