3.6.90 \(\int \frac {x (A+B x^2)}{(a+b x^2)^{5/2}} \, dx\) [590]

Optimal. Leaf size=44 \[ \frac {-A b+a B}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {B}{b^2 \sqrt {a+b x^2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {455, 45} \begin {gather*} -\frac {A b-a B}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {B}{b^2 \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(A*b - a*B)/(b^2*(a + b*x^2)^(3/2)) - B/(b^2*Sqrt[a + b*x^2])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 34, normalized size = 0.77 \begin {gather*} \frac {-A b-2 a B-3 b B x^2}{3 b^2 \left (a+b x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.09, size = 52, normalized size = 1.18

method	result	size
gosper	\(-\frac {3 b B \,x^{2}+A b +2 B a}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}}\)	\(30\)
trager	\(-\frac {3 b B \,x^{2}+A b +2 B a}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}}\)	\(30\)
default	\(B \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )-\frac {A}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 50, normalized size = 1.14 \begin {gather*} -\frac {B x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {2 \, B a}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} - \frac {A}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.40, size = 52, normalized size = 1.18 \begin {gather*} -\frac {{\left (3 \, B b x^{2} + 2 \, B a + A b\right )} \sqrt {b x^{2} + a}}{3 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (37) = 74\).
time = 0.39, size = 143, normalized size = 3.25 \begin {gather*} \begin {cases} - \frac {A b}{3 a b^{2} \sqrt {a + b x^{2}} + 3 b^{3} x^{2} \sqrt {a + b x^{2}}} - \frac {2 B a}{3 a b^{2} \sqrt {a + b x^{2}} + 3 b^{3} x^{2} \sqrt {a + b x^{2}}} - \frac {3 B b x^{2}}{3 a b^{2} \sqrt {a + b x^{2}} + 3 b^{3} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{2}}{2} + \frac {B x^{4}}{4}}{a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-A*b/(3*a*b**2*sqrt(a + b*x**2) + 3*b**3*x**2*sqrt(a + b*x**2)) - 2*B*a/(3*a*b**2*sqrt(a + b*x**2)
+ 3*b**3*x**2*sqrt(a + b*x**2)) - 3*B*b*x**2/(3*a*b**2*sqrt(a + b*x**2) + 3*b**3*x**2*sqrt(a + b*x**2)), Ne(b,
0)), ((A*x**2/2 + B*x**4/4)/a**(5/2), True))

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Giac [A]
time = 1.42, size = 32, normalized size = 0.73 \begin {gather*} -\frac {3 \, {\left (b x^{2} + a\right )} B - B a + A b}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.28, size = 32, normalized size = 0.73 \begin {gather*} -\frac {A\,b-B\,a+3\,B\,\left (b\,x^2+a\right )}{3\,b^2\,{\left (b\,x^2+a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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