3.8.5 \(\int \frac {x^m}{(a+b x)^2} \, dx\) [705]

Optimal. Leaf size=29 \[ \frac {x^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {b x}{a}\right )}{a^2 (1+m)} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {66} \begin {gather*} \frac {x^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {b x}{a}\right )}{a^2 (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m/(a + b*x)^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 29, normalized size = 1.00 \begin {gather*} \frac {x^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {b x}{a}\right )}{a^2 (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m/(a + b*x)^2,x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 4.01, size = 57, normalized size = 1.97 \begin {gather*} \frac {\left (a-a m \text {LerchPhi}\left [\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a},1,1+m\right ]-b m x \text {LerchPhi}\left [\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a},1,1+m\right ]\right ) x^{1+m}}{a^2 \left (a+b x\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^m/(a + b*x)^2,x]')

[Out]

(a - a m LerchPhi[b x exp_polar[I Pi] / a, 1, 1 + m] - b m x LerchPhi[b x exp_polar[I Pi] / a, 1, 1 + m]) x ^
(1 + m) / (a ^ 2 (a + b x))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (b x +a \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/(b*x + a)^2, x)

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Fricas [F]
time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 262, normalized size = 9.03 \begin {gather*} - \frac {a m^{2} x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {a m x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} + \frac {a m x x^{m} \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} + \frac {a x x^{m} \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {b m^{2} x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {b m x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*m**2*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m +
2)) - a*m*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m
+ 2)) + a*m*x*x**m*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2)) + a*x*x**m*gamma(m + 1)/(a**3*gamm
a(m + 2) + a**2*b*x*gamma(m + 2)) - b*m**2*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a
**3*gamma(m + 2) + a**2*b*x*gamma(m + 2)) - b*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m +
1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2))

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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^m}{{\left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(a + b*x)^2,x)

[Out]

int(x^m/(a + b*x)^2, x)

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