3.8.6 \(\int \frac {x^m}{(a+b x)^3} \, dx\) [706]

Optimal. Leaf size=29 \[ \frac {x^{1+m} \, _2F_1\left (3,1+m;2+m;-\frac {b x}{a}\right )}{a^3 (1+m)} \]

[Out]

x^(1+m)*hypergeom([3, 1+m],[2+m],-b*x/a)/a^3/(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 (3,m+1;m+2;-\frac {b x}{a}\right )}{a^3 (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m/(a + b*x)^3,x]

[Out]

(x^(1 + m)*Hypergeometric2F1[3, 1 + m, 2 + m, -((b*x)/a)])/(a^3*(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)^3} \, dx &=\frac {x^{1+m} \, _2F_1\left (3,1+m;2+m;-\frac {b x}{a}\right )}{a^3 (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 (3,1+m;2+m;-\frac {b x}{a}\right )}{a^3 (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m/(a + b*x)^3,x]

[Out]

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (b x +a \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m/(b*x+a)^3,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)^3,x, algorithm="maxima")

[Out]

integrate(x^m/(b*x + a)^3, x)

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Fricas [F]
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)^3,x, algorithm="fricas")

[Out]

integral(x^m/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3), x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.63, size = 717, normalized size = 24.72 \begin {gather*} \frac {a^{2} m^{3} x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{2 a^{5} \Gamma \left (m + 2\right ) + 4 a^{4} b x \Gamma \left (m + 2\right ) + 2 a^{3} b^{2} x^{2} \Gamma \left (m + 2\right )} - \frac {a^{2} m^{2} x x^{m} \Gamma \left (m + 1\right )}{2 a^{5} \Gamma \left (m + 2\right ) + 4 a^{4} b x \Gamma \left (m + 2\right ) + 2 a^{3} b^{2} x^{2} \Gamma \left (m + 2\right )} - \frac {a^{2} m x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{2 a^{5} \Gamma \left (m + 2\right ) + 4 a^{4} b x \Gamma \left (m + 2\right ) + 2 a^{3} b^{2} x^{2} \Gamma \left (m + 2\right )} + \frac {a^{2} m x x^{m} \Gamma \left (m + 1\right )}{2 a^{5} \Gamma \left (m + 2\right ) + 4 a^{4} b x \Gamma \left (m + 2\right ) + 2 a^{3} b^{2} x^{2} \Gamma \left (m + 2\right )} + \frac {2 a^{2} x x^{m} \Gamma \left (m + 1\right )}{2 a^{5} \Gamma \left (m + 2\right ) + 4 a^{4} b x \Gamma \left (m + 2\right ) + 2 a^{3} b^{2} x^{2} \Gamma \left (m + 2\right )} + \frac {2 a b m^{3} x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{2 a^{5} \Gamma \left (m + 2\right ) + 4 a^{4} b x \Gamma \left (m + 2\right ) + 2 a^{3} b^{2} x^{2} \Gamma \left (m + 2\right )} - \frac {a b m^{2} x^{2} x^{m} \Gamma \left (m + 1\right )}{2 a^{5} \Gamma \left (m + 2\right ) + 4 a^{4} b x \Gamma \left (m + 2\right ) + 2 a^{3} b^{2} x^{2} \Gamma \left (m + 2\right )} - \frac {2 a b m x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{2 a^{5} \Gamma \left (m + 2\right ) + 4 a^{4} b x \Gamma \left (m + 2\right ) + 2 a^{3} b^{2} x^{2} \Gamma \left (m + 2\right )} + \frac {a b x^{2} x^{m} \Gamma \left (m + 1\right )}{2 a^{5} \Gamma \left (m + 2\right ) + 4 a^{4} b x \Gamma \left (m + 2\right ) + 2 a^{3} b^{2} x^{2} \Gamma \left (m + 2\right )} + \frac {b^{2} m^{3} x^{3} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{2 a^{5} \Gamma \left (m + 2\right ) + 4 a^{4} b x \Gamma \left (m + 2\right ) + 2 a^{3} b^{2} x^{2} \Gamma \left (m + 2\right )} - \frac {b^{2} m x^{3} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{2 a^{5} \Gamma \left (m + 2\right ) + 4 a^{4} b x \Gamma \left (m + 2\right ) + 2 a^{3} b^{2} x^{2} \Gamma \left (m + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/(b*x+a)**3,x)

[Out]

a**2*m**3*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamm
a(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - a**2*m**2*x*x**m*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*ga
mma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - a**2*m*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m
+ 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + a**2*m*x*x**m*gamma(m +
1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + 2*a**2*x*x**m*gamma(m +
1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + 2*a*b*m**3*x**2*x**m*lerc
hphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**
2*x**2*gamma(m + 2)) - a*b*m**2*x**2*x**m*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3
*b**2*x**2*gamma(m + 2)) - 2*a*b*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*ga
mma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + a*b*x**2*x**m*gamma(m + 1)/(2*a**5*gam
ma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + b**2*m**3*x**3*x**m*lerchphi(b*x*exp_po
lar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m
+ 2)) - b**2*m*x**3*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*
b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2))

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Giac [F]
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)^3,x, algorithm="giac")

[Out]

integrate(x^m/(b*x + a)^3, x)

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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 )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(a + b*x)^3,x)

[Out]

int(x^m/(a + b*x)^3, x)

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