3.39 \(\int \frac {A+B x}{(a+b x^2)^{5/2}} \, dx\)

Optimal. Leaf size=51 \[ \frac {2 A x}{3 a^2 \sqrt {a+b x^2}}+\frac {A b x-a B}{3 a b \left (a+b x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {639, 191} \[ \frac {2 A x}{3 a^2 \sqrt {a+b x^2}}-\frac {a B-A b x}{3 a b \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(a + b*x^2)^(5/2),x]

[Out]

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

Rule 191

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

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 43, normalized size = 0.84 \[ \frac {-a^2 B+3 a A b x+2 A b^2 x^3}{3 a^2 b \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.61, size = 62, normalized size = 1.22 \[ \frac {{\left (2 \, A b^{2} x^{3} + 3 \, A a b x - B a^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.48, size = 37, normalized size = 0.73 \[ \frac {{\left (\frac {2 \, A b x^{2}}{a^{2}} + \frac {3 \, A}{a}\right )} x - \frac {B}{b}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 40, normalized size = 0.78 \[ \frac {2 A \,x^{3} b^{2}+3 A x a b -B \,a^{2}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.37, size = 48, normalized size = 0.94 \[ \frac {2 \, A x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {A x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.93, size = 41, normalized size = 0.80 \[ \frac {2\,A\,b\,x\,\left (b\,x^2+a\right )-B\,a^2+A\,a\,b\,x}{3\,a^2\,b\,{\left (b\,x^2+a\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(a + b*x^2)^(5/2),x)

[Out]

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

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sympy [B]  time = 13.20, size = 146, normalized size = 2.86 \[ A \left (\frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (\begin {cases} - \frac {1}{3 a b \sqrt {a + b x^{2}} + 3 b^{2} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*(3*a*x/(3*a**(7/2)*sqrt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x**2/a)) + 2*b*x**3/(3*a**(7/2)*sqrt(1
+ b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x**2/a))) + B*Piecewise((-1/(3*a*b*sqrt(a + b*x**2) + 3*b**2*x**2*s
qrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(5/2)), True))

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