3.38 \(\int x^3 \sqrt {a x+b x^3} \, dx\)

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

[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 = 0.19, 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 = {2021, 2024, 2011, 329, 220} \[ -\frac {20 a^2 \sqrt {a x+b x^3}}{231 b^2}+\frac {10 a^{11/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a x+b x^3}}+\frac {2}{11} x^4 \sqrt {a x+b x^3}+\frac {4 a x^2 \sqrt {a x+b x^3}}{77 b} \]

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 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 329

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 2011

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 2021

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 2024

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 ) \operatorname {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]  time = 0.06, size = 95, normalized size = 0.58 \[ \frac {2 \sqrt {x \left (a+b x^2\right )} \left (\sqrt {\frac {b x^2}{a}+1} \left (-5 a^2+2 a b x^2+7 b^2 x^4\right )+5 a^2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{77 b^2 \sqrt {\frac {b x^2}{a}+1}} \]

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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fricas [F]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b x^{3} + a x} x^{3}, x\right ) \]

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]

integral(sqrt(b*x^3 + a*x)*x^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{3} + a x} x^{3}\,{d x} \]

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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maple [A]  time = 0.10, size = 168, normalized size = 1.03 \[ \frac {2 \sqrt {b \,x^{3}+a x}\, x^{4}}{11}+\frac {4 \sqrt {b \,x^{3}+a x}\, a \,x^{2}}{77 b}+\frac {10 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a^{3} \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{231 \sqrt {b \,x^{3}+a x}\, b^{3}}-\frac {20 \sqrt {b \,x^{3}+a x}\, a^{2}}{231 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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+(-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)*(-1/(-a*b)^(1/
2)*b*x)^(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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{3} + a x} x^{3}\,{d x} \]

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

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

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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