Optimal. Leaf size=184 \[ -\frac {6 B^3 n^3 (c+d x)}{(b c-a d) (a+b x)}-\frac {6 B^2 n^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d) (a+b x)}-\frac {3 B n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d) (a+b x)}-\frac {(c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d) (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2573, 2549,
2342, 2341} \begin {gather*} -\frac {6 B^2 n^2 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x) (b c-a d)}-\frac {3 B n (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x) (b c-a d)}-\frac {(c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x) (b c-a d)}-\frac {6 B^3 n^3 (c+d x)}{(a+b x) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2341
Rule 2342
Rule 2549
Rule 2573
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^2} \, dx &=\int \left (\frac {A^3}{(a+b x)^2}+\frac {3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2}+\frac {3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2}+\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2}\right ) \, dx\\ &=-\frac {A^3}{b (a+b x)}+\left (3 A^2 B\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx+\left (3 A B^2\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx+B^3 \int \frac {\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx\\ &=-\frac {A^3}{b (a+b x)}-\frac {3 A^2 B (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {3 A B^2 (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {B^3 (c+d x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}+\left (3 A^2 B n\right ) \int \frac {1}{(a+b x)^2} \, dx+\left (6 A B^2 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx+\left (3 B^3 n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx\\ &=-\frac {A^3}{b (a+b x)}-\frac {3 A^2 B n}{b (a+b x)}-\frac {3 A^2 B (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {6 A B^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {3 A B^2 (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {3 B^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {B^3 (c+d x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}+\left (6 A B^2 n^2\right ) \int \frac {1}{(a+b x)^2} \, dx+\left (6 B^3 n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx\\ &=-\frac {A^3}{b (a+b x)}-\frac {3 A^2 B n}{b (a+b x)}-\frac {6 A B^2 n^2}{b (a+b x)}-\frac {3 A^2 B (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {6 A B^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {6 B^3 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {3 A B^2 (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {3 B^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {B^3 (c+d x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}+\left (6 B^3 n^3\right ) \int \frac {1}{(a+b x)^2} \, dx\\ &=-\frac {A^3}{b (a+b x)}-\frac {3 A^2 B n}{b (a+b x)}-\frac {6 A B^2 n^2}{b (a+b x)}-\frac {6 B^3 n^3}{b (a+b x)}-\frac {3 A^2 B (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {6 A B^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {6 B^3 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {3 A B^2 (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {3 B^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {B^3 (c+d x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(524\) vs. \(2(184)=368\).
time = 0.47, size = 524, normalized size = 2.85 \begin {gather*} \frac {-B^3 d n^3 (a+b x) \log ^3(a+b x)+B^3 d n^3 (a+b x) \log ^3(c+d x)+3 B^2 d n^2 (a+b x) \log ^2(c+d x) \left (A+B n+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+3 B^2 d n^2 (a+b x) \log ^2(a+b x) \left (A+B n+B n \log (c+d x)+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+3 B d n (a+b x) \log (c+d x) \left (A^2+2 A B n+2 B^2 n^2+2 B (A+B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-(b c-a d) \left (A^3+3 A^2 B n+6 A B^2 n^2+6 B^3 n^3+3 B \left (A^2+2 A B n+2 B^2 n^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 B^2 (A+B n) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-3 B d n (a+b x) \log (a+b x) \left (A^2+2 A B n+2 B^2 n^2+B^2 n^2 \log ^2(c+d x)+2 B (A+B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+2 B n \log (c+d x) \left (A+B n+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )}{b (b c-a d) (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 14.08, size = 69354, normalized size = 376.92
method | result | size |
risch | \(\text {Expression too large to display}\) | \(69354\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1092 vs.
\(2 (187) = 374\).
time = 0.36, size = 1092, normalized size = 5.93 \begin {gather*} -3 \, {\left (\frac {d n e \log \left (b x + a\right )}{b^{2} c - a b d} - \frac {d n e \log \left (d x + c\right )}{b^{2} c - a b d} + \frac {n e}{b^{2} x + a b}\right )} A^{2} B e^{\left (-1\right )} - \frac {B^{3} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3}}{b^{2} x + a b} - 3 \, {\left (2 \, {\left (\frac {d n e \log \left (b x + a\right )}{b^{2} c - a b d} - \frac {d n e \log \left (d x + c\right )}{b^{2} c - a b d} + \frac {n e}{b^{2} x + a b}\right )} e^{\left (-1\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) - \frac {{\left ({\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right )^{2} + {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (d x + c\right )^{2} - 2 \, {\left (b c n^{2} - a d n^{2}\right )} e^{2} - 2 \, {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (b d n^{2} x e^{2} + a d n^{2} e^{2} - {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} e^{\left (-2\right )}}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x}\right )} A B^{2} - {\left (3 \, {\left (\frac {d n e \log \left (b x + a\right )}{b^{2} c - a b d} - \frac {d n e \log \left (d x + c\right )}{b^{2} c - a b d} + \frac {n e}{b^{2} x + a b}\right )} e^{\left (-1\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} - {\left (\frac {3 \, {\left ({\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right )^{2} + {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (d x + c\right )^{2} - 2 \, {\left (b c n^{2} - a d n^{2}\right )} e^{2} - 2 \, {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (b d n^{2} x e^{2} + a d n^{2} e^{2} - {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} e^{\left (-1\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x} - \frac {{\left ({\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right )^{3} - {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (d x + c\right )^{3} - 3 \, {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right )^{2} - 3 \, {\left (b d n^{3} x e^{3} + a d n^{3} e^{3} - {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} + 6 \, {\left (b c n^{3} - a d n^{3}\right )} e^{3} + 6 \, {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, b d n^{3} x e^{3} + 2 \, a d n^{3} e^{3} + {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right )^{2} - 2 \, {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} e^{\left (-2\right )}}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x}\right )} e^{\left (-1\right )}\right )} B^{3} - \frac {3 \, A B^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2}}{b^{2} x + a b} - \frac {3 \, A^{2} B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{b^{2} x + a b} - \frac {A^{3}}{b^{2} x + a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 673 vs.
\(2 (187) = 374\).
time = 0.42, size = 673, normalized size = 3.66 \begin {gather*} -\frac {6 \, {\left (B^{3} b c - B^{3} a d\right )} n^{3} + {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (b x + a\right )^{3} - {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (d x + c\right )^{3} + {\left (A^{3} + 3 \, A^{2} B + 3 \, A B^{2} + B^{3}\right )} b c - {\left (A^{3} + 3 \, A^{2} B + 3 \, A B^{2} + B^{3}\right )} a d + 6 \, {\left ({\left (A B^{2} + B^{3}\right )} b c - {\left (A B^{2} + B^{3}\right )} a d\right )} n^{2} + 3 \, {\left (B^{3} b c n^{3} + {\left (A B^{2} + B^{3}\right )} b c n^{2} + {\left (B^{3} b d n^{3} + {\left (A B^{2} + B^{3}\right )} b d n^{2}\right )} x\right )} \log \left (b x + a\right )^{2} + 3 \, {\left (B^{3} b c n^{3} + {\left (A B^{2} + B^{3}\right )} b c n^{2} + {\left (B^{3} b d n^{3} + {\left (A B^{2} + B^{3}\right )} b d n^{2}\right )} x + {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} + 3 \, {\left ({\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} b c - {\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} a d\right )} n + 3 \, {\left (2 \, B^{3} b c n^{3} + 2 \, {\left (A B^{2} + B^{3}\right )} b c n^{2} + {\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} b c n + {\left (2 \, B^{3} b d n^{3} + 2 \, {\left (A B^{2} + B^{3}\right )} b d n^{2} + {\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} b d n\right )} x\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, B^{3} b c n^{3} + 2 \, {\left (A B^{2} + B^{3}\right )} b c n^{2} + {\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} b c n + {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (b x + a\right )^{2} + {\left (2 \, B^{3} b d n^{3} + 2 \, {\left (A B^{2} + B^{3}\right )} b d n^{2} + {\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} b d n\right )} x + 2 \, {\left (B^{3} b c n^{3} + {\left (A B^{2} + B^{3}\right )} b c n^{2} + {\left (B^{3} b d n^{3} + {\left (A B^{2} + B^{3}\right )} b d n^{2}\right )} x\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[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 [B]
time = 6.06, size = 474, normalized size = 2.58 \begin {gather*} -\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (\frac {3\,B\,b\,d\,A^2\,x^2+3\,B\,\left (a\,d+b\,c\right )\,A^2\,x+3\,B\,a\,c\,A^2}{b\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}+\frac {6\,d\,\left (n\,B^3+A\,B^2\right )\,\left (b^2\,n\,x^2\,\left (a\,d-b\,c\right )+\frac {a\,b\,c\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b\,n\,x\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{d}\right )}{b^2\,\left (a\,d-b\,c\right )\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}\right )-\frac {A^3+3\,A^2\,B\,n+6\,A\,B^2\,n^2+6\,B^3\,n^3}{x\,b^2+a\,b}-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2\,\left (\frac {3\,A\,B^2}{x\,b^2+a\,b}+\frac {3\,B^3\,n}{x\,b^2+a\,b}-\frac {3\,d\,\left (n\,B^3+A\,B^2\right )}{b\,\left (a\,d-b\,c\right )}\right )-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^3\,\left (\frac {B^3}{b\,\left (a+b\,x\right )}-\frac {B^3\,d}{b\,\left (a\,d-b\,c\right )}\right )-\frac {B\,d\,n\,\mathrm {atan}\left (\frac {B\,d\,n\,\left (\frac {c\,b^2+a\,d\,b}{b}+2\,b\,d\,x\right )\,\left (A^2+2\,A\,B\,n+2\,B^2\,n^2\right )\,3{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (3\,d\,A^2\,B\,n+6\,d\,A\,B^2\,n^2+6\,d\,B^3\,n^3\right )}\right )\,\left (A^2+2\,A\,B\,n+2\,B^2\,n^2\right )\,6{}\mathrm {i}}{b\,\left (a\,d-b\,c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________