3.4.76 \(\int \frac {x^2 (A+B x)}{(a+c x^2)^{5/2}} \, dx\) [376]

Optimal. Leaf size=53 \[ -\frac {x^2 (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 B}{3 c^2 \sqrt {a+c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {819, 267} \begin {gather*} -\frac {x^2 (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 B}{3 c^2 \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 267

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

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(
a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c*(p + 1))), x] - Dist[m*((c*d*f + a*e*g)/(2*a*c*(p + 1))), Int[(d + e*
x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplif
y[m + 2*p + 3], 0] && LtQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.37, size = 44, normalized size = 0.83 \begin {gather*} \frac {-2 a^2 B-3 a B c x^2+A c^2 x^3}{3 a c^2 \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(45)=90\).
time = 0.51, size = 92, normalized size = 1.74

method	result	size
gosper	\(\frac {A \,c^{2} x^{3}-3 a B c \,x^{2}-2 a^{2} B}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} c^{2} a}\)	\(41\)
trager	\(\frac {A \,c^{2} x^{3}-3 a B c \,x^{2}-2 a^{2} B}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} c^{2} a}\)	\(41\)
default	\(B \left (-\frac {x^{2}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\right )+A \left (-\frac {x}{2 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )}{2 c}\right )\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 70, normalized size = 1.32 \begin {gather*} -\frac {B x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {A x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {A x}{3 \, \sqrt {c x^{2} + a} a c} - \frac {2 \, B a}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.56, size = 63, normalized size = 1.19 \begin {gather*} \frac {{\left (A c^{2} x^{3} - 3 \, B a c x^{2} - 2 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a c^{4} x^{4} + 2 \, a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (46) = 92\).
time = 5.19, size = 141, normalized size = 2.66 \begin {gather*} \frac {A x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {3}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + B \left (\begin {cases} - \frac {2 a}{3 a c^{2} \sqrt {a + c x^{2}} + 3 c^{3} x^{2} \sqrt {a + c x^{2}}} - \frac {3 c x^{2}}{3 a c^{2} \sqrt {a + c x^{2}} + 3 c^{3} x^{2} \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*x**3/(3*a**(5/2)*sqrt(1 + c*x**2/a) + 3*a**(3/2)*c*x**2*sqrt(1 + c*x**2/a)) + B*Piecewise((-2*a/(3*a*c**2*sq
rt(a + c*x**2) + 3*c**3*x**2*sqrt(a + c*x**2)) - 3*c*x**2/(3*a*c**2*sqrt(a + c*x**2) + 3*c**3*x**2*sqrt(a + c*
x**2)), Ne(c, 0)), (x**4/(4*a**(5/2)), True))

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Giac [A]
time = 0.65, size = 36, normalized size = 0.68 \begin {gather*} \frac {{\left (\frac {A x}{a} - \frac {3 \, B}{c}\right )} x^{2} - \frac {2 \, B a}{c^{2}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.13, size = 51, normalized size = 0.96 \begin {gather*} \frac {B\,a^2-3\,B\,a\,\left (c\,x^2+a\right )+A\,c\,x\,\left (c\,x^2+a\right )-A\,a\,c\,x}{3\,a\,c^2\,{\left (c\,x^2+a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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