3.6.57 \(\int \frac {A+B x^3}{(e x)^{5/2} (a+b x^3)^{3/2}} \, dx\) [557]

Optimal. Leaf size=67 \[ -\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^3}}-\frac {2 (2 A b-a B) (e x)^{3/2}}{3 a^2 e^4 \sqrt {a+b x^3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 67, 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 = {464, 270} \begin {gather*} -\frac {2 (e x)^{3/2} (2 A b-a B)}{3 a^2 e^4 \sqrt {a+b x^3}}-\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A)/(3*a*e*(e*x)^(3/2)*Sqrt[a + b*x^3]) - (2*(2*A*b - a*B)*(e*x)^(3/2))/(3*a^2*e^4*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 464

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

Rubi steps

\begin {align*} \int \frac {A+B x^3}{(e x)^{5/2} \left (a+b x^3\right )^{3/2}} \, dx &=-\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^3}}-\frac {(2 A b-a B) \int \frac {\sqrt {e x}}{\left (a+b x^3\right )^{3/2}} \, dx}{a e^3}\\ &=-\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^3}}-\frac {2 (2 A b-a B) (e x)^{3/2}}{3 a^2 e^4 \sqrt {a+b x^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.26, size = 44, normalized size = 0.66 \begin {gather*} \frac {2 x \left (-a A-2 A b x^3+a B x^3\right )}{3 a^2 (e x)^{5/2} \sqrt {a+b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.34, size = 44, normalized size = 0.66

method	result	size
gosper	\(-\frac {2 x \left (2 A b \,x^{3}-B a \,x^{3}+A a \right )}{3 \sqrt {b \,x^{3}+a}\, a^{2} \left (e x \right )^{\frac {5}{2}}}\)	\(39\)
default	\(-\frac {2 \left (2 A b \,x^{3}-B a \,x^{3}+A a \right )}{3 x \sqrt {b \,x^{3}+a}\, a^{2} e^{2} \sqrt {e x}}\)	\(44\)
risch	\(-\frac {2 A \sqrt {b \,x^{3}+a}}{3 a^{2} x \,e^{2} \sqrt {e x}}-\frac {2 \left (A b -B a \right ) x^{2}}{3 a^{2} e^{2} \sqrt {e x}\, \sqrt {b \,x^{3}+a}}\)	\(61\)
elliptic	\(\frac {\sqrt {\left (b \,x^{3}+a \right ) e x}\, \left (-\frac {2 x^{2} \left (A b -B a \right )}{3 e^{2} a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b e x}}-\frac {2 A \sqrt {b e \,x^{4}+a e x}}{3 e^{3} a^{2} x^{2}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{3}+a}}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 59, normalized size = 0.88 \begin {gather*} -\frac {2}{3} \, {\left (A {\left (\frac {b x^{\frac {3}{2}}}{\sqrt {b x^{3} + a} a^{2}} + \frac {\sqrt {b x^{3} + a}}{a^{2} x^{\frac {3}{2}}}\right )} - \frac {B x^{\frac {3}{2}}}{\sqrt {b x^{3} + a} a}\right )} e^{\left (-\frac {5}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 8.14, size = 51, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left ({\left (B a - 2 \, A b\right )} x^{3} - A a\right )} \sqrt {b x^{3} + a} \sqrt {x} e^{\left (-\frac {5}{2}\right )}}{3 \, {\left (a^{2} b x^{5} + a^{3} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 76.34, size = 90, normalized size = 1.34 \begin {gather*} A \left (- \frac {2}{3 a \sqrt {b} e^{\frac {5}{2}} x^{3} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {4 \sqrt {b}}{3 a^{2} e^{\frac {5}{2}} \sqrt {\frac {a}{b x^{3}} + 1}}\right ) + \frac {2 B}{3 a \sqrt {b} e^{\frac {5}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 4.75, size = 70, normalized size = 1.04 \begin {gather*} -\frac {\left (\frac {2\,A}{3\,a\,b\,e^2}+\frac {x^3\,\left (4\,A\,b-2\,B\,a\right )}{3\,a^2\,b\,e^2}\right )\,\sqrt {b\,x^3+a}}{x^4\,\sqrt {e\,x}+\frac {a\,x\,\sqrt {e\,x}}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________