\(\int \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx\) [11]

Optimal result

Integrand size = 22, antiderivative size = 74 \[ \int \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {a x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

[Out]

a*x*((b*x^3+a)^2)^(1/2)/(b*x^3+a)+1/4*b*x^4*((b*x^3+a)^2)^(1/2)/(b*x^3+a)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1357} \[ \int \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {a x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

[In]

Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

(a*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x^3) + (b*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3))

Rule 1357

Int[((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^p/(b + 2*c*x
^n)^(2*p), Int[(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (2 a b+2 b^2 x^3\right ) \, dx}{2 a b+2 b^2 x^3} \\ & = \frac {a x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.49 \[ \int \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {\sqrt {\left (a+b x^3\right )^2} \left (4 a x+b x^4\right )}{4 \left (a+b x^3\right )} \]

[In]

Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

(Sqrt[(a + b*x^3)^2]*(4*a*x + b*x^4))/(4*(a + b*x^3))

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.45

method	result	size
gosper	\(\frac {x \left (b \,x^{3}+4 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{4 b \,x^{3}+4 a}\)	\(33\)
default	\(\frac {x \left (b \,x^{3}+4 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{4 b \,x^{3}+4 a}\)	\(33\)
risch	\(\frac {a x \sqrt {\left (b \,x^{3}+a \right )^{2}}}{b \,x^{3}+a}+\frac {b \,x^{4} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{4 b \,x^{3}+4 a}\)	\(51\)

[In]

int(((b*x^3+a)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*x*(b*x^3+4*a)*((b*x^3+a)^2)^(1/2)/(b*x^3+a)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.14 \[ \int \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {1}{4} \, b x^{4} + a x \]

[In]

integrate(((b*x^3+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/4*b*x^4 + a*x

Sympy [F]

\[ \int \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\int \sqrt {\left (a + b x^{3}\right )^{2}}\, dx \]

[In]

integrate(((b*x**3+a)**2)**(1/2),x)

[Out]

Integral(sqrt((a + b*x**3)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.14 \[ \int \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {1}{4} \, b x^{4} + a x \]

[In]

integrate(((b*x^3+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/4*b*x^4 + a*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.27 \[ \int \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {1}{4} \, {\left (b x^{4} + 4 \, a x\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \]

[In]

integrate(((b*x^3+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/4*(b*x^4 + 4*a*x)*sgn(b*x^3 + a)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\int \sqrt {{\left (b\,x^3+a\right )}^2} \,d x \]

[In]

int(((a + b*x^3)^2)^(1/2),x)

[Out]

int(((a + b*x^3)^2)^(1/2), x)