3.10.76 \(\int \frac {(a+b x)^n (c+d x)^{-n}}{x^3} \, dx\) [976]

Optimal. Leaf size=117 \[ -\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 a c x^2}-\frac {(b c-a d) (a d (1+n)+b (c-c n)) (a+b x)^{1+n} (c+d x)^{-1-n} \, _2F_1\left (2,1+n;2+n;\frac {c (a+b x)}{a (c+d x)}\right )}{2 a^3 c (1+n)} \]

[Out]

-1/2*(b*x+a)^(1+n)*(d*x+c)^(1-n)/a/c/x^2-1/2*(-a*d+b*c)*(a*d*(1+n)+b*(-c*n+c))*(b*x+a)^(1+n)*(d*x+c)^(-1-n)*hy
pergeom([2, 1+n],[2+n],c*(b*x+a)/a/(d*x+c))/a^3/c/(1+n)

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Rubi [A]
time = 0.03, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {98, 133} \begin {gather*} -\frac {(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} (a d (n+1)+b (c-c n)) \, _2F_1\left (2,n+1;n+2;\frac {c (a+b x)}{a (c+d x)}\right )}{2 a^3 c (n+1)}-\frac {(a+b x)^{n+1} (c+d x)^{1-n}}{2 a c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n/(x^3*(c + d*x)^n),x]

[Out]

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

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 133

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*c - a
*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 99, normalized size = 0.85 \begin {gather*} \frac {(a+b x)^{1+n} (c+d x)^{-1-n} \left (-\frac {a^2 (c+d x)^2}{x^2}+\frac {(b c-a d) (b c (-1+n)-a d (1+n)) \, _2F_1\left (2,1+n;2+n;\frac {c (a+b x)}{a (c+d x)}\right )}{1+n}\right )}{2 a^3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n/(x^3*(c + d*x)^n),x]

[Out]

((a + b*x)^(1 + n)*(c + d*x)^(-1 - n)*(-((a^2*(c + d*x)^2)/x^2) + ((b*c - a*d)*(b*c*(-1 + n) - a*d*(1 + n))*Hy
pergeometric2F1[2, 1 + n, 2 + n, (c*(a + b*x))/(a*(c + d*x))])/(1 + n)))/(2*a^3*c)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n} \left (d x +c \right )^{-n}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^n/x^3/((d*x+c)^n),x)

[Out]

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

[Out]

integrate((b*x + a)^n/((d*x + c)^n*x^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((b*x+a)^n/x^3/((d*x+c)^n),x, algorithm="fricas")

[Out]

integral((b*x + a)^n/((d*x + c)^n*x^3), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**n/x**3/((d*x+c)**n),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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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((b*x+a)^n/x^3/((d*x+c)^n),x, algorithm="giac")

[Out]

integrate((b*x + a)^n/((d*x + c)^n*x^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^n}{x^3\,{\left (c+d\,x\right )}^n} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^n/(x^3*(c + d*x)^n),x)

[Out]

int((a + b*x)^n/(x^3*(c + d*x)^n), x)

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