3.1044 \(\int \frac {(a+b x^2)^p}{x^3} \, dx\)

Optimal. Leaf size=42 \[ \frac {b \left (a+b x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac {b x^2}{a}+1\right )}{2 a^2 (p+1)} \]

[Out]

1/2*b*(b*x^2+a)^(1+p)*hypergeom([2, 1+p],[2+p],1+b*x^2/a)/a^2/(1+p)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 65} \[ \frac {b \left (a+b x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac {b x^2}{a}+1\right )}{2 a^2 (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^p/x^3,x]

[Out]

(b*(a + b*x^2)^(1 + p)*Hypergeometric2F1[2, 1 + p, 2 + p, 1 + (b*x^2)/a])/(2*a^2*(1 + p))

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rule 266

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^p}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^p}{x^2} \, dx,x,x^2\right )\\ &=\frac {b \left (a+b x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;1+\frac {b x^2}{a}\right )}{2 a^2 (1+p)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 42, normalized size = 1.00 \[ \frac {b \left (a+b x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac {b x^2}{a}+1\right )}{2 a^2 (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^p/x^3,x]

[Out]

(b*(a + b*x^2)^(1 + p)*Hypergeometric2F1[2, 1 + p, 2 + p, 1 + (b*x^2)/a])/(2*a^2*(1 + p))

________________________________________________________________________________________

fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{p}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^p/x^3,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^p/x^3, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{p}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^p/x^3,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p/x^3, x)

________________________________________________________________________________________

maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^p/x^3,x)

[Out]

int((b*x^2+a)^p/x^3,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{p}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^p/x^3,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^p/x^3, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,x^2+a\right )}^p}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^p/x^3,x)

[Out]

int((a + b*x^2)^p/x^3, x)

________________________________________________________________________________________

sympy [C]  time = 5.13, size = 42, normalized size = 1.00 \[ - \frac {b^{p} x^{2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{2} \Gamma \left (2 - p\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**p/x**3,x)

[Out]

-b**p*x**(2*p)*gamma(1 - p)*hyper((-p, 1 - p), (2 - p,), a*exp_polar(I*pi)/(b*x**2))/(2*x**2*gamma(2 - p))

________________________________________________________________________________________