3.950 $$\int \frac{(a+b x)^m}{(a^2-b^2 x^2)^3} \, dx$$

Optimal. Leaf size=46 $-\frac{(a+b x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac{a+b x}{2 a}\right )}{8 a^3 b (2-m)}$

[Out]

-((a + b*x)^(-2 + m)*Hypergeometric2F1[3, -2 + m, -1 + m, (a + b*x)/(2*a)])/(8*a^3*b*(2 - m))

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Rubi [A]  time = 0.0151704, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {627, 68} $-\frac{(a+b x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac{a+b x}{2 a}\right )}{8 a^3 b (2-m)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^m/(a^2 - b^2*x^2)^3,x]

[Out]

-((a + b*x)^(-2 + m)*Hypergeometric2F1[3, -2 + m, -1 + m, (a + b*x)/(2*a)])/(8*a^3*b*(2 - m))

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{(a+b x)^m}{\left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac{(a+b x)^{-3+m}}{(a-b x)^3} \, dx\\ &=-\frac{(a+b x)^{-2+m} \, _2F_1\left (3,-2+m;-1+m;\frac{a+b x}{2 a}\right )}{8 a^3 b (2-m)}\\ \end{align*}

Mathematica [B]  time = 0.185065, size = 153, normalized size = 3.33 $\frac{(a+b x)^m \left (\frac{8 a^3}{(m-2) (a+b x)^2}+\frac{12 a^2}{(m-1) (a+b x)}+\frac{6 (a+b x) \, _2F_1\left (1,m+1;m+2;\frac{a+b x}{2 a}\right )}{m+1}+\frac{3 (a+b x) \, _2F_1\left (2,m+1;m+2;\frac{a+b x}{2 a}\right )}{m+1}+\frac{(a+b x) \, _2F_1\left (3,m+1;m+2;\frac{a+b x}{2 a}\right )}{m+1}+\frac{12 a}{m}\right )}{64 a^6 b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a + b*x)^m*((12*a)/m + (8*a^3)/((-2 + m)*(a + b*x)^2) + (12*a^2)/((-1 + m)*(a + b*x)) + (6*(a + b*x)*Hyperge
ometric2F1[1, 1 + m, 2 + m, (a + b*x)/(2*a)])/(1 + m) + (3*(a + b*x)*Hypergeometric2F1[2, 1 + m, 2 + m, (a + b
*x)/(2*a)])/(1 + m) + ((a + b*x)*Hypergeometric2F1[3, 1 + m, 2 + m, (a + b*x)/(2*a)])/(1 + m)))/(64*a^6*b)

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Maple [F]  time = 0.536, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{3}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (b x + a\right )}^{m}}{{\left (b^{2} x^{2} - a^{2}\right )}^{3}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (b x + a\right )}^{m}}{b^{6} x^{6} - 3 \, a^{2} b^{4} x^{4} + 3 \, a^{4} b^{2} x^{2} - a^{6}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(b*x + a)^m/(b^6*x^6 - 3*a^2*b^4*x^4 + 3*a^4*b^2*x^2 - a^6), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\left (a + b x\right )^{m}}{- a^{6} + 3 a^{4} b^{2} x^{2} - 3 a^{2} b^{4} x^{4} + b^{6} x^{6}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral((a + b*x)**m/(-a**6 + 3*a**4*b**2*x**2 - 3*a**2*b**4*x**4 + b**6*x**6), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b x + a\right )}^{m}}{{\left (b^{2} x^{2} - a^{2}\right )}^{3}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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