Optimal. Leaf size=101 \[ -\frac {(a-b x)^{1-n} (a+b x)^{2+n}}{3 a^2 x^3}-\frac {4 b^3 (1+2 n) (a-b x)^{1-n} (a+b x)^{-1+n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{3 a^2 (1-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {98, 133}
\begin {gather*} -\frac {4 b^3 (2 n+1) (a+b x)^{n-1} (a-b x)^{1-n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{3 a^2 (1-n)}-\frac {(a+b x)^{n+2} (a-b x)^{1-n}}{3 a^2 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 98
Rule 133
Rubi steps
\begin {align*} \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^4} \, dx &=-\frac {(a-b x)^{1-n} (a+b x)^{2+n}}{3 a^2 x^3}+\frac {(b (1+2 n)) \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx}{3 a}\\ &=-\frac {(a-b x)^{1-n} (a+b x)^{2+n}}{3 a^2 x^3}-\frac {4 b^3 (1+2 n) (a-b x)^{1-n} (a+b x)^{-1+n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{3 a^2 (1-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 88, normalized size = 0.87 \begin {gather*} \frac {(a-b x)^{1-n} (a+b x)^{-1+n} \left (-\left ((-1+n) (a+b x)^3\right )+4 b^3 (1+2 n) x^3 \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )\right )}{3 a^2 (-1+n) x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{1+n} \left (-b x +a \right )^{-n}}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{n+1}}{x^4\,{\left (a-b\,x\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________