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

Optimal. Leaf size=134 \[ -\frac{3 a (4 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{7/2}}+\frac{3 \sqrt{x} \sqrt{a+b x} (4 A b-5 a B)}{4 b^3}-\frac{x^{3/2} \sqrt{a+b x} (4 A b-5 a B)}{2 a b^2}+\frac{2 x^{5/2} (A b-a B)}{a b \sqrt{a+b x}} \]

[Out]

(2*(A*b - a*B)*x^(5/2))/(a*b*Sqrt[a + b*x]) + (3*(4*A*b - 5*a*B)*Sqrt[x]*Sqrt[a
+ b*x])/(4*b^3) - ((4*A*b - 5*a*B)*x^(3/2)*Sqrt[a + b*x])/(2*a*b^2) - (3*a*(4*A*
b - 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(4*b^(7/2))

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Rubi [A]  time = 0.149402, antiderivative size = 134, 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 \[ -\frac{3 a (4 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{7/2}}+\frac{3 \sqrt{x} \sqrt{a+b x} (4 A b-5 a B)}{4 b^3}-\frac{x^{3/2} \sqrt{a+b x} (4 A b-5 a B)}{2 a b^2}+\frac{2 x^{5/2} (A b-a B)}{a b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*x^(5/2))/(a*b*Sqrt[a + b*x]) + (3*(4*A*b - 5*a*B)*Sqrt[x]*Sqrt[a
+ b*x])/(4*b^3) - ((4*A*b - 5*a*B)*x^(3/2)*Sqrt[a + b*x])/(2*a*b^2) - (3*a*(4*A*
b - 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(4*b^(7/2))

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Rubi in Sympy [A]  time = 14.465, size = 126, normalized size = 0.94 \[ - \frac{3 a \left (4 A b - 5 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{4 b^{\frac{7}{2}}} + \frac{3 \sqrt{x} \sqrt{a + b x} \left (4 A b - 5 B a\right )}{4 b^{3}} + \frac{2 x^{\frac{5}{2}} \left (A b - B a\right )}{a b \sqrt{a + b x}} - \frac{x^{\frac{3}{2}} \sqrt{a + b x} \left (4 A b - 5 B a\right )}{2 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a*(4*A*b - 5*B*a)*atanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/(4*b**(7/2)) + 3*sqrt(
x)*sqrt(a + b*x)*(4*A*b - 5*B*a)/(4*b**3) + 2*x**(5/2)*(A*b - B*a)/(a*b*sqrt(a +
b*x)) - x**(3/2)*sqrt(a + b*x)*(4*A*b - 5*B*a)/(2*a*b**2)

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Mathematica [A]  time = 0.167547, size = 96, normalized size = 0.72 \[ \frac{\sqrt{x} \left (-15 a^2 B+a b (12 A-5 B x)+2 b^2 x (2 A+B x)\right )}{4 b^3 \sqrt{a+b x}}+\frac{3 a (5 a B-4 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{4 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(-15*a^2*B + a*b*(12*A - 5*B*x) + 2*b^2*x*(2*A + B*x)))/(4*b^3*Sqrt[a +
b*x]) + (3*a*(-4*A*b + 5*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(4*b^(7/2
))

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Maple [B]  time = 0.024, size = 244, normalized size = 1.8 \[ -{\frac{1}{8} \left ( -4\,B{x}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+12\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) xa{b}^{2}-8\,A\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}-15\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{2}b+10\,Ba\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+12\,A{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-24\,A\sqrt{x \left ( bx+a \right ) }a{b}^{3/2}-15\,B{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +30\,B{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ) \sqrt{x}{b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(-4*B*x^2*b^(5/2)*(x*(b*x+a))^(1/2)+12*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2
)+2*b*x+a)/b^(1/2))*x*a*b^2-8*A*(x*(b*x+a))^(1/2)*x*b^(5/2)-15*B*ln(1/2*(2*(x*(b
*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x*a^2*b+10*B*a*(x*(b*x+a))^(1/2)*x*b^(3/2
)+12*A*a^2*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-24*A*(x*(b*x+
a))^(1/2)*a*b^(3/2)-15*B*a^3*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2
))+30*B*a^2*(x*(b*x+a))^(1/2)*b^(1/2))/b^(7/2)*x^(1/2)/(x*(b*x+a))^(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((B*x + A)*x^(3/2)/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242215, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, B a^{2} - 4 \, A a b\right )} \sqrt{b x + a} \sqrt{x} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (2 \, B b^{2} x^{3} -{\left (5 \, B a b - 4 \, A b^{2}\right )} x^{2} - 3 \,{\left (5 \, B a^{2} - 4 \, A a b\right )} x\right )} \sqrt{b}}{8 \, \sqrt{b x + a} b^{\frac{7}{2}} \sqrt{x}}, \frac{3 \,{\left (5 \, B a^{2} - 4 \, A a b\right )} \sqrt{b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (2 \, B b^{2} x^{3} -{\left (5 \, B a b - 4 \, A b^{2}\right )} x^{2} - 3 \,{\left (5 \, B a^{2} - 4 \, A a b\right )} x\right )} \sqrt{-b}}{4 \, \sqrt{b x + a} \sqrt{-b} b^{3} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(3*(5*B*a^2 - 4*A*a*b)*sqrt(b*x + a)*sqrt(x)*log(-2*sqrt(b*x + a)*b*sqrt(x
) + (2*b*x + a)*sqrt(b)) - 2*(2*B*b^2*x^3 - (5*B*a*b - 4*A*b^2)*x^2 - 3*(5*B*a^2
- 4*A*a*b)*x)*sqrt(b))/(sqrt(b*x + a)*b^(7/2)*sqrt(x)), 1/4*(3*(5*B*a^2 - 4*A*a
*b)*sqrt(b*x + a)*sqrt(x)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (2*B*b^2*
x^3 - (5*B*a*b - 4*A*b^2)*x^2 - 3*(5*B*a^2 - 4*A*a*b)*x)*sqrt(-b))/(sqrt(b*x + a
)*sqrt(-b)*b^3*sqrt(x))]

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Sympy [A]  time = 174.608, size = 182, normalized size = 1.36 \[ A \left (\frac{3 \sqrt{a} \sqrt{x}}{b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} + \frac{x^{\frac{3}{2}}}{\sqrt{a} b \sqrt{1 + \frac{b x}{a}}}\right ) + B \left (- \frac{15 a^{\frac{3}{2}} \sqrt{x}}{4 b^{3} \sqrt{1 + \frac{b x}{a}}} - \frac{5 \sqrt{a} x^{\frac{3}{2}}}{4 b^{2} \sqrt{1 + \frac{b x}{a}}} + \frac{15 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{7}{2}}} + \frac{x^{\frac{5}{2}}}{2 \sqrt{a} b \sqrt{1 + \frac{b x}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(3*sqrt(a)*sqrt(x)/(b**2*sqrt(1 + b*x/a)) - 3*a*asinh(sqrt(b)*sqrt(x)/sqrt(a))
/b**(5/2) + x**(3/2)/(sqrt(a)*b*sqrt(1 + b*x/a))) + B*(-15*a**(3/2)*sqrt(x)/(4*b
**3*sqrt(1 + b*x/a)) - 5*sqrt(a)*x**(3/2)/(4*b**2*sqrt(1 + b*x/a)) + 15*a**2*asi
nh(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(7/2)) + x**(5/2)/(2*sqrt(a)*b*sqrt(1 + b*x/a)
))

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GIAC/XCAS [A]  time = 0.249064, size = 244, normalized size = 1.82 \[ \frac{1}{4} \, \sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} B{\left | b \right |}}{b^{5}} - \frac{9 \, B a b^{9}{\left | b \right |} - 4 \, A b^{10}{\left | b \right |}}{b^{14}}\right )} - \frac{3 \,{\left (5 \, B a^{2} \sqrt{b}{\left | b \right |} - 4 \, A a b^{\frac{3}{2}}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{8 \, b^{5}} - \frac{4 \,{\left (B a^{3} \sqrt{b}{\left | b \right |} - A a^{2} b^{\frac{3}{2}}{\left | b \right |}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*(2*(b*x + a)*B*abs(b)/b^5 - (9*B*a*b^9
*abs(b) - 4*A*b^10*abs(b))/b^14) - 3/8*(5*B*a^2*sqrt(b)*abs(b) - 4*A*a*b^(3/2)*a
bs(b))*ln((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^5 - 4*(B*a^3*sq
rt(b)*abs(b) - A*a^2*b^(3/2)*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b
- a*b))^2 + a*b)*b^4)