\(\int x \sqrt {a x+b x^3} \, dx\) [40]

Optimal result

Integrand size = 15, antiderivative size = 137 \[ \int x \sqrt {a x+b x^3} \, dx=\frac {4 a \sqrt {a x+b x^3}}{21 b}+\frac {2}{7} x^2 \sqrt {a x+b x^3}-\frac {2 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{21 b^{5/4} \sqrt {a x+b x^3}} \]

[Out]

4/21*a*(b*x^3+a*x)^(1/2)/b+2/7*x^2*(b*x^3+a*x)^(1/2)-2/21*a^(7/4)*(cos(2*arctan(b^(1/4)*x^(1/2)/a^(1/4)))^2)^(
1/2)/cos(2*arctan(b^(1/4)*x^(1/2)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(a^(
1/2)+x*b^(1/2))*x^(1/2)*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(5/4)/(b*x^3+a*x)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2046, 2049, 2036, 335, 226} \[ \int x \sqrt {a x+b x^3} \, dx=-\frac {2 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{21 b^{5/4} \sqrt {a x+b x^3}}+\frac {4 a \sqrt {a x+b x^3}}{21 b}+\frac {2}{7} x^2 \sqrt {a x+b x^3} \]

[In]

Int[x*Sqrt[a*x + b*x^3],x]

[Out]

(4*a*Sqrt[a*x + b*x^3])/(21*b) + (2*x^2*Sqrt[a*x + b*x^3])/7 - (2*a^(7/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(
a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(21*b^(5/4)*Sqrt[a*x
+ b*x^3])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2036

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 2046

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b
*x^n)^p/(c*(m + n*p + 1))), x] + Dist[a*(n - j)*(p/(c^j*(m + n*p + 1))), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2049

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n +
1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{7} x^2 \sqrt {a x+b x^3}+\frac {1}{7} (2 a) \int \frac {x^2}{\sqrt {a x+b x^3}} \, dx \\ & = \frac {4 a \sqrt {a x+b x^3}}{21 b}+\frac {2}{7} x^2 \sqrt {a x+b x^3}-\frac {\left (2 a^2\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{21 b} \\ & = \frac {4 a \sqrt {a x+b x^3}}{21 b}+\frac {2}{7} x^2 \sqrt {a x+b x^3}-\frac {\left (2 a^2 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{21 b \sqrt {a x+b x^3}} \\ & = \frac {4 a \sqrt {a x+b x^3}}{21 b}+\frac {2}{7} x^2 \sqrt {a x+b x^3}-\frac {\left (4 a^2 \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{21 b \sqrt {a x+b x^3}} \\ & = \frac {4 a \sqrt {a x+b x^3}}{21 b}+\frac {2}{7} x^2 \sqrt {a x+b x^3}-\frac {2 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 b^{5/4} \sqrt {a x+b x^3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.58 \[ \int x \sqrt {a x+b x^3} \, dx=\frac {2 \sqrt {x \left (a+b x^2\right )} \left (\left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}}-a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{7 b \sqrt {1+\frac {b x^2}{a}}} \]

[In]

Integrate[x*Sqrt[a*x + b*x^3],x]

[Out]

(2*Sqrt[x*(a + b*x^2)]*((a + b*x^2)*Sqrt[1 + (b*x^2)/a] - a*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^2)/a)]))/
(7*b*Sqrt[1 + (b*x^2)/a])

Maple [A] (verified)

Time = 2.36 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {2 x^{2} \sqrt {b \,x^{3}+a x}}{7}+\frac {4 a \sqrt {b \,x^{3}+a x}}{21 b}-\frac {2 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{2} \sqrt {b \,x^{3}+a x}}\)	\(146\)
elliptic	\(\frac {2 x^{2} \sqrt {b \,x^{3}+a x}}{7}+\frac {4 a \sqrt {b \,x^{3}+a x}}{21 b}-\frac {2 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{2} \sqrt {b \,x^{3}+a x}}\)	\(146\)
risch	\(\frac {2 \left (3 b \,x^{2}+2 a \right ) x \left (b \,x^{2}+a \right )}{21 b \sqrt {x \left (b \,x^{2}+a \right )}}-\frac {2 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{2} \sqrt {b \,x^{3}+a x}}\)	\(147\)

[In]

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

[Out]

2/7*x^2*(b*x^3+a*x)^(1/2)+4/21*a*(b*x^3+a*x)^(1/2)/b-2/21*a^2/b^2*(-a*b)^(1/2)*((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2
)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/2)/(b*x^3+a*x)^(1/2)*EllipticF(
((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.36 \[ \int x \sqrt {a x+b x^3} \, dx=-\frac {2 \, {\left (2 \, a^{2} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (3 \, b^{2} x^{2} + 2 \, a b\right )} \sqrt {b x^{3} + a x}\right )}}{21 \, b^{2}} \]

[In]

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

[Out]

-2/21*(2*a^2*sqrt(b)*weierstrassPInverse(-4*a/b, 0, x) - (3*b^2*x^2 + 2*a*b)*sqrt(b*x^3 + a*x))/b^2

Sympy [F]

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

[In]

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

[Out]

Integral(x*sqrt(x*(a + b*x**2)), x)

Maxima [F]

\[ \int x \sqrt {a x+b x^3} \, dx=\int { \sqrt {b x^{3} + a x} x \,d x } \]

[In]

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

[Out]

integrate(sqrt(b*x^3 + a*x)*x, x)

Giac [F]

\[ \int x \sqrt {a x+b x^3} \, dx=\int { \sqrt {b x^{3} + a x} x \,d x } \]

[In]

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

[Out]

integrate(sqrt(b*x^3 + a*x)*x, x)

Mupad [F(-1)]

Timed out. \[ \int x \sqrt {a x+b x^3} \, dx=\int x\,\sqrt {b\,x^3+a\,x} \,d x \]

[In]

int(x*(a*x + b*x^3)^(1/2),x)

[Out]

int(x*(a*x + b*x^3)^(1/2), x)