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

Optimal. Leaf size=127 \[ -\frac{(3 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 b c-a d)}{d^2 (b c-a d)}-\frac{2 c (a+b x)^{3/2}}{d \sqrt{c+d x} (b c-a d)} \]

[Out]

(-2*c*(a + b*x)^(3/2))/(d*(b*c - a*d)*Sqrt[c + d*x]) + ((3*b*c - a*d)*Sqrt[a + b
*x]*Sqrt[c + d*x])/(d^2*(b*c - a*d)) - ((3*b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a +
b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(5/2))

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Rubi [A]  time = 0.178197, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{(3 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 b c-a d)}{d^2 (b c-a d)}-\frac{2 c (a+b x)^{3/2}}{d \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*(a + b*x)^(3/2))/(d*(b*c - a*d)*Sqrt[c + d*x]) + ((3*b*c - a*d)*Sqrt[a + b
*x]*Sqrt[c + d*x])/(d^2*(b*c - a*d)) - ((3*b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a +
b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(5/2))

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Rubi in Sympy [A]  time = 18.6678, size = 110, normalized size = 0.87 \[ \frac{2 c \left (a + b x\right )^{\frac{3}{2}}}{d \sqrt{c + d x} \left (a d - b c\right )} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - 3 b c\right )}{d^{2} \left (a d - b c\right )} + \frac{\left (a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{b} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*(a + b*x)**(3/2)/(d*sqrt(c + d*x)*(a*d - b*c)) + sqrt(a + b*x)*sqrt(c + d*x)
*(a*d - 3*b*c)/(d**2*(a*d - b*c)) + (a*d - 3*b*c)*atanh(sqrt(d)*sqrt(a + b*x)/(s
qrt(b)*sqrt(c + d*x)))/(sqrt(b)*d**(5/2))

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Mathematica [A]  time = 0.133551, size = 95, normalized size = 0.75 \[ \frac{(a d-3 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 \sqrt{b} d^{5/2}}+\frac{\sqrt{a+b x} (3 c+d x)}{d^2 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(3*c + d*x))/(d^2*Sqrt[c + d*x]) + ((-3*b*c + a*d)*Log[b*c + a*d
+ 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2*Sqrt[b]*d^(5/2))

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Maple [B]  time = 0.029, size = 264, normalized size = 2.1 \[{\frac{1}{2\,{d}^{2}}\sqrt{bx+a} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) xa{d}^{2}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xbcd+\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) acd-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) b{c}^{2}+2\,xd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+6\,c\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(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^2-3*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-3*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*x*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+6*c
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(b*d)^(1/2)/((b*x+a)*(d*x+c))^(1/2)/d^2/(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(sqrt(b*x + a)*x/(d*x + c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304979, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{b d} \sqrt{b x + a}{\left (d x + 3 \, c\right )} \sqrt{d x + c} -{\left (3 \, b c^{2} - a c d +{\left (3 \, b c d - a d^{2}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{4 \,{\left (d^{3} x + c d^{2}\right )} \sqrt{b d}}, \frac{2 \, \sqrt{-b d} \sqrt{b x + a}{\left (d x + 3 \, c\right )} \sqrt{d x + c} -{\left (3 \, b c^{2} - a c d +{\left (3 \, b c d - a d^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{2 \,{\left (d^{3} x + c d^{2}\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*sqrt(b*d)*sqrt(b*x + a)*(d*x + 3*c)*sqrt(d*x + c) - (3*b*c^2 - a*c*d + (
3*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)))/((d^3*x + c*d^2)*sqrt(b*d)), 1/2*(2*sqrt(-b*d)*sqrt(b*x + a)*(d*
x + 3*c)*sqrt(d*x + c) - (3*b*c^2 - a*c*d + (3*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)))/((d^3*x + c*d^2
)*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*(b*x+a)**(1/2)/(d*x+c)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.236759, size = 255, normalized size = 2.01 \[ \frac{\frac{{\left (3 \, b c{\left | b \right |} - a d{\left | b \right |}\right )} \sqrt{b d}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{b^{5} c d^{4} - a b^{4} d^{5}} + \frac{{\left (\frac{{\left (b x + a\right )} b^{2} d^{2}{\left | b \right |}}{b^{6} c d^{4} - a b^{5} d^{5}} + \frac{3 \, b^{3} c d{\left | b \right |} - a b^{2} d^{2}{\left | b \right |}}{b^{6} c d^{4} - a b^{5} d^{5}}\right )} \sqrt{b x + a}}{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}}}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((3*b*c*abs(b) - a*d*abs(b))*sqrt(b*d)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqr
t(b^2*c + (b*x + a)*b*d - a*b*d)))/(b^5*c*d^4 - a*b^4*d^5) + ((b*x + a)*b^2*d^2*
abs(b)/(b^6*c*d^4 - a*b^5*d^5) + (3*b^3*c*d*abs(b) - a*b^2*d^2*abs(b))/(b^6*c*d^
4 - a*b^5*d^5))*sqrt(b*x + a)/sqrt(b^2*c + (b*x + a)*b*d - a*b*d))/b