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

Optimal. Leaf size=108 \[ \frac{3 \sqrt{c} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}-\frac{3 \sqrt{c+d x} (b c-a d)}{a^2 \sqrt{a+b x}}-\frac{(c+d x)^{3/2}}{a x \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.19986, antiderivative size = 108, 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 \[ \frac{3 \sqrt{c} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}-\frac{3 \sqrt{c+d x} (b c-a d)}{a^2 \sqrt{a+b x}}-\frac{(c+d x)^{3/2}}{a x \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.9517, size = 94, normalized size = 0.87 \[ \frac{2 \left (c + d x\right )^{\frac{3}{2}}}{a x \sqrt{a + b x}} - \frac{3 c \sqrt{a + b x} \sqrt{c + d x}}{a^{2} x} - \frac{3 \sqrt{c} \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.339743, size = 128, normalized size = 1.19 \[ \frac{-\frac{2 \sqrt{a} \sqrt{c+d x} (a c-2 a d x+3 b c x)}{x \sqrt{a+b x}}+3 \sqrt{c} \log (x) (a d-b c)+3 \sqrt{c} (b c-a d) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{2 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.036, size = 298, normalized size = 2.8 \[ -{\frac{1}{2\,{a}^{2}x}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}abcd-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{2}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}cd-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xab{c}^{2}-4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dax\sqrt{ac}+6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }bcx\sqrt{ac}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ca\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.316171, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (b^{2} c - a b d\right )} x^{2} +{\left (a b c - a^{2} d\right )} x\right )} \sqrt{\frac{c}{a}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a^{2} c +{\left (a b c + a^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{c}{a}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \,{\left (a c +{\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{4 \,{\left (a^{2} b x^{2} + a^{3} x\right )}}, \frac{3 \,{\left ({\left (b^{2} c - a b d\right )} x^{2} +{\left (a b c - a^{2} d\right )} x\right )} \sqrt{-\frac{c}{a}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} a \sqrt{-\frac{c}{a}}}\right ) - 2 \,{\left (a c +{\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (a^{2} b x^{2} + a^{3} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: TypeError