3.64 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x} \, dx\)

Optimal. Leaf size=251 \[ \frac{b^5 x^{15} \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac{5 a b^4 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^4 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )} \]

[Out]

(5*a^4*b*x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (5*a^3*b^2*x^6*S
qrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (10*a^2*b^3*x^9*Sqrt[a^2 + 2*a
*b*x^3 + b^2*x^6])/(9*(a + b*x^3)) + (5*a*b^4*x^12*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^
6])/(12*(a + b*x^3)) + (b^5*x^15*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(15*(a + b*x^3
)) + (a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(a + b*x^3)

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Rubi [A]  time = 0.171591, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{b^5 x^{15} \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac{5 a b^4 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^4 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x,x]

[Out]

(5*a^4*b*x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (5*a^3*b^2*x^6*S
qrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (10*a^2*b^3*x^9*Sqrt[a^2 + 2*a
*b*x^3 + b^2*x^6])/(9*(a + b*x^3)) + (5*a*b^4*x^12*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^
6])/(12*(a + b*x^3)) + (b^5*x^15*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(15*(a + b*x^3
)) + (a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(a + b*x^3)

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Rubi in Sympy [A]  time = 26.4812, size = 178, normalized size = 0.71 \[ \frac{a^{5} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (x \right )}}{a + b x^{3}} + \frac{a^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{3} + \frac{a^{3} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{6} + \frac{a^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{9} + \frac{a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{12} + \frac{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x,x)

[Out]

a**5*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)*log(x)/(a + b*x**3) + a**4*sqrt(a**2 +
2*a*b*x**3 + b**2*x**6)/3 + a**3*(a + b*x**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6
)/6 + a**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/9 + a*(a + b*x**3)*(a**2 + 2*a
*b*x**3 + b**2*x**6)**(3/2)/12 + (a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/15

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Mathematica [A]  time = 0.0505826, size = 82, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (180 a^5 \log (x)+b x^3 \left (300 a^4+300 a^3 b x^3+200 a^2 b^2 x^6+75 a b^3 x^9+12 b^4 x^{12}\right )\right )}{180 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x,x]

[Out]

(Sqrt[(a + b*x^3)^2]*(b*x^3*(300*a^4 + 300*a^3*b*x^3 + 200*a^2*b^2*x^6 + 75*a*b^
3*x^9 + 12*b^4*x^12) + 180*a^5*Log[x]))/(180*(a + b*x^3))

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Maple [A]  time = 0.012, size = 79, normalized size = 0.3 \[{\frac{12\,{b}^{5}{x}^{15}+75\,a{b}^{4}{x}^{12}+200\,{a}^{2}{b}^{3}{x}^{9}+300\,{a}^{3}{b}^{2}{x}^{6}+300\,{a}^{4}b{x}^{3}+180\,{a}^{5}\ln \left ( x \right ) }{180\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x,x)

[Out]

1/180*((b*x^3+a)^2)^(5/2)*(12*b^5*x^15+75*a*b^4*x^12+200*a^2*b^3*x^9+300*a^3*b^2
*x^6+300*a^4*b*x^3+180*a^5*ln(x))/(b*x^3+a)^5

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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^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272122, size = 74, normalized size = 0.29 \[ \frac{1}{15} \, b^{5} x^{15} + \frac{5}{12} \, a b^{4} x^{12} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{5}{3} \, a^{3} b^{2} x^{6} + \frac{5}{3} \, a^{4} b x^{3} + a^{5} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x,x, algorithm="fricas")

[Out]

1/15*b^5*x^15 + 5/12*a*b^4*x^12 + 10/9*a^2*b^3*x^9 + 5/3*a^3*b^2*x^6 + 5/3*a^4*b
*x^3 + a^5*log(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.296822, size = 140, normalized size = 0.56 \[ \frac{1}{15} \, b^{5} x^{15}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{12} \, a b^{4} x^{12}{\rm sign}\left (b x^{3} + a\right ) + \frac{10}{9} \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{3} \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{3} \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a^{5}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x,x, algorithm="giac")

[Out]

1/15*b^5*x^15*sign(b*x^3 + a) + 5/12*a*b^4*x^12*sign(b*x^3 + a) + 10/9*a^2*b^3*x
^9*sign(b*x^3 + a) + 5/3*a^3*b^2*x^6*sign(b*x^3 + a) + 5/3*a^4*b*x^3*sign(b*x^3
+ a) + a^5*ln(abs(x))*sign(b*x^3 + a)