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

Optimal. Leaf size=171 \[ \frac{\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{7/2}}-\frac{d \sqrt{a+b x} (b c-15 a d)}{4 a c^3 \sqrt{c+d x}}-\frac{\sqrt{a+b x} (b c-5 a d)}{4 a c^2 x \sqrt{c+d x}}-\frac{\sqrt{a+b x}}{2 c x^2 \sqrt{c+d x}} \]

[Out]

-(d*(b*c - 15*a*d)*Sqrt[a + b*x])/(4*a*c^3*Sqrt[c + d*x]) - Sqrt[a + b*x]/(2*c*x
^2*Sqrt[c + d*x]) - ((b*c - 5*a*d)*Sqrt[a + b*x])/(4*a*c^2*x*Sqrt[c + d*x]) + ((
b^2*c^2 + 6*a*b*c*d - 15*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[
c + d*x])])/(4*a^(3/2)*c^(7/2))

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Rubi [A]  time = 0.517432, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{7/2}}-\frac{d \sqrt{a+b x} (b c-15 a d)}{4 a c^3 \sqrt{c+d x}}-\frac{\sqrt{a+b x} (b c-5 a d)}{4 a c^2 x \sqrt{c+d x}}-\frac{\sqrt{a+b x}}{2 c x^2 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(b*c - 15*a*d)*Sqrt[a + b*x])/(4*a*c^3*Sqrt[c + d*x]) - Sqrt[a + b*x]/(2*c*x
^2*Sqrt[c + d*x]) - ((b*c - 5*a*d)*Sqrt[a + b*x])/(4*a*c^2*x*Sqrt[c + d*x]) + ((
b^2*c^2 + 6*a*b*c*d - 15*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[
c + d*x])])/(4*a^(3/2)*c^(7/2))

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Rubi in Sympy [A]  time = 73.024, size = 155, normalized size = 0.91 \[ - \frac{\sqrt{a + b x}}{2 c x^{2} \sqrt{c + d x}} + \frac{\sqrt{a + b x} \left (5 a d - b c\right )}{4 a c^{2} x \sqrt{c + d x}} + \frac{d \sqrt{a + b x} \left (15 a d - b c\right )}{4 a c^{3} \sqrt{c + d x}} - \frac{\left (15 a^{2} d^{2} - 6 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)/(2*c*x**2*sqrt(c + d*x)) + sqrt(a + b*x)*(5*a*d - b*c)/(4*a*c**2*
x*sqrt(c + d*x)) + d*sqrt(a + b*x)*(15*a*d - b*c)/(4*a*c**3*sqrt(c + d*x)) - (15
*a**2*d**2 - 6*a*b*c*d - b**2*c**2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c
+ d*x)))/(4*a**(3/2)*c**(7/2))

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Mathematica [A]  time = 0.208688, size = 173, normalized size = 1.01 \[ \frac{-\log (x) \left (-15 a^2 d^2+6 a b c d+b^2 c^2\right )+\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\frac{2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \left (a \left (-2 c^2+5 c d x+15 d^2 x^2\right )-b c x (c+d x)\right )}{x^2 \sqrt{c+d x}}}{8 a^{3/2} c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*(-(b*c*x*(c + d*x)) + a*(-2*c^2 + 5*c*d*x + 15
*d^2*x^2)))/(x^2*Sqrt[c + d*x]) - (b^2*c^2 + 6*a*b*c*d - 15*a^2*d^2)*Log[x] + (b
^2*c^2 + 6*a*b*c*d - 15*a^2*d^2)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*S
qrt[a + b*x]*Sqrt[c + d*x]])/(8*a^(3/2)*c^(7/2))

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Maple [B]  time = 0.045, size = 467, normalized size = 2.7 \[ -{\frac{1}{8\,{c}^{3}a{x}^{2}}\sqrt{bx+a} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}{d}^{3}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}abc{d}^{2}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{3}{b}^{2}{c}^{2}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}c{d}^{2}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}ab{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{b}^{2}{c}^{3}-30\,{x}^{2}a{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,{x}^{2}bcd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-10\,xacd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,xb{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+4\,a{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(b*x+a)^(1/2)/a/c^3*(15*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^3*a^2*d^3-6*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^3*a*b*c*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^3*b^2*c^2*d+15*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^2*a^2*c*d^2-6*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^2*a*b*c^2*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)*x^2*b^2*c^3-30*x^2*a*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*x^2*b*c*d
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-10*x*a*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)+2*x*b*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*a*c^2*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2))/(a*c)^(1/2)/x^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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.383254, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (2 \, a c^{2} +{\left (b c d - 15 \, a d^{2}\right )} x^{2} +{\left (b c^{2} - 5 \, a c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} +{\left ({\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} + 6 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2}\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{16 \,{\left (a c^{3} d x^{3} + a c^{4} x^{2}\right )} \sqrt{a c}}, -\frac{2 \,{\left (2 \, a c^{2} +{\left (b c d - 15 \, a d^{2}\right )} x^{2} +{\left (b c^{2} - 5 \, a c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} -{\left ({\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} + 6 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{8 \,{\left (a c^{3} d x^{3} + a c^{4} x^{2}\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(4*(2*a*c^2 + (b*c*d - 15*a*d^2)*x^2 + (b*c^2 - 5*a*c*d)*x)*sqrt(a*c)*sqr
t(b*x + a)*sqrt(d*x + c) + ((b^2*c^2*d + 6*a*b*c*d^2 - 15*a^2*d^3)*x^3 + (b^2*c^
3 + 6*a*b*c^2*d - 15*a^2*c*d^2)*x^2)*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)*sqrt(a*c))/x^2))/((a*c^3*d*x^3 + a*c^4*x^2)*sqrt(a*c)
), -1/8*(2*(2*a*c^2 + (b*c*d - 15*a*d^2)*x^2 + (b*c^2 - 5*a*c*d)*x)*sqrt(-a*c)*s
qrt(b*x + a)*sqrt(d*x + c) - ((b^2*c^2*d + 6*a*b*c*d^2 - 15*a^2*d^3)*x^3 + (b^2*
c^3 + 6*a*b*c^2*d - 15*a^2*c*d^2)*x^2)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-
a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)))/((a*c^3*d*x^3 + a*c^4*x^2)*sqrt(-a*c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError