3.546 \(\int x^{3/2} (a+b x)^{5/2} \, dx\)

Optimal. Leaf size=140 \[ \frac{3 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{5/2}}-\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^2}+\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b}+\frac{1}{16} a^2 x^{5/2} \sqrt{a+b x}+\frac{1}{8} a x^{5/2} (a+b x)^{3/2}+\frac{1}{5} x^{5/2} (a+b x)^{5/2} \]

[Out]

(-3*a^4*Sqrt[x]*Sqrt[a + b*x])/(128*b^2) + (a^3*x^(3/2)*Sqrt[a + b*x])/(64*b) +
(a^2*x^(5/2)*Sqrt[a + b*x])/16 + (a*x^(5/2)*(a + b*x)^(3/2))/8 + (x^(5/2)*(a + b
*x)^(5/2))/5 + (3*a^5*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(128*b^(5/2))

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Rubi [A]  time = 0.118121, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{5/2}}-\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^2}+\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b}+\frac{1}{16} a^2 x^{5/2} \sqrt{a+b x}+\frac{1}{8} a x^{5/2} (a+b x)^{3/2}+\frac{1}{5} x^{5/2} (a+b x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*(a + b*x)^(5/2),x]

[Out]

(-3*a^4*Sqrt[x]*Sqrt[a + b*x])/(128*b^2) + (a^3*x^(3/2)*Sqrt[a + b*x])/(64*b) +
(a^2*x^(5/2)*Sqrt[a + b*x])/16 + (a*x^(5/2)*(a + b*x)^(3/2))/8 + (x^(5/2)*(a + b
*x)^(5/2))/5 + (3*a^5*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(128*b^(5/2))

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Rubi in Sympy [A]  time = 18.7515, size = 138, normalized size = 0.99 \[ \frac{3 a^{5} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{128 b^{\frac{5}{2}}} + \frac{3 a^{4} \sqrt{x} \sqrt{a + b x}}{128 b^{2}} + \frac{a^{3} \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{64 b^{2}} + \frac{a^{2} \sqrt{x} \left (a + b x\right )^{\frac{5}{2}}}{80 b^{2}} - \frac{3 a \sqrt{x} \left (a + b x\right )^{\frac{7}{2}}}{40 b^{2}} + \frac{x^{\frac{3}{2}} \left (a + b x\right )^{\frac{7}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**5*atanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/(128*b**(5/2)) + 3*a**4*sqrt(x)*sqrt
(a + b*x)/(128*b**2) + a**3*sqrt(x)*(a + b*x)**(3/2)/(64*b**2) + a**2*sqrt(x)*(a
+ b*x)**(5/2)/(80*b**2) - 3*a*sqrt(x)*(a + b*x)**(7/2)/(40*b**2) + x**(3/2)*(a
+ b*x)**(7/2)/(5*b)

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Mathematica [A]  time = 0.0711898, size = 100, normalized size = 0.71 \[ \frac{15 a^5 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )+\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (-15 a^4+10 a^3 b x+248 a^2 b^2 x^2+336 a b^3 x^3+128 b^4 x^4\right )}{640 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(-15*a^4 + 10*a^3*b*x + 248*a^2*b^2*x^2 + 336*a*b
^3*x^3 + 128*b^4*x^4) + 15*a^5*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(640*b^(5
/2))

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Maple [A]  time = 0.009, size = 138, normalized size = 1. \[{\frac{1}{5\,b}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,a}{40\,{b}^{2}}\sqrt{x} \left ( bx+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}}{80\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{{a}^{3}}{64\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3\,{a}^{4}}{128\,{b}^{2}}\sqrt{x}\sqrt{bx+a}}+{\frac{3\,{a}^{5}}{256}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(b*x+a)^(5/2),x)

[Out]

1/5/b*x^(3/2)*(b*x+a)^(7/2)-3/40*a/b^2*x^(1/2)*(b*x+a)^(7/2)+1/80*a^2/b^2*(b*x+a
)^(5/2)*x^(1/2)+1/64*a^3/b^2*(b*x+a)^(3/2)*x^(1/2)+3/128*a^4*x^(1/2)*(b*x+a)^(1/
2)/b^2+3/256*a^5/b^(5/2)*(x*(b*x+a))^(1/2)/(b*x+a)^(1/2)/x^(1/2)*ln((1/2*a+b*x)/
b^(1/2)+(b*x^2+a*x)^(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)^(5/2)*x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224471, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{5} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (128 \, b^{4} x^{4} + 336 \, a b^{3} x^{3} + 248 \, a^{2} b^{2} x^{2} + 10 \, a^{3} b x - 15 \, a^{4}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{1280 \, b^{\frac{5}{2}}}, \frac{15 \, a^{5} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (128 \, b^{4} x^{4} + 336 \, a b^{3} x^{3} + 248 \, a^{2} b^{2} x^{2} + 10 \, a^{3} b x - 15 \, a^{4}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{640 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1280*(15*a^5*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)) + 2*(128*b^
4*x^4 + 336*a*b^3*x^3 + 248*a^2*b^2*x^2 + 10*a^3*b*x - 15*a^4)*sqrt(b*x + a)*sqr
t(b)*sqrt(x))/b^(5/2), 1/640*(15*a^5*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x)))
+ (128*b^4*x^4 + 336*a*b^3*x^3 + 248*a^2*b^2*x^2 + 10*a^3*b*x - 15*a^4)*sqrt(b*x
+ a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^2)]

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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(x**(3/2)*(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 24.6496, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x