\(\int \frac {x^m}{(a+b \sqrt {x})^2} \, dx\) [2262]

Optimal result

Integrand size = 15, antiderivative size = 37 \[ \int \frac {x^m}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,2 (1+m),3+2 m,-\frac {b \sqrt {x}}{a}\right )}{a^2 (1+m)} \]

[Out]

x^(1+m)*hypergeom([2, 2+2*m],[3+2*m],-b*x^(1/2)/a)/a^2/(1+m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {348, 66} \[ \int \frac {x^m}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (2,2 (m+1),2 m+3,-\frac {b \sqrt {x}}{a}\right )}{a^2 (m+1)} \]

[In]

Int[x^m/(a + b*Sqrt[x])^2,x]

[Out]

(x^(1 + m)*Hypergeometric2F1[2, 2*(1 + m), 3 + 2*m, -((b*Sqrt[x])/a)])/(a^2*(1 + m))

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 348

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

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^{-1+2 (1+m)}}{(a+b x)^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {x^{1+m} \, _2F_1\left (2,2 (1+m);3+2 m;-\frac {b \sqrt {x}}{a}\right )}{a^2 (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int \frac {x^m}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,2 (1+m),1+2 (1+m),-\frac {b \sqrt {x}}{a}\right )}{a^2 (1+m)} \]

[In]

Integrate[x^m/(a + b*Sqrt[x])^2,x]

[Out]

(x^(1 + m)*Hypergeometric2F1[2, 2*(1 + m), 1 + 2*(1 + m), -((b*Sqrt[x])/a)])/(a^2*(1 + m))

Maple [F]

\[\int \frac {x^{m}}{\left (a +b \sqrt {x}\right )^{2}}d x\]

[In]

int(x^m/(a+b*x^(1/2))^2,x)

[Out]

int(x^m/(a+b*x^(1/2))^2,x)

Fricas [F]

\[ \int \frac {x^m}{\left (a+b \sqrt {x}\right )^2} \, dx=\int { \frac {x^{m}}{{\left (b \sqrt {x} + a\right )}^{2}} \,d x } \]

[In]

integrate(x^m/(a+b*x^(1/2))^2,x, algorithm="fricas")

[Out]

integral(-(2*a*b*sqrt(x)*x^m - (b^2*x + a^2)*x^m)/(b^4*x^2 - 2*a^2*b^2*x + a^4), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.04 (sec) , antiderivative size = 478, normalized size of antiderivative = 12.92 \[ \int \frac {x^m}{\left (a+b \sqrt {x}\right )^2} \, dx=- \frac {8 a m^{2} x^{m + 1} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} - \frac {12 a m x^{m + 1} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} + \frac {4 a m x^{m + 1} \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} - \frac {4 a x^{m + 1} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} + \frac {4 a x^{m + 1} \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} - \frac {8 b m^{2} \sqrt {x} x^{m + 1} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} - \frac {12 b m \sqrt {x} x^{m + 1} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} - \frac {4 b \sqrt {x} x^{m + 1} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} \]

[In]

integrate(x**m/(a+b*x**(1/2))**2,x)

[Out]

-8*a*m**2*x**(m + 1)*lerchphi(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m + 2)*gamma(2*m + 2)/(a**3*gamma(2*m + 3) + a
**2*b*sqrt(x)*gamma(2*m + 3)) - 12*a*m*x**(m + 1)*lerchphi(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m + 2)*gamma(2*m
+ 2)/(a**3*gamma(2*m + 3) + a**2*b*sqrt(x)*gamma(2*m + 3)) + 4*a*m*x**(m + 1)*gamma(2*m + 2)/(a**3*gamma(2*m +
3) + a**2*b*sqrt(x)*gamma(2*m + 3)) - 4*a*x**(m + 1)*lerchphi(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m + 2)*gamma(
2*m + 2)/(a**3*gamma(2*m + 3) + a**2*b*sqrt(x)*gamma(2*m + 3)) + 4*a*x**(m + 1)*gamma(2*m + 2)/(a**3*gamma(2*m
+ 3) + a**2*b*sqrt(x)*gamma(2*m + 3)) - 8*b*m**2*sqrt(x)*x**(m + 1)*lerchphi(b*sqrt(x)*exp_polar(I*pi)/a, 1,
2*m + 2)*gamma(2*m + 2)/(a**3*gamma(2*m + 3) + a**2*b*sqrt(x)*gamma(2*m + 3)) - 12*b*m*sqrt(x)*x**(m + 1)*lerc
hphi(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m + 2)*gamma(2*m + 2)/(a**3*gamma(2*m + 3) + a**2*b*sqrt(x)*gamma(2*m +
3)) - 4*b*sqrt(x)*x**(m + 1)*lerchphi(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m + 2)*gamma(2*m + 2)/(a**3*gamma(2*m
+ 3) + a**2*b*sqrt(x)*gamma(2*m + 3))

Maxima [F]

\[ \int \frac {x^m}{\left (a+b \sqrt {x}\right )^2} \, dx=\int { \frac {x^{m}}{{\left (b \sqrt {x} + a\right )}^{2}} \,d x } \]

[In]

integrate(x^m/(a+b*x^(1/2))^2,x, algorithm="maxima")

[Out]

-(2*m + 1)*integrate(x^m/(a*b*sqrt(x) + a^2), x) + 2*x*x^m/(a*b*sqrt(x) + a^2)

Giac [F]

\[ \int \frac {x^m}{\left (a+b \sqrt {x}\right )^2} \, dx=\int { \frac {x^{m}}{{\left (b \sqrt {x} + a\right )}^{2}} \,d x } \]

[In]

integrate(x^m/(a+b*x^(1/2))^2,x, algorithm="giac")

[Out]

integrate(x^m/(b*sqrt(x) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m}{\left (a+b \sqrt {x}\right )^2} \, dx=\int \frac {x^m}{{\left (a+b\,\sqrt {x}\right )}^2} \,d x \]

[In]

int(x^m/(a + b*x^(1/2))^2,x)

[Out]

int(x^m/(a + b*x^(1/2))^2, x)