3.3.14 \(\int \frac {x^5 (A+B x^3)}{\sqrt {a+b x^3}} \, dx\) [214]

Optimal. Leaf size=73 \[ -\frac {2 a (A b-a B) \sqrt {a+b x^3}}{3 b^3}+\frac {2 (A b-2 a B) \left (a+b x^3\right )^{3/2}}{9 b^3}+\frac {2 B \left (a+b x^3\right )^{5/2}}{15 b^3} \]

[Out]

2/9*(A*b-2*B*a)*(b*x^3+a)^(3/2)/b^3+2/15*B*(b*x^3+a)^(5/2)/b^3-2/3*a*(A*b-B*a)*(b*x^3+a)^(1/2)/b^3

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Rubi [A]
time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 78} \begin {gather*} \frac {2 \left (a+b x^3\right )^{3/2} (A b-2 a B)}{9 b^3}-\frac {2 a \sqrt {a+b x^3} (A b-a B)}{3 b^3}+\frac {2 B \left (a+b x^3\right )^{5/2}}{15 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5*(A + B*x^3))/Sqrt[a + b*x^3],x]

[Out]

(-2*a*(A*b - a*B)*Sqrt[a + b*x^3])/(3*b^3) + (2*(A*b - 2*a*B)*(a + b*x^3)^(3/2))/(9*b^3) + (2*B*(a + b*x^3)^(5
/2))/(15*b^3)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^5 \left (A+B x^3\right )}{\sqrt {a+b x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x (A+B x)}{\sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a (-A b+a B)}{b^2 \sqrt {a+b x}}+\frac {(A b-2 a B) \sqrt {a+b x}}{b^2}+\frac {B (a+b x)^{3/2}}{b^2}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 a (A b-a B) \sqrt {a+b x^3}}{3 b^3}+\frac {2 (A b-2 a B) \left (a+b x^3\right )^{3/2}}{9 b^3}+\frac {2 B \left (a+b x^3\right )^{5/2}}{15 b^3}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 56, normalized size = 0.77 \begin {gather*} \frac {2 \sqrt {a+b x^3} \left (-10 a A b+8 a^2 B+5 A b^2 x^3-4 a b B x^3+3 b^2 B x^6\right )}{45 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(A + B*x^3))/Sqrt[a + b*x^3],x]

[Out]

(2*Sqrt[a + b*x^3]*(-10*a*A*b + 8*a^2*B + 5*A*b^2*x^3 - 4*a*b*B*x^3 + 3*b^2*B*x^6))/(45*b^3)

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Maple [A]
time = 0.30, size = 92, normalized size = 1.26

method	result	size
gosper	\(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (-3 b^{2} B \,x^{6}-5 A \,b^{2} x^{3}+4 B a b \,x^{3}+10 a b A -8 a^{2} B \right )}{45 b^{3}}\)	\(53\)
trager	\(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (-3 b^{2} B \,x^{6}-5 A \,b^{2} x^{3}+4 B a b \,x^{3}+10 a b A -8 a^{2} B \right )}{45 b^{3}}\)	\(53\)
risch	\(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (-3 b^{2} B \,x^{6}-5 A \,b^{2} x^{3}+4 B a b \,x^{3}+10 a b A -8 a^{2} B \right )}{45 b^{3}}\)	\(53\)
elliptic	\(\frac {2 B \,x^{6} \sqrt {b \,x^{3}+a}}{15 b}+\frac {2 \left (A -\frac {4 a B}{5 b}\right ) x^{3} \sqrt {b \,x^{3}+a}}{9 b}-\frac {4 a \left (A -\frac {4 a B}{5 b}\right ) \sqrt {b \,x^{3}+a}}{9 b^{2}}\)	\(70\)
default	\(B \left (\frac {2 x^{6} \sqrt {b \,x^{3}+a}}{15 b}-\frac {8 a \,x^{3} \sqrt {b \,x^{3}+a}}{45 b^{2}}+\frac {16 a^{2} \sqrt {b \,x^{3}+a}}{45 b^{3}}\right )+A \left (\frac {2 x^{3} \sqrt {b \,x^{3}+a}}{9 b}-\frac {4 a \sqrt {b \,x^{3}+a}}{9 b^{2}}\right )\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(B*x^3+A)/(b*x^3+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

B*(2/15*x^6*(b*x^3+a)^(1/2)/b-8/45*a/b^2*x^3*(b*x^3+a)^(1/2)+16/45*a^2*(b*x^3+a)^(1/2)/b^3)+A*(2/9*x^3*(b*x^3+
a)^(1/2)/b-4/9*a*(b*x^3+a)^(1/2)/b^2)

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Maxima [A]
time = 0.27, size = 83, normalized size = 1.14 \begin {gather*} \frac {2}{45} \, B {\left (\frac {3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}}}{b^{3}} - \frac {10 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a}{b^{3}} + \frac {15 \, \sqrt {b x^{3} + a} a^{2}}{b^{3}}\right )} + \frac {2}{9} \, A {\left (\frac {{\left (b x^{3} + a\right )}^{\frac {3}{2}}}{b^{2}} - \frac {3 \, \sqrt {b x^{3} + a} a}{b^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^3+A)/(b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

2/45*B*(3*(b*x^3 + a)^(5/2)/b^3 - 10*(b*x^3 + a)^(3/2)*a/b^3 + 15*sqrt(b*x^3 + a)*a^2/b^3) + 2/9*A*((b*x^3 + a
)^(3/2)/b^2 - 3*sqrt(b*x^3 + a)*a/b^2)

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Fricas [A]
time = 2.07, size = 52, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (3 \, B b^{2} x^{6} - {\left (4 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 10 \, A a b\right )} \sqrt {b x^{3} + a}}{45 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^3+A)/(b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

2/45*(3*B*b^2*x^6 - (4*B*a*b - 5*A*b^2)*x^3 + 8*B*a^2 - 10*A*a*b)*sqrt(b*x^3 + a)/b^3

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Sympy [A]
time = 0.37, size = 124, normalized size = 1.70 \begin {gather*} \begin {cases} - \frac {4 A a \sqrt {a + b x^{3}}}{9 b^{2}} + \frac {2 A x^{3} \sqrt {a + b x^{3}}}{9 b} + \frac {16 B a^{2} \sqrt {a + b x^{3}}}{45 b^{3}} - \frac {8 B a x^{3} \sqrt {a + b x^{3}}}{45 b^{2}} + \frac {2 B x^{6} \sqrt {a + b x^{3}}}{15 b} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{6}}{6} + \frac {B x^{9}}{9}}{\sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(B*x**3+A)/(b*x**3+a)**(1/2),x)

[Out]

Piecewise((-4*A*a*sqrt(a + b*x**3)/(9*b**2) + 2*A*x**3*sqrt(a + b*x**3)/(9*b) + 16*B*a**2*sqrt(a + b*x**3)/(45
*b**3) - 8*B*a*x**3*sqrt(a + b*x**3)/(45*b**2) + 2*B*x**6*sqrt(a + b*x**3)/(15*b), Ne(b, 0)), ((A*x**6/6 + B*x
**9/9)/sqrt(a), True))

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Giac [A]
time = 1.40, size = 70, normalized size = 0.96 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a} {\left (B a^{2} - A a b\right )}}{3 \, b^{3}} + \frac {2 \, {\left (3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} B - 10 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} B a + 5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} A b\right )}}{45 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^3+A)/(b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

2/3*sqrt(b*x^3 + a)*(B*a^2 - A*a*b)/b^3 + 2/45*(3*(b*x^3 + a)^(5/2)*B - 10*(b*x^3 + a)^(3/2)*B*a + 5*(b*x^3 +
a)^(3/2)*A*b)/b^3

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Mupad [B]
time = 2.65, size = 52, normalized size = 0.71 \begin {gather*} \frac {2\,\sqrt {b\,x^3+a}\,\left (8\,B\,a^2-4\,B\,a\,b\,x^3-10\,A\,a\,b+3\,B\,b^2\,x^6+5\,A\,b^2\,x^3\right )}{45\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(A + B*x^3))/(a + b*x^3)^(1/2),x)

[Out]

(2*(a + b*x^3)^(1/2)*(8*B*a^2 + 5*A*b^2*x^3 + 3*B*b^2*x^6 - 10*A*a*b - 4*B*a*b*x^3))/(45*b^3)

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