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

Optimal. Leaf size=122 \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{7/2}}+\frac{5 a^3 \sqrt{x} \sqrt{a+b x}}{64 b^3}-\frac{5 a^2 x^{3/2} \sqrt{a+b x}}{96 b^2}+\frac{a x^{5/2} \sqrt{a+b x}}{24 b}+\frac{1}{4} x^{7/2} \sqrt{a+b x} \]

[Out]

(5*a^3*Sqrt[x]*Sqrt[a + b*x])/(64*b^3) - (5*a^2*x^(3/2)*Sqrt[a + b*x])/(96*b^2)
+ (a*x^(5/2)*Sqrt[a + b*x])/(24*b) + (x^(7/2)*Sqrt[a + b*x])/4 - (5*a^4*ArcTanh[
(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(64*b^(7/2))

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Rubi [A]  time = 0.106371, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{7/2}}+\frac{5 a^3 \sqrt{x} \sqrt{a+b x}}{64 b^3}-\frac{5 a^2 x^{3/2} \sqrt{a+b x}}{96 b^2}+\frac{a x^{5/2} \sqrt{a+b x}}{24 b}+\frac{1}{4} x^{7/2} \sqrt{a+b x} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*Sqrt[a + b*x],x]

[Out]

(5*a^3*Sqrt[x]*Sqrt[a + b*x])/(64*b^3) - (5*a^2*x^(3/2)*Sqrt[a + b*x])/(96*b^2)
+ (a*x^(5/2)*Sqrt[a + b*x])/(24*b) + (x^(7/2)*Sqrt[a + b*x])/4 - (5*a^4*ArcTanh[
(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(64*b^(7/2))

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Rubi in Sympy [A]  time = 14.98, size = 117, normalized size = 0.96 \[ - \frac{5 a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{64 b^{\frac{7}{2}}} - \frac{5 a^{3} \sqrt{x} \sqrt{a + b x}}{64 b^{3}} + \frac{5 a^{2} \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{32 b^{3}} - \frac{5 a x^{\frac{3}{2}} \left (a + b x\right )^{\frac{3}{2}}}{24 b^{2}} + \frac{x^{\frac{5}{2}} \left (a + b x\right )^{\frac{3}{2}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(b*x+a)**(1/2),x)

[Out]

-5*a**4*atanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/(64*b**(7/2)) - 5*a**3*sqrt(x)*sqrt
(a + b*x)/(64*b**3) + 5*a**2*sqrt(x)*(a + b*x)**(3/2)/(32*b**3) - 5*a*x**(3/2)*(
a + b*x)**(3/2)/(24*b**2) + x**(5/2)*(a + b*x)**(3/2)/(4*b)

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Mathematica [A]  time = 0.0670688, size = 89, normalized size = 0.73 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (15 a^3-10 a^2 b x+8 a b^2 x^2+48 b^3 x^3\right )-15 a^4 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{192 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(15*a^3 - 10*a^2*b*x + 8*a*b^2*x^2 + 48*b^3*x^3)
- 15*a^4*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(192*b^(7/2))

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Maple [A]  time = 0.025, size = 120, normalized size = 1. \[{\frac{1}{4\,b}{x}^{{\frac{5}{2}}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,a}{24\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}}{32\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}-{\frac{5\,{a}^{3}}{64\,{b}^{3}}\sqrt{x}\sqrt{bx+a}}-{\frac{5\,{a}^{4}}{128}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x+a)^(1/2),x)

[Out]

1/4/b*x^(5/2)*(b*x+a)^(3/2)-5/24*a/b^2*x^(3/2)*(b*x+a)^(3/2)+5/32*a^2/b^3*x^(1/2
)*(b*x+a)^(3/2)-5/64*a^3*x^(1/2)*(b*x+a)^(1/2)/b^3-5/128*a^4/b^(7/2)*(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(sqrt(b*x + a)*x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224356, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{4} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (48 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} - 10 \, a^{2} b x + 15 \, a^{3}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{384 \, b^{\frac{7}{2}}}, -\frac{15 \, a^{4} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (48 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} - 10 \, a^{2} b x + 15 \, a^{3}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{192 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(15*a^4*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)) + 2*(48*b^3
*x^3 + 8*a*b^2*x^2 - 10*a^2*b*x + 15*a^3)*sqrt(b*x + a)*sqrt(b)*sqrt(x))/b^(7/2)
, -1/192*(15*a^4*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (48*b^3*x^3 + 8*a*
b^2*x^2 - 10*a^2*b*x + 15*a^3)*sqrt(b*x + a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^3)]

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Sympy [A]  time = 106.69, size = 153, normalized size = 1.25 \[ \frac{5 a^{\frac{7}{2}} \sqrt{x}}{64 b^{3} \sqrt{1 + \frac{b x}{a}}} + \frac{5 a^{\frac{5}{2}} x^{\frac{3}{2}}}{192 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{a^{\frac{3}{2}} x^{\frac{5}{2}}}{96 b \sqrt{1 + \frac{b x}{a}}} + \frac{7 \sqrt{a} x^{\frac{7}{2}}}{24 \sqrt{1 + \frac{b x}{a}}} - \frac{5 a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{7}{2}}} + \frac{b x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(b*x+a)**(1/2),x)

[Out]

5*a**(7/2)*sqrt(x)/(64*b**3*sqrt(1 + b*x/a)) + 5*a**(5/2)*x**(3/2)/(192*b**2*sqr
t(1 + b*x/a)) - a**(3/2)*x**(5/2)/(96*b*sqrt(1 + b*x/a)) + 7*sqrt(a)*x**(7/2)/(2
4*sqrt(1 + b*x/a)) - 5*a**4*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(64*b**(7/2)) + b*x**
(9/2)/(4*sqrt(a)*sqrt(1 + b*x/a))

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GIAC/XCAS [A]  time = 12.6743, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x