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

Optimal. Leaf size=119 \[ -\frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a} c^{5/2}}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 c^2 x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 c x^2} \]

[Out]

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

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Rubi [A]  time = 0.199418, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a} c^{5/2}}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 c^2 x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.363, size = 105, normalized size = 0.88 \[ - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{2 c x^{2}} + \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )}{4 c^{2} x} - \frac{3 \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 \sqrt{a} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.122525, size = 139, normalized size = 1.17 \[ \frac{3 x^2 \log (x) (b c-a d)^2-3 x^2 (b c-a d)^2 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (-2 a c+3 a d x-5 b c x)}{8 \sqrt{a} c^{5/2} x^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-2*a*c - 5*b*c*x + 3*a*d*x) + 3*
(b*c - a*d)^2*x^2*Log[x] - 3*(b*c - a*d)^2*x^2*Log[2*a*c + b*c*x + a*d*x + 2*Sqr
t[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*Sqrt[a]*c^(5/2)*x^2)

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Maple [B]  time = 0.032, size = 255, normalized size = 2.1 \[ -{\frac{1}{8\,{c}^{2}{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{2}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}abcd+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{2}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dax\sqrt{ac}+10\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }bcx\sqrt{ac}+4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ca\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c^2*(3*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*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*d+3*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^2-6*((b*x+a)*(d*x+c))^(1/2)*d*a*x*(a*c)^(1/2)+10*(
(b*x+a)*(d*x+c))^(1/2)*b*c*x*(a*c)^(1/2)+4*((b*x+a)*(d*x+c))^(1/2)*c*a*(a*c)^(1/
2))/((b*x+a)*(d*x+c))^(1/2)/x^2/(a*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((b*x + a)^(3/2)/(sqrt(d*x + c)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300625, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (2 \, a c +{\left (5 \, b c - 3 \, a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{16 \, \sqrt{a c} c^{2} x^{2}}, -\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (2 \, a c +{\left (5 \, b c - 3 \, a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{8 \, \sqrt{-a c} c^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*(b^2*c^2 - 2*a*b*c*d + a^2*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) - 4*(2*a*c + (5*b*c - 3*a*d)*
x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*c^2*x^2), -1/8*(3*(b^2*c^2
- 2*a*b*c*d + a^2*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)) + 2*(2*a*c + (5*b*c - 3*a*d)*x)*sqrt(-a*c)*sqrt(b*x
+ a)*sqrt(d*x + c))/(sqrt(-a*c)*c^2*x^2)]

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

[Out]

Exception raised: TypeError