Integrand size = 33, antiderivative size = 156 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {B (b c-a d)^3 i^3 n x}{4 b^3}-\frac {B (b c-a d)^2 i^3 n (c+d x)^2}{8 b^2 d}-\frac {B (b c-a d) i^3 n (c+d x)^3}{12 b d}-\frac {B (b c-a d)^4 i^3 n \log (a+b x)}{4 b^4 d}+\frac {i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d} \]
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Time = 0.07 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2547, 21, 45} \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {i^3 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d}-\frac {B i^3 n (b c-a d)^4 \log (a+b x)}{4 b^4 d}-\frac {B i^3 n x (b c-a d)^3}{4 b^3}-\frac {B i^3 n (c+d x)^2 (b c-a d)^2}{8 b^2 d}-\frac {B i^3 n (c+d x)^3 (b c-a d)}{12 b d} \]
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Rule 21
Rule 45
Rule 2547
Rubi steps \begin{align*} \text {integral}& = \frac {i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d}-\frac {(B (b c-a d) n) \int \frac {(c i+d i x)^4}{(a+b x) (c+d x)} \, dx}{4 d i} \\ & = \frac {i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d}-\frac {\left (B (b c-a d) i^3 n\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{4 d} \\ & = \frac {i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d}-\frac {\left (B (b c-a d) i^3 n\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{4 d} \\ & = -\frac {B (b c-a d)^3 i^3 n x}{4 b^3}-\frac {B (b c-a d)^2 i^3 n (c+d x)^2}{8 b^2 d}-\frac {B (b c-a d) i^3 n (c+d x)^3}{12 b d}-\frac {B (b c-a d)^4 i^3 n \log (a+b x)}{4 b^4 d}+\frac {i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.79 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {i^3 \left (-\frac {B (b c-a d) n \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )}{6 b^4}+(c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{4 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(651\) vs. \(2(146)=292\).
Time = 4.81 (sec) , antiderivative size = 652, normalized size of antiderivative = 4.18
method | result | size |
parallelrisch | \(\frac {24 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c \,d^{3} i^{3} n +6 A \,x^{4} b^{4} d^{4} i^{3} n +6 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{4} i^{3} n -6 B \ln \left (b x +a \right ) a^{4} d^{4} i^{3} n^{2}-6 B \ln \left (b x +a \right ) b^{4} c^{4} i^{3} n^{2}+36 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{2} d^{2} i^{3} n +12 B \,x^{2} a \,b^{3} c \,d^{3} i^{3} n^{2}+24 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{3} d \,i^{3} n -24 B x \,a^{2} b^{2} c \,d^{3} i^{3} n^{2}+36 B x a \,b^{3} c^{2} d^{2} i^{3} n^{2}+24 B \ln \left (b x +a \right ) a^{3} b c \,d^{3} i^{3} n^{2}-36 B \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{2} i^{3} n^{2}+24 B \ln \left (b x +a \right ) a \,b^{3} c^{3} d \,i^{3} n^{2}+21 B \,a^{3} b c \,d^{3} i^{3} n^{2}-24 B \,a^{2} b^{2} c^{2} d^{2} i^{3} n^{2}-9 B a \,b^{3} c^{3} d \,i^{3} n^{2}-60 A a \,b^{3} c^{3} d \,i^{3} n +6 B \,x^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} d^{4} i^{3} n +2 B \,x^{3} a \,b^{3} d^{4} i^{3} n^{2}-2 B \,x^{3} b^{4} c \,d^{3} i^{3} n^{2}+24 A \,x^{3} b^{4} c \,d^{3} i^{3} n -3 B \,x^{2} a^{2} b^{2} d^{4} i^{3} n^{2}-9 B \,x^{2} b^{4} c^{2} d^{2} i^{3} n^{2}+36 A \,x^{2} b^{4} c^{2} d^{2} i^{3} n +6 B x \,a^{3} b \,d^{4} i^{3} n^{2}-18 B x \,b^{4} c^{3} d \,i^{3} n^{2}+24 A x \,b^{4} c^{3} d \,i^{3} n -6 B \,a^{4} d^{4} i^{3} n^{2}+18 B \,b^{4} c^{4} i^{3} n^{2}-24 A \,b^{4} c^{4} i^{3} n}{24 b^{4} d n}\) | \(652\) |
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Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (146) = 292\).
Time = 0.38 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.75 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {6 \, A b^{4} d^{4} i^{3} x^{4} - 6 \, B b^{4} c^{4} i^{3} n \log \left (d x + c\right ) + 6 \, {\left (4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} i^{3} n \log \left (b x + a\right ) + 2 \, {\left (12 \, A b^{4} c d^{3} i^{3} - {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i^{3} n\right )} x^{3} + 3 \, {\left (12 \, A b^{4} c^{2} d^{2} i^{3} - {\left (3 \, B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} i^{3} n\right )} x^{2} + 6 \, {\left (4 \, A b^{4} c^{3} d i^{3} - {\left (3 \, B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 4 \, B a^{2} b^{2} c d^{3} - B a^{3} b d^{4}\right )} i^{3} n\right )} x + 6 \, {\left (B b^{4} d^{4} i^{3} x^{4} + 4 \, B b^{4} c d^{3} i^{3} x^{3} + 6 \, B b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \, B b^{4} c^{3} d i^{3} x\right )} \log \left (e\right ) + 6 \, {\left (B b^{4} d^{4} i^{3} n x^{4} + 4 \, B b^{4} c d^{3} i^{3} n x^{3} + 6 \, B b^{4} c^{2} d^{2} i^{3} n x^{2} + 4 \, B b^{4} c^{3} d i^{3} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{24 \, b^{4} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 945 vs. \(2 (136) = 272\).
Time = 78.68 (sec) , antiderivative size = 945, normalized size of antiderivative = 6.06 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\begin {cases} c^{3} i^{3} x \left (A + B \log {\left (e \left (\frac {a}{c}\right )^{n} \right )}\right ) & \text {for}\: b = 0 \wedge d = 0 \\A c^{3} i^{3} x + \frac {3 A c^{2} d i^{3} x^{2}}{2} + A c d^{2} i^{3} x^{3} + \frac {A d^{3} i^{3} x^{4}}{4} + \frac {B c^{4} i^{3} \log {\left (e \left (\frac {a}{c + d x}\right )^{n} \right )}}{4 d} + \frac {B c^{3} i^{3} n x}{4} + B c^{3} i^{3} x \log {\left (e \left (\frac {a}{c + d x}\right )^{n} \right )} + \frac {3 B c^{2} d i^{3} n x^{2}}{8} + \frac {3 B c^{2} d i^{3} x^{2} \log {\left (e \left (\frac {a}{c + d x}\right )^{n} \right )}}{2} + \frac {B c d^{2} i^{3} n x^{3}}{4} + B c d^{2} i^{3} x^{3} \log {\left (e \left (\frac {a}{c + d x}\right )^{n} \right )} + \frac {B d^{3} i^{3} n x^{4}}{16} + \frac {B d^{3} i^{3} x^{4} \log {\left (e \left (\frac {a}{c + d x}\right )^{n} \right )}}{4} & \text {for}\: b = 0 \\c^{3} i^{3} \left (A x + \frac {B a \log {\left (e \left (\frac {a}{c} + \frac {b x}{c}\right )^{n} \right )}}{b} - B n x + B x \log {\left (e \left (\frac {a}{c} + \frac {b x}{c}\right )^{n} \right )}\right ) & \text {for}\: d = 0 \\A c^{3} i^{3} x + \frac {3 A c^{2} d i^{3} x^{2}}{2} + A c d^{2} i^{3} x^{3} + \frac {A d^{3} i^{3} x^{4}}{4} - \frac {B a^{4} d^{3} i^{3} n \log {\left (\frac {c}{d} + x \right )}}{4 b^{4}} - \frac {B a^{4} d^{3} i^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{4 b^{4}} + \frac {B a^{3} c d^{2} i^{3} n \log {\left (\frac {c}{d} + x \right )}}{b^{3}} + \frac {B a^{3} c d^{2} i^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{b^{3}} + \frac {B a^{3} d^{3} i^{3} n x}{4 b^{3}} - \frac {3 B a^{2} c^{2} d i^{3} n \log {\left (\frac {c}{d} + x \right )}}{2 b^{2}} - \frac {3 B a^{2} c^{2} d i^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{2 b^{2}} - \frac {B a^{2} c d^{2} i^{3} n x}{b^{2}} - \frac {B a^{2} d^{3} i^{3} n x^{2}}{8 b^{2}} + \frac {B a c^{3} i^{3} n \log {\left (\frac {c}{d} + x \right )}}{b} + \frac {B a c^{3} i^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{b} + \frac {3 B a c^{2} d i^{3} n x}{2 b} + \frac {B a c d^{2} i^{3} n x^{2}}{2 b} + \frac {B a d^{3} i^{3} n x^{3}}{12 b} - \frac {B c^{4} i^{3} n \log {\left (\frac {c}{d} + x \right )}}{4 d} - \frac {3 B c^{3} i^{3} n x}{4} + B c^{3} i^{3} x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )} - \frac {3 B c^{2} d i^{3} n x^{2}}{8} + \frac {3 B c^{2} d i^{3} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{2} - \frac {B c d^{2} i^{3} n x^{3}}{12} + B c d^{2} i^{3} x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )} + \frac {B d^{3} i^{3} x^{4} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{4} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (146) = 292\).
Time = 0.21 (sec) , antiderivative size = 479, normalized size of antiderivative = 3.07 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {1}{4} \, B d^{3} i^{3} x^{4} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{4} \, A d^{3} i^{3} x^{4} + B c d^{2} i^{3} x^{3} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A c d^{2} i^{3} x^{3} + \frac {3}{2} \, B c^{2} d i^{3} x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {3}{2} \, A c^{2} d i^{3} x^{2} - \frac {1}{24} \, B d^{3} i^{3} n {\left (\frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} + \frac {1}{2} \, B c d^{2} i^{3} n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - \frac {3}{2} \, B c^{2} d i^{3} n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + B c^{3} i^{3} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B c^{3} i^{3} x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A c^{3} i^{3} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 1402 vs. \(2 (146) = 292\).
Time = 0.80 (sec) , antiderivative size = 1402, normalized size of antiderivative = 8.99 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]
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Time = 1.55 (sec) , antiderivative size = 588, normalized size of antiderivative = 3.77 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=x^3\,\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{12\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{12\,b}\right )-x^2\,\left (\frac {\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )\,\left (4\,a\,d+4\,b\,c\right )}{8\,b\,d}-\frac {c\,d\,i^3\,\left (4\,A\,a\,d+6\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{2\,b}+\frac {A\,a\,c\,d^2\,i^3}{2\,b}\right )+\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,c^3\,i^3\,x+\frac {3\,B\,c^2\,d\,i^3\,x^2}{2}+B\,c\,d^2\,i^3\,x^3+\frac {B\,d^3\,i^3\,x^4}{4}\right )+x\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}-\frac {c\,d\,i^3\,\left (4\,A\,a\,d+6\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{b}+\frac {A\,a\,c\,d^2\,i^3}{b}\right )}{4\,b\,d}+\frac {c^2\,i^3\,\left (12\,A\,a\,d+8\,A\,b\,c+3\,B\,a\,d\,n-3\,B\,b\,c\,n\right )}{2\,b}-\frac {a\,c\,\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )}{b\,d}\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,n\,a^4\,d^3\,i^3-4\,B\,n\,a^3\,b\,c\,d^2\,i^3+6\,B\,n\,a^2\,b^2\,c^2\,d\,i^3-4\,B\,n\,a\,b^3\,c^3\,i^3\right )}{4\,b^4}+\frac {A\,d^3\,i^3\,x^4}{4}-\frac {B\,c^4\,i^3\,n\,\ln \left (c+d\,x\right )}{4\,d} \]
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