3.1.38 \(\int x^3 \sqrt {a x+b x^3} \, dx\) [38]

Optimal. Leaf size=163 \[ -\frac {20 a^2 \sqrt {a x+b x^3}}{231 b^2}+\frac {4 a x^2 \sqrt {a x+b x^3}}{77 b}+\frac {2}{11} x^4 \sqrt {a x+b x^3}+\frac {10 a^{11/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a x+b x^3}} \]

[Out]

-20/231*a^2*(b*x^3+a*x)^(1/2)/b^2+4/77*a*x^2*(b*x^3+a*x)^(1/2)/b+2/11*x^4*(b*x^3+a*x)^(1/2)+10/231*a^(11/4)*(c
os(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^(9
/4)/(b*x^3+a*x)^(1/2)

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Rubi [A]
time = 0.13, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2046, 2049, 2036, 335, 226} \begin {gather*} \frac {10 a^{11/4} \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 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a x+b x^3}}-\frac {20 a^2 \sqrt {a x+b x^3}}{231 b^2}+\frac {2}{11} x^4 \sqrt {a x+b x^3}+\frac {4 a x^2 \sqrt {a x+b x^3}}{77 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a*x + b*x^3],x]

[Out]

(-20*a^2*Sqrt[a*x + b*x^3])/(231*b^2) + (4*a*x^2*Sqrt[a*x + b*x^3])/(77*b) + (2*x^4*Sqrt[a*x + b*x^3])/11 + (1
0*a^(11/4)*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])/(231*b^(9/4)*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*} \int x^3 \sqrt {a x+b x^3} \, dx &=\frac {2}{11} x^4 \sqrt {a x+b x^3}+\frac {1}{11} (2 a) \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx\\ &=\frac {4 a x^2 \sqrt {a x+b x^3}}{77 b}+\frac {2}{11} x^4 \sqrt {a x+b x^3}-\frac {\left (10 a^2\right ) \int \frac {x^2}{\sqrt {a x+b x^3}} \, dx}{77 b}\\ &=-\frac {20 a^2 \sqrt {a x+b x^3}}{231 b^2}+\frac {4 a x^2 \sqrt {a x+b x^3}}{77 b}+\frac {2}{11} x^4 \sqrt {a x+b x^3}+\frac {\left (10 a^3\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{231 b^2}\\ &=-\frac {20 a^2 \sqrt {a x+b x^3}}{231 b^2}+\frac {4 a x^2 \sqrt {a x+b x^3}}{77 b}+\frac {2}{11} x^4 \sqrt {a x+b x^3}+\frac {\left (10 a^3 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{231 b^2 \sqrt {a x+b x^3}}\\ &=-\frac {20 a^2 \sqrt {a x+b x^3}}{231 b^2}+\frac {4 a x^2 \sqrt {a x+b x^3}}{77 b}+\frac {2}{11} x^4 \sqrt {a x+b x^3}+\frac {\left (20 a^3 \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{231 b^2 \sqrt {a x+b x^3}}\\ &=-\frac {20 a^2 \sqrt {a x+b x^3}}{231 b^2}+\frac {4 a x^2 \sqrt {a x+b x^3}}{77 b}+\frac {2}{11} x^4 \sqrt {a x+b x^3}+\frac {10 a^{11/4} \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 )}{231 b^{9/4} \sqrt {a x+b x^3}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.04, size = 95, normalized size = 0.58 \begin {gather*} \frac {2 \sqrt {x \left (a+b x^2\right )} \left (\sqrt {1+\frac {b x^2}{a}} \left (-5 a^2+2 a b x^2+7 b^2 x^4\right )+5 a^2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{77 b^2 \sqrt {1+\frac {b x^2}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[a*x + b*x^3],x]

[Out]

(2*Sqrt[x*(a + b*x^2)]*(Sqrt[1 + (b*x^2)/a]*(-5*a^2 + 2*a*b*x^2 + 7*b^2*x^4) + 5*a^2*Hypergeometric2F1[-1/2, 1
/4, 5/4, -((b*x^2)/a)]))/(77*b^2*Sqrt[1 + (b*x^2)/a])

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Maple [A]
time = 0.35, size = 168, normalized size = 1.03

method	result	size
risch	\(-\frac {2 \left (-21 b^{2} x^{4}-6 a b \,x^{2}+10 a^{2}\right ) x \left (b \,x^{2}+a \right )}{231 b^{2} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {10 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{231 b^{3} \sqrt {b \,x^{3}+a x}}\)	\(158\)
default	\(\frac {2 x^{4} \sqrt {b \,x^{3}+a x}}{11}+\frac {4 a \,x^{2} \sqrt {b \,x^{3}+a x}}{77 b}-\frac {20 a^{2} \sqrt {b \,x^{3}+a x}}{231 b^{2}}+\frac {10 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{231 b^{3} \sqrt {b \,x^{3}+a x}}\)	\(168\)
elliptic	\(\frac {2 x^{4} \sqrt {b \,x^{3}+a x}}{11}+\frac {4 a \,x^{2} \sqrt {b \,x^{3}+a x}}{77 b}-\frac {20 a^{2} \sqrt {b \,x^{3}+a x}}{231 b^{2}}+\frac {10 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{231 b^{3} \sqrt {b \,x^{3}+a x}}\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*x^4*(b*x^3+a*x)^(1/2)+4/77*a*x^2*(b*x^3+a*x)^(1/2)/b-20/231*a^2*(b*x^3+a*x)^(1/2)/b^2+10/231*a^3/b^3*(-a*
b)^(1/2)*((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2)*(-2*(x-1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*
b)^(1/2))^(1/2)/(b*x^3+a*x)^(1/2)*EllipticF(((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*x^3 + a*x)*x^3, x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.49, size = 59, normalized size = 0.36 \begin {gather*} \frac {2 \, {\left (10 \, a^{3} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (21 \, b^{3} x^{4} + 6 \, a b^{2} x^{2} - 10 \, a^{2} b\right )} \sqrt {b x^{3} + a x}\right )}}{231 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*(10*a^3*sqrt(b)*weierstrassPInverse(-4*a/b, 0, x) + (21*b^3*x^4 + 6*a*b^2*x^2 - 10*a^2*b)*sqrt(b*x^3 + a
*x))/b^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \sqrt {x \left (a + b x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*sqrt(x*(a + b*x**2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*x^3 + a*x)*x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,\sqrt {b\,x^3+a\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a*x + b*x^3)^(1/2),x)

[Out]

int(x^3*(a*x + b*x^3)^(1/2), x)

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