Optimal. Leaf size=410 \[ \frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h k n t}-\frac {p r \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{h k}+\frac {2 n p r t \text {Li}_3\left (\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{3 h k n t}-\frac {2 n^2 p r t^2 \text {Li}_4\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{h k}+\frac {2 n q r t \text {Li}_3\left (\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{3 h k n t}-\frac {2 n^2 q r t^2 \text {Li}_4\left (\frac {d (g+h x)}{d g-c h}\right )}{h k} \]
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Rubi [A] time = 0.47, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2499, 2396, 2433, 2374, 2383, 6589} \[ -\frac {p r \text {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{h k}+\frac {2 n p r t \text {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac {2 n^2 p r t^2 \text {PolyLog}\left (4,\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \text {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{h k}+\frac {2 n q r t \text {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac {2 n^2 q r t^2 \text {PolyLog}\left (4,\frac {d (g+h x)}{d g-c h}\right )}{h k}+\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h k n t}-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{3 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{3 h k n t} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2383
Rule 2396
Rule 2433
Rule 2499
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^2}{g k+h k x} \, dx &=\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {(b p r) \int \frac {\left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{a+b x} \, dx}{3 h k n t}-\frac {(d q r) \int \frac {\left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{c+d x} \, dx}{3 h k n t}\\ &=-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac {(p r) \int \frac {\log \left (\frac {h (a+b x)}{-b g+a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^2}{g+h x} \, dx}{k}+\frac {(q r) \int \frac {\log \left (\frac {h (c+d x)}{-d g+c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^2}{g+h x} \, dx}{k}\\ &=-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac {(p r) \operatorname {Subst}\left (\int \frac {\left (s+t \log \left (51 x^n\right )\right )^2 \log \left (\frac {h \left (\frac {-b g+a h}{h}+\frac {b x}{h}\right )}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h k}+\frac {(q r) \operatorname {Subst}\left (\int \frac {\left (s+t \log \left (51 x^n\right )\right )^2 \log \left (\frac {h \left (\frac {-d g+c h}{h}+\frac {d x}{h}\right )}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h k}\\ &=-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {p r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}+\frac {(2 n p r t) \operatorname {Subst}\left (\int \frac {\left (s+t \log \left (51 x^n\right )\right ) \text {Li}_2\left (-\frac {b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h k}+\frac {(2 n q r t) \operatorname {Subst}\left (\int \frac {\left (s+t \log \left (51 x^n\right )\right ) \text {Li}_2\left (-\frac {d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h k}\\ &=-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {p r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}+\frac {2 n p r t \left (s+t \log \left (51 (g+h x)^n\right )\right ) \text {Li}_3\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}+\frac {2 n q r t \left (s+t \log \left (51 (g+h x)^n\right )\right ) \text {Li}_3\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}-\frac {\left (2 n^2 p r t^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h k}-\frac {\left (2 n^2 q r t^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h k}\\ &=-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {p r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}+\frac {2 n p r t \left (s+t \log \left (51 (g+h x)^n\right )\right ) \text {Li}_3\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}+\frac {2 n q r t \left (s+t \log \left (51 (g+h x)^n\right )\right ) \text {Li}_3\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}-\frac {2 n^2 p r t^2 \text {Li}_4\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {2 n^2 q r t^2 \text {Li}_4\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}\\ \end {align*}
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Mathematica [B] time = 7.49, size = 22595, normalized size = 55.11 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (t^{2} \log \left ({\left (h x + g\right )}^{n} i\right )^{2} + 2 \, s t \log \left ({\left (h x + g\right )}^{n} i\right ) + s^{2}\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k x + g k}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (t \log \left ({\left (h x + g\right )}^{n} i\right ) + s\right )}^{2} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k x + g k}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {\left (t \ln \left (i \left (h x +g \right )^{n}\right )+s \right )^{2} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{h k x +g k}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,{\left (s+t\,\ln \left (i\,{\left (g+h\,x\right )}^n\right )\right )}^2}{g\,k+h\,k\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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