3.483 \(\int \frac{(a+b x)^{3/2} (A+B x)}{x^{5/2}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{2 (a+b x)^{3/2} (3 a B+2 A b)}{3 a \sqrt{x}}+\frac{b \sqrt{x} \sqrt{a+b x} (3 a B+2 A b)}{a}+\sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.143367, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 (a+b x)^{3/2} (3 a B+2 A b)}{3 a \sqrt{x}}+\frac{b \sqrt{x} \sqrt{a+b x} (3 a B+2 A b)}{a}+\sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.8721, size = 112, normalized size = 0.96 \[ - \frac{2 A \left (a + b x\right )^{\frac{5}{2}}}{3 a x^{\frac{3}{2}}} + 2 \sqrt{b} \left (A b + \frac{3 B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )} + \frac{b \sqrt{x} \sqrt{a + b x} \left (2 A b + 3 B a\right )}{a} - \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (A b + \frac{3 B a}{2}\right )}{3 a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.114963, size = 79, normalized size = 0.68 \[ \sqrt{b} (3 a B+2 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-\frac{\sqrt{a+b x} (2 a (A+3 B x)+b x (8 A-3 B x))}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/x^(5/2),x]

[Out]

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

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Maple [A]  time = 0.019, size = 151, normalized size = 1.3 \[{\frac{1}{6}\sqrt{bx+a} \left ( 6\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}{b}^{3/2}+9\,B\sqrt{b}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) a{x}^{2}+6\,B{x}^{2}b\sqrt{x \left ( bx+a \right ) }-16\,Axb\sqrt{x \left ( bx+a \right ) }-12\,Bxa\sqrt{x \left ( bx+a \right ) }-4\,Aa\sqrt{x \left ( bx+a \right ) } \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/x^(5/2),x)

[Out]

1/6*(b*x+a)^(1/2)/x^(3/2)*(6*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1
/2))*x^2*b^(3/2)+9*B*b^(1/2)*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2
))*a*x^2+6*B*x^2*b*(x*(b*x+a))^(1/2)-16*A*x*b*(x*(b*x+a))^(1/2)-12*B*x*a*(x*(b*x
+a))^(1/2)-4*A*a*(x*(b*x+a))^(1/2))/(x*(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((B*x + A)*(b*x + a)^(3/2)/x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248888, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (3 \, B a + 2 \, A b\right )} \sqrt{b} x^{2} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (3 \, B b x^{2} - 2 \, A a - 2 \,{\left (3 \, B a + 4 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{6 \, x^{2}}, \frac{3 \,{\left (3 \, B a + 2 \, A b\right )} \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) +{\left (3 \, B b x^{2} - 2 \, A a - 2 \,{\left (3 \, B a + 4 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 170.571, size = 168, normalized size = 1.44 \[ A \left (- \frac{2 a \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x} - \frac{8 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3} - b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )} + 2 b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{b x} + 1} + 1 \right )}\right ) + B \left (- \frac{2 a^{\frac{3}{2}}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} - \frac{\sqrt{a} b \sqrt{x}}{\sqrt{1 + \frac{b x}{a}}} + 3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} + \frac{b^{2} x^{\frac{3}{2}}}{\sqrt{a} \sqrt{1 + \frac{b x}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)*(B*x+A)/x**(5/2),x)

[Out]

A*(-2*a*sqrt(b)*sqrt(a/(b*x) + 1)/(3*x) - 8*b**(3/2)*sqrt(a/(b*x) + 1)/3 - b**(3
/2)*log(a/(b*x)) + 2*b**(3/2)*log(sqrt(a/(b*x) + 1) + 1)) + B*(-2*a**(3/2)/(sqrt
(x)*sqrt(1 + b*x/a)) - sqrt(a)*b*sqrt(x)/sqrt(1 + b*x/a) + 3*a*sqrt(b)*asinh(sqr
t(b)*sqrt(x)/sqrt(a)) + b**2*x**(3/2)/(sqrt(a)*sqrt(1 + b*x/a)))

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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^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError