3.579 \(\int \frac{\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x} \, dx\)

Optimal. Leaf size=173 \[ -\frac{a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{n}-\frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^n}{2 \sqrt{c} \sqrt{a+b x^n+c x^{2 n}}}\right )}{16 c^{3/2} n}+\frac{\left (8 a c+b^2+2 b c x^n\right ) \sqrt{a+b x^n+c x^{2 n}}}{8 c n}+\frac{\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n} \]

[Out]

((b^2 + 8*a*c + 2*b*c*x^n)*Sqrt[a + b*x^n + c*x^(2*n)])/(8*c*n) + (a + b*x^n + c
*x^(2*n))^(3/2)/(3*n) - (a^(3/2)*ArcTanh[(2*a + b*x^n)/(2*Sqrt[a]*Sqrt[a + b*x^n
+ c*x^(2*n)])])/n - (b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^n)/(2*Sqrt[c]*Sqrt[a +
b*x^n + c*x^(2*n)])])/(16*c^(3/2)*n)

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Rubi [A]  time = 0.398101, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{n}-\frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^n}{2 \sqrt{c} \sqrt{a+b x^n+c x^{2 n}}}\right )}{16 c^{3/2} n}+\frac{\left (8 a c+b^2+2 b c x^n\right ) \sqrt{a+b x^n+c x^{2 n}}}{8 c n}+\frac{\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n} \]

Antiderivative was successfully verified.

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

[Out]

((b^2 + 8*a*c + 2*b*c*x^n)*Sqrt[a + b*x^n + c*x^(2*n)])/(8*c*n) + (a + b*x^n + c
*x^(2*n))^(3/2)/(3*n) - (a^(3/2)*ArcTanh[(2*a + b*x^n)/(2*Sqrt[a]*Sqrt[a + b*x^n
+ c*x^(2*n)])])/n - (b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^n)/(2*Sqrt[c]*Sqrt[a +
b*x^n + c*x^(2*n)])])/(16*c^(3/2)*n)

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Rubi in Sympy [A]  time = 48.8707, size = 151, normalized size = 0.87 \[ - \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a + b x^{n}}{2 \sqrt{a} \sqrt{a + b x^{n} + c x^{2 n}}} \right )}}{n} - \frac{b \left (- 12 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{n}}{2 \sqrt{c} \sqrt{a + b x^{n} + c x^{2 n}}} \right )}}{16 c^{\frac{3}{2}} n} + \frac{\left (a + b x^{n} + c x^{2 n}\right )^{\frac{3}{2}}}{3 n} + \frac{\sqrt{a + b x^{n} + c x^{2 n}} \left (4 a c + \frac{b^{2}}{2} + b c x^{n}\right )}{4 c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(3/2)*atanh((2*a + b*x**n)/(2*sqrt(a)*sqrt(a + b*x**n + c*x**(2*n))))/n - b*
(-12*a*c + b**2)*atanh((b + 2*c*x**n)/(2*sqrt(c)*sqrt(a + b*x**n + c*x**(2*n))))
/(16*c**(3/2)*n) + (a + b*x**n + c*x**(2*n))**(3/2)/(3*n) + sqrt(a + b*x**n + c*
x**(2*n))*(4*a*c + b**2/2 + b*c*x**n)/(4*c*n)

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Mathematica [A]  time = 0.720881, size = 161, normalized size = 0.93 \[ -\frac{a^{3/2} \log \left (2 \sqrt{a} \sqrt{a+x^n \left (b+c x^n\right )}+2 a+b x^n\right )}{n}+a^{3/2} \log (x)-\frac{b \left (b^2-12 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x^n \left (b+c x^n\right )}+b+2 c x^n\right )}{16 c^{3/2} n}+\frac{\sqrt{a+x^n \left (b+c x^n\right )} \left (8 c \left (4 a+c x^{2 n}\right )+3 b^2+14 b c x^n\right )}{24 c n} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + x^n*(b + c*x^n)]*(3*b^2 + 14*b*c*x^n + 8*c*(4*a + c*x^(2*n))))/(24*c*n
) + a^(3/2)*Log[x] - (a^(3/2)*Log[2*a + b*x^n + 2*Sqrt[a]*Sqrt[a + x^n*(b + c*x^
n)]])/n - (b*(b^2 - 12*a*c)*Log[b + 2*c*x^n + 2*Sqrt[c]*Sqrt[a + x^n*(b + c*x^n)
]])/(16*c^(3/2)*n)

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Maple [A]  time = 0.051, size = 209, normalized size = 1.2 \[{\frac{8\,{c}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}+14\,b{{\rm e}^{n\ln \left ( x \right ) }}c+32\,ac+3\,{b}^{2}}{24\,cn}\sqrt{a+b{{\rm e}^{n\ln \left ( x \right ) }}+c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}}-{\frac{{b}^{3}}{16\,n}\ln \left ({1 \left ({\frac{b}{2}}+c{{\rm e}^{n\ln \left ( x \right ) }} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{a+b{{\rm e}^{n\ln \left ( x \right ) }}+c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,ab}{4\,n}\ln \left ({1 \left ({\frac{b}{2}}+c{{\rm e}^{n\ln \left ( x \right ) }} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{a+b{{\rm e}^{n\ln \left ( x \right ) }}+c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{1}{n}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( 2\,a+b{{\rm e}^{n\ln \left ( x \right ) }}+2\,\sqrt{a}\sqrt{a+b{{\rm e}^{n\ln \left ( x \right ) }}+c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(8*c^2*exp(n*ln(x))^2+14*b*exp(n*ln(x))*c+32*a*c+3*b^2)*(a+b*exp(n*ln(x))+c
*exp(n*ln(x))^2)^(1/2)/c/n-1/16/c^(3/2)/n*b^3*ln((1/2*b+c*exp(n*ln(x)))/c^(1/2)+
(a+b*exp(n*ln(x))+c*exp(n*ln(x))^2)^(1/2))+3/4/c^(1/2)/n*a*b*ln((1/2*b+c*exp(n*l
n(x)))/c^(1/2)+(a+b*exp(n*ln(x))+c*exp(n*ln(x))^2)^(1/2))-1/n*a^(3/2)*ln((2*a+b*
exp(n*ln(x))+2*a^(1/2)*(a+b*exp(n*ln(x))+c*exp(n*ln(x))^2)^(1/2))/exp(n*ln(x)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.418184, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(48*a^(3/2)*c^(3/2)*log(-(8*a*b*x^n + 8*a^2 + (b^2 + 4*a*c)*x^(2*n) - 4*(s
qrt(a)*b*x^n + 2*a^(3/2))*sqrt(c*x^(2*n) + b*x^n + a))/x^(2*n)) - 3*(b^3 - 12*a*
b*c)*log(-8*c^(5/2)*x^(2*n) - 8*b*c^(3/2)*x^n - 4*(2*c^2*x^n + b*c)*sqrt(c*x^(2*
n) + b*x^n + a) - (b^2 + 4*a*c)*sqrt(c)) + 4*(8*c^(5/2)*x^(2*n) + 14*b*c^(3/2)*x
^n + (3*b^2 + 32*a*c)*sqrt(c))*sqrt(c*x^(2*n) + b*x^n + a))/(c^(3/2)*n), 1/48*(2
4*a^(3/2)*sqrt(-c)*c*log(-(8*a*b*x^n + 8*a^2 + (b^2 + 4*a*c)*x^(2*n) - 4*(sqrt(a
)*b*x^n + 2*a^(3/2))*sqrt(c*x^(2*n) + b*x^n + a))/x^(2*n)) - 3*(b^3 - 12*a*b*c)*
arctan(1/2*(2*sqrt(-c)*c*x^n + b*sqrt(-c))/(sqrt(c*x^(2*n) + b*x^n + a)*c)) + 2*
(8*sqrt(-c)*c^2*x^(2*n) + 14*b*sqrt(-c)*c*x^n + (3*b^2 + 32*a*c)*sqrt(-c))*sqrt(
c*x^(2*n) + b*x^n + a))/(sqrt(-c)*c*n), -1/96*(96*sqrt(-a)*a*c^(3/2)*arctan(1/2*
(b*x^n + 2*a)/(sqrt(c*x^(2*n) + b*x^n + a)*sqrt(-a))) + 3*(b^3 - 12*a*b*c)*log(-
8*c^(5/2)*x^(2*n) - 8*b*c^(3/2)*x^n - 4*(2*c^2*x^n + b*c)*sqrt(c*x^(2*n) + b*x^n
+ a) - (b^2 + 4*a*c)*sqrt(c)) - 4*(8*c^(5/2)*x^(2*n) + 14*b*c^(3/2)*x^n + (3*b^
2 + 32*a*c)*sqrt(c))*sqrt(c*x^(2*n) + b*x^n + a))/(c^(3/2)*n), -1/48*(48*sqrt(-a
)*a*sqrt(-c)*c*arctan(1/2*(b*x^n + 2*a)/(sqrt(c*x^(2*n) + b*x^n + a)*sqrt(-a)))
+ 3*(b^3 - 12*a*b*c)*arctan(1/2*(2*sqrt(-c)*c*x^n + b*sqrt(-c))/(sqrt(c*x^(2*n)
+ b*x^n + a)*c)) - 2*(8*sqrt(-c)*c^2*x^(2*n) + 14*b*sqrt(-c)*c*x^n + (3*b^2 + 32
*a*c)*sqrt(-c))*sqrt(c*x^(2*n) + b*x^n + a))/(sqrt(-c)*c*n)]

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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((a+b*x**n+c*x**(2*n))**(3/2)/x,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac{3}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^(2*n) + b*x^n + a)^(3/2)/x, x)