3.482 \(\int \frac{(a+b x)^{3/2} (A+B x)}{x^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ \frac{\sqrt{x} (a+b x)^{3/2} (a B+4 A b)}{2 a}+\frac{3}{4} \sqrt{x} \sqrt{a+b x} (a B+4 A b)+\frac{3 a (a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 \sqrt{b}}-\frac{2 A (a+b x)^{5/2}}{a \sqrt{x}} \]

[Out]

(3*(4*A*b + a*B)*Sqrt[x]*Sqrt[a + b*x])/4 + ((4*A*b + a*B)*Sqrt[x]*(a + b*x)^(3/
2))/(2*a) - (2*A*(a + b*x)^(5/2))/(a*Sqrt[x]) + (3*a*(4*A*b + a*B)*ArcTanh[(Sqrt
[b]*Sqrt[x])/Sqrt[a + b*x]])/(4*Sqrt[b])

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Rubi [A]  time = 0.145777, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{x} (a+b x)^{3/2} (a B+4 A b)}{2 a}+\frac{3}{4} \sqrt{x} \sqrt{a+b x} (a B+4 A b)+\frac{3 a (a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 \sqrt{b}}-\frac{2 A (a+b x)^{5/2}}{a \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(4*A*b + a*B)*Sqrt[x]*Sqrt[a + b*x])/4 + ((4*A*b + a*B)*Sqrt[x]*(a + b*x)^(3/
2))/(2*a) - (2*A*(a + b*x)^(5/2))/(a*Sqrt[x]) + (3*a*(4*A*b + a*B)*ArcTanh[(Sqrt
[b]*Sqrt[x])/Sqrt[a + b*x]])/(4*Sqrt[b])

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Rubi in Sympy [A]  time = 11.5328, size = 107, normalized size = 0.93 \[ - \frac{2 A \left (a + b x\right )^{\frac{5}{2}}}{a \sqrt{x}} + \frac{3 a \left (4 A b + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{b}} + \sqrt{x} \sqrt{a + b x} \left (3 A b + \frac{3 B a}{4}\right ) + \frac{\sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (4 A b + B a\right )}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**(3/2),x)

[Out]

-2*A*(a + b*x)**(5/2)/(a*sqrt(x)) + 3*a*(4*A*b + B*a)*atanh(sqrt(a + b*x)/(sqrt(
b)*sqrt(x)))/(4*sqrt(b)) + sqrt(x)*sqrt(a + b*x)*(3*A*b + 3*B*a/4) + sqrt(x)*(a
+ b*x)**(3/2)*(4*A*b + B*a)/(2*a)

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Mathematica [A]  time = 0.141015, size = 82, normalized size = 0.71 \[ \frac{1}{4} \left (\frac{\sqrt{a+b x} (a (5 B x-8 A)+2 b x (2 A+B x))}{\sqrt{x}}+\frac{3 a (a B+4 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a + b*x]*(2*b*x*(2*A + B*x) + a*(-8*A + 5*B*x)))/Sqrt[x] + (3*a*(4*A*b +
a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/Sqrt[b])/4

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Maple [A]  time = 0.02, size = 158, normalized size = 1.4 \[{\frac{1}{8}\sqrt{bx+a} \left ( 4\,B{b}^{3/2}\sqrt{x \left ( bx+a \right ) }{x}^{2}+12\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) bAx+8\,\sqrt{x \left ( bx+a \right ) }A{b}^{3/2}x+3\,B{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x+10\,B\sqrt{x \left ( bx+a \right ) }a\sqrt{b}x-16\,Aa\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b*x+a)^(1/2)*(4*B*b^(3/2)*(x*(b*x+a))^(1/2)*x^2+12*a*ln(1/2*(2*(x*(b*x+a))^
(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b*A*x+8*(x*(b*x+a))^(1/2)*A*b^(3/2)*x+3*B*a^2*ln
(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x+10*B*(x*(b*x+a))^(1/2)*a*b
^(1/2)*x-16*A*a*(x*(b*x+a))^(1/2)*b^(1/2))/x^(1/2)/(x*(b*x+a))^(1/2)/b^(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)*(b*x + a)^(3/2)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247431, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B a^{2} + 4 \, A a b\right )} x \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (2 \, B b x^{2} - 8 \, A a +{\left (5 \, B a + 4 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{8 \, \sqrt{b} x}, \frac{3 \,{\left (B a^{2} + 4 \, A a b\right )} x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (2 \, B b x^{2} - 8 \, A a +{\left (5 \, B a + 4 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{4 \, \sqrt{-b} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(3*(B*a^2 + 4*A*a*b)*x*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b))
+ 2*(2*B*b*x^2 - 8*A*a + (5*B*a + 4*A*b)*x)*sqrt(b*x + a)*sqrt(b)*sqrt(x))/(sqr
t(b)*x), 1/4*(3*(B*a^2 + 4*A*a*b)*x*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) +
(2*B*b*x^2 - 8*A*a + (5*B*a + 4*A*b)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x))/(sqrt(-
b)*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((b*x+a)**(3/2)*(B*x+A)/x**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError