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

Optimal. Leaf size=144 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}+\frac{\sqrt{x} (a+b x)^{5/2} (a B+6 A b)}{3 a}+\frac{5}{12} \sqrt{x} (a+b x)^{3/2} (a B+6 A b)+\frac{5}{8} a \sqrt{x} \sqrt{a+b x} (a B+6 A b)-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}} \]

[Out]

(5*a*(6*A*b + a*B)*Sqrt[x]*Sqrt[a + b*x])/8 + (5*(6*A*b + a*B)*Sqrt[x]*(a + b*x)
^(3/2))/12 + ((6*A*b + a*B)*Sqrt[x]*(a + b*x)^(5/2))/(3*a) - (2*A*(a + b*x)^(7/2
))/(a*Sqrt[x]) + (5*a^2*(6*A*b + a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/
(8*Sqrt[b])

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Rubi [A]  time = 0.176545, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}+\frac{\sqrt{x} (a+b x)^{5/2} (a B+6 A b)}{3 a}+\frac{5}{12} \sqrt{x} (a+b x)^{3/2} (a B+6 A b)+\frac{5}{8} a \sqrt{x} \sqrt{a+b x} (a B+6 A b)-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(5*a*(6*A*b + a*B)*Sqrt[x]*Sqrt[a + b*x])/8 + (5*(6*A*b + a*B)*Sqrt[x]*(a + b*x)
^(3/2))/12 + ((6*A*b + a*B)*Sqrt[x]*(a + b*x)^(5/2))/(3*a) - (2*A*(a + b*x)^(7/2
))/(a*Sqrt[x]) + (5*a^2*(6*A*b + a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/
(8*Sqrt[b])

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Rubi in Sympy [A]  time = 14.8557, size = 138, normalized size = 0.96 \[ - \frac{2 A \left (a + b x\right )^{\frac{7}{2}}}{a \sqrt{x}} + \frac{5 a^{2} \left (6 A b + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{8 \sqrt{b}} + \frac{5 a \sqrt{x} \sqrt{a + b x} \left (6 A b + B a\right )}{8} + \sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (\frac{5 A b}{2} + \frac{5 B a}{12}\right ) + \frac{\sqrt{x} \left (a + b x\right )^{\frac{5}{2}} \left (6 A b + B a\right )}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*(a + b*x)**(7/2)/(a*sqrt(x)) + 5*a**2*(6*A*b + B*a)*atanh(sqrt(b)*sqrt(x)/s
qrt(a + b*x))/(8*sqrt(b)) + 5*a*sqrt(x)*sqrt(a + b*x)*(6*A*b + B*a)/8 + sqrt(x)*
(a + b*x)**(3/2)*(5*A*b/2 + 5*B*a/12) + sqrt(x)*(a + b*x)**(5/2)*(6*A*b + B*a)/(
3*a)

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Mathematica [A]  time = 0.17596, size = 105, normalized size = 0.73 \[ \frac{\sqrt{a+b x} \left (a^2 (33 B x-48 A)+2 a b x (27 A+13 B x)+4 b^2 x^2 (3 A+2 B x)\right )}{24 \sqrt{x}}+\frac{5 a^2 (a B+6 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{8 \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(4*b^2*x^2*(3*A + 2*B*x) + 2*a*b*x*(27*A + 13*B*x) + a^2*(-48*A +
33*B*x)))/(24*Sqrt[x]) + (5*a^2*(6*A*b + a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a +
b*x]])/(8*Sqrt[b])

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Maple [A]  time = 0.021, size = 202, normalized size = 1.4 \[{\frac{1}{48}\sqrt{bx+a} \left ( 16\,B{x}^{3}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+24\,{b}^{5/2}A\sqrt{x \left ( bx+a \right ) }{x}^{2}+52\,B{b}^{3/2}a\sqrt{x \left ( bx+a \right ) }{x}^{2}+90\,bA{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x+108\,{b}^{3/2}A\sqrt{x \left ( bx+a \right ) }ax+15\,B{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x+66\,B{a}^{2}\sqrt{x \left ( bx+a \right ) }x\sqrt{b}-96\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(b*x+a)^(1/2)*(16*B*x^3*b^(5/2)*(x*(b*x+a))^(1/2)+24*b^(5/2)*A*(x*(b*x+a))^
(1/2)*x^2+52*B*b^(3/2)*a*(x*(b*x+a))^(1/2)*x^2+90*b*A*a^2*ln(1/2*(2*(x*(b*x+a))^
(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x+108*b^(3/2)*A*(x*(b*x+a))^(1/2)*a*x+15*B*a^3*l
n(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x+66*B*a^2*(x*(b*x+a))^(1/2
)*x*b^(1/2)-96*A*a^2*(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)^(5/2)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244613, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} x \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, B b^{2} x^{3} - 48 \, A a^{2} + 2 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} x^{2} + 3 \,{\left (11 \, B a^{2} + 18 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{48 \, \sqrt{b} x}, \frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (8 \, B b^{2} x^{3} - 48 \, A a^{2} + 2 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} x^{2} + 3 \,{\left (11 \, B a^{2} + 18 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*(B*a^3 + 6*A*a^2*b)*x*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt
(b)) + 2*(8*B*b^2*x^3 - 48*A*a^2 + 2*(13*B*a*b + 6*A*b^2)*x^2 + 3*(11*B*a^2 + 18
*A*a*b)*x)*sqrt(b*x + a)*sqrt(b)*sqrt(x))/(sqrt(b)*x), 1/24*(15*(B*a^3 + 6*A*a^2
*b)*x*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (8*B*b^2*x^3 - 48*A*a^2 + 2*(
13*B*a*b + 6*A*b^2)*x^2 + 3*(11*B*a^2 + 18*A*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)**(5/2)*(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)^(5/2)/x^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError