3.6.62 \(\int \frac {\sqrt {e x} (A+B x^3)}{(a+b x^3)^{5/2}} \, dx\) [562]

Optimal. Leaf size=79 \[ \frac {2 (A b-a B) (e x)^{3/2}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac {2 (2 A b+a B) (e x)^{3/2}}{9 a^2 b e \sqrt {a+b x^3}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {468, 270} \begin {gather*} \frac {2 (e x)^{3/2} (a B+2 A b)}{9 a^2 b e \sqrt {a+b x^3}}+\frac {2 (e x)^{3/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[e*x]*(A + B*x^3))/(a + b*x^3)^(5/2),x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

Rubi steps

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

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Mathematica [A]
time = 0.35, size = 44, normalized size = 0.56 \begin {gather*} \frac {2 x \sqrt {e x} \left (3 a A+2 A b x^3+a B x^3\right )}{9 a^2 \left (a+b x^3\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[e*x]*(A + B*x^3))/(a + b*x^3)^(5/2),x]

[Out]

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

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Maple [A]
time = 0.30, size = 39, normalized size = 0.49

method	result	size
gosper	\(\frac {2 x \left (2 A b \,x^{3}+B a \,x^{3}+3 A a \right ) \sqrt {e x}}{9 \left (b \,x^{3}+a \right )^{\frac {3}{2}} a^{2}}\)	\(39\)
default	\(\frac {2 x \left (2 A b \,x^{3}+B a \,x^{3}+3 A a \right ) \sqrt {e x}}{9 \left (b \,x^{3}+a \right )^{\frac {3}{2}} a^{2}}\)	\(39\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{3}+a \right ) e x}\, \left (\frac {2 x \left (A b -B a \right ) \sqrt {b e \,x^{4}+a e x}}{9 a \,b^{3} \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {2 e \,x^{2} \left (2 A b +B a \right )}{9 b \,a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b e x}}\right )}{e x \sqrt {b \,x^{3}+a}}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*x*(2*A*b*x^3+B*a*x^3+3*A*a)*(e*x)^(1/2)/(b*x^3+a)^(3/2)/a^2

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Maxima [A]
time = 0.29, size = 54, normalized size = 0.68 \begin {gather*} \frac {2}{9} \, {\left (\frac {B x^{\frac {9}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a} - \frac {A {\left (b - \frac {3 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} x^{\frac {9}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}\right )} e^{\frac {1}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 4.48, size = 59, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left ({\left (B a + 2 \, A b\right )} x^{4} + 3 \, A a x\right )} \sqrt {b x^{3} + a} \sqrt {x} e^{\frac {1}{2}}}{9 \, {\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*((B*a + 2*A*b)*x^4 + 3*A*a*x)*sqrt(b*x^3 + a)*sqrt(x)*e^(1/2)/(a^2*b^2*x^6 + 2*a^3*b*x^3 + a^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (71) = 142\).
time = 71.09, size = 168, normalized size = 2.13 \begin {gather*} A \left (\frac {6 a \sqrt {e} x^{\frac {3}{2}}}{9 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{3}}{a}} + 9 a^{\frac {5}{2}} b x^{3} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {4 b \sqrt {e} x^{\frac {9}{2}}}{9 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{3}}{a}} + 9 a^{\frac {5}{2}} b x^{3} \sqrt {1 + \frac {b x^{3}}{a}}}\right ) + \frac {2 B \sqrt {e} x^{\frac {9}{2}}}{9 a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}} + 9 a^{\frac {3}{2}} b x^{3} \sqrt {1 + \frac {b x^{3}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*(6*a*sqrt(e)*x**(3/2)/(9*a**(7/2)*sqrt(1 + b*x**3/a) + 9*a**(5/2)*b*x**3*sqrt(1 + b*x**3/a)) + 4*b*sqrt(e)*x
**(9/2)/(9*a**(7/2)*sqrt(1 + b*x**3/a) + 9*a**(5/2)*b*x**3*sqrt(1 + b*x**3/a))) + 2*B*sqrt(e)*x**(9/2)/(9*a**(
5/2)*sqrt(1 + b*x**3/a) + 9*a**(3/2)*b*x**3*sqrt(1 + b*x**3/a))

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Giac [A]
time = 1.60, size = 51, normalized size = 0.65 \begin {gather*} \frac {2 \, x^{\frac {3}{2}} {\left (\frac {3 \, A}{a} + \frac {{\left (B a^{5} b^{5} + 2 \, A a^{4} b^{6}\right )} x^{3}}{a^{6} b^{5}}\right )} e^{\frac {1}{2}}}{9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*x^(3/2)*(3*A/a + (B*a^5*b^5 + 2*A*a^4*b^6)*x^3/(a^6*b^5))*e^(1/2)/(b*x^3 + a)^(3/2)

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Mupad [B]
time = 4.63, size = 73, normalized size = 0.92 \begin {gather*} \frac {\left (\frac {2\,A\,x\,\sqrt {e\,x}}{3\,a\,b^2}+\frac {x^4\,\sqrt {e\,x}\,\left (4\,A\,b+2\,B\,a\right )}{9\,a^2\,b^2}\right )\,\sqrt {b\,x^3+a}}{x^6+\frac {a^2}{b^2}+\frac {2\,a\,x^3}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((2*A*x*(e*x)^(1/2))/(3*a*b^2) + (x^4*(e*x)^(1/2)*(4*A*b + 2*B*a))/(9*a^2*b^2))*(a + b*x^3)^(1/2))/(x^6 + a^2
/b^2 + (2*a*x^3)/b)

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