Optimal. Leaf size=230 \[ -\frac {B d i^2 (c+d x)}{b^2 g^3 (a+b x)}-\frac {B i^2 (c+d x)^2}{4 b g^3 (a+b x)^2}-\frac {d i^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {i^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}-\frac {d^2 i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3}+\frac {B d^2 i^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.21, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2562, 2380,
2341, 2379, 2438} \begin {gather*} \frac {B d^2 i^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3}-\frac {d^2 i^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^3}-\frac {d i^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (a+b x)}-\frac {i^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b g^3 (a+b x)^2}-\frac {B d i^2 (c+d x)}{b^2 g^3 (a+b x)}-\frac {B i^2 (c+d x)^2}{4 b g^3 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2341
Rule 2379
Rule 2380
Rule 2438
Rule 2562
Rubi steps
\begin {align*} \int \frac {(16 c+16 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac {256 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)^3}+\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)^2}+\frac {256 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}\right ) \, dx\\ &=\frac {\left (256 d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^3}+\frac {(512 d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 g^3}+\frac {\left (256 (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^3}\\ &=-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}-\frac {\left (256 B d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^3 g^3}+\frac {(512 B d (b c-a d)) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac {\left (128 B (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^3}\\ &=-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {\left (512 B d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac {\left (128 B (b c-a d)^3\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^3}-\frac {\left (256 B d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 e g^3}\\ &=-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {\left (512 B d (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^3}+\frac {\left (128 B (b c-a d)^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^3}-\frac {\left (256 B d^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 e g^3}\\ &=-\frac {64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {384 B d^2 \log (a+b x)}{b^3 g^3}-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {384 B d^2 \log (c+d x)}{b^3 g^3}-\frac {\left (256 B d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g^3}+\frac {\left (256 B d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g^3}\\ &=-\frac {64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {384 B d^2 \log (a+b x)}{b^3 g^3}-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {384 B d^2 \log (c+d x)}{b^3 g^3}+\frac {256 B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac {\left (256 B d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^3}-\frac {\left (256 B d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^3}\\ &=-\frac {64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {384 B d^2 \log (a+b x)}{b^3 g^3}-\frac {128 B d^2 \log ^2(a+b x)}{b^3 g^3}-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {384 B d^2 \log (c+d x)}{b^3 g^3}+\frac {256 B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac {\left (256 B d^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^3}\\ &=-\frac {64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {384 B d^2 \log (a+b x)}{b^3 g^3}-\frac {128 B d^2 \log ^2(a+b x)}{b^3 g^3}-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {384 B d^2 \log (c+d x)}{b^3 g^3}+\frac {256 B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}+\frac {256 B d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.21, size = 244, normalized size = 1.06 \begin {gather*} \frac {i^2 \left (-\frac {B (b c-a d)^2}{(a+b x)^2}+\frac {6 B d (-b c+a d)}{a+b x}-6 B d^2 \log (a+b x)-\frac {2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}+\frac {8 d (-b c+a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}+4 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 B d^2 \log (c+d x)-2 B d^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{4 b^3 g^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(739\) vs.
\(2(226)=452\).
time = 1.41, size = 740, normalized size = 3.22
method | result | size |
derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (\frac {i^{2} d^{4} A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e \left (a d -c b \right ) g^{3} b^{3}}+\frac {i^{2} d^{2} e A}{2 \left (a d -c b \right ) g^{3} b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {i^{2} d^{4} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \left (a d -c b \right ) g^{3} b^{3}}+\frac {i^{2} d^{3} A}{\left (a d -c b \right ) g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {i^{2} d^{4} B \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e \left (a d -c b \right ) g^{3} b^{3}}+\frac {i^{2} d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e \left (a d -c b \right ) g^{3} b^{3}}+\frac {i^{2} d^{2} e B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (a d -c b \right ) g^{3} b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {i^{2} d^{2} e B}{4 \left (a d -c b \right ) g^{3} b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {i^{2} d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \left (a d -c b \right ) g^{3} b^{3}}+\frac {i^{2} d^{3} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\left (a d -c b \right ) g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {i^{2} d^{3} B}{\left (a d -c b \right ) g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}\right )}{d^{2}}\) | \(740\) |
default | \(-\frac {e \left (a d -c b \right ) \left (\frac {i^{2} d^{4} A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e \left (a d -c b \right ) g^{3} b^{3}}+\frac {i^{2} d^{2} e A}{2 \left (a d -c b \right ) g^{3} b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {i^{2} d^{4} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \left (a d -c b \right ) g^{3} b^{3}}+\frac {i^{2} d^{3} A}{\left (a d -c b \right ) g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {i^{2} d^{4} B \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e \left (a d -c b \right ) g^{3} b^{3}}+\frac {i^{2} d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e \left (a d -c b \right ) g^{3} b^{3}}+\frac {i^{2} d^{2} e B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (a d -c b \right ) g^{3} b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {i^{2} d^{2} e B}{4 \left (a d -c b \right ) g^{3} b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {i^{2} d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \left (a d -c b \right ) g^{3} b^{3}}+\frac {i^{2} d^{3} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\left (a d -c b \right ) g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {i^{2} d^{3} B}{\left (a d -c b \right ) g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}\right )}{d^{2}}\) | \(740\) |
risch | \(\text {Expression too large to display}\) | \(2261\) |
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 = -0.00 \begin {gather*} \int \frac {{\left (c\,i+d\,i\,x\right )}^2\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{{\left (a\,g+b\,g\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________