Optimal. Leaf size=140 \[ -\frac {B (b c-a d)^2 g i x}{6 b d}+\frac {g i (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b}+\frac {(b c-a d) g i (a+b x)^2 \left (A-B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b^2}+\frac {B (b c-a d)^3 g i \log (c+d x)}{6 b^2 d^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2560, 2548, 21,
45} \begin {gather*} \frac {g i (a+b x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{6 b^2}+\frac {g i (a+b x)^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b}+\frac {B g i (b c-a d)^3 \log (c+d x)}{6 b^2 d^2}-\frac {B g i x (b c-a d)^2}{6 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 2548
Rule 2560
Rubi steps
\begin {align*} \int (3 c+3 d x) (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (3 a c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+3 (b c+a d) g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+3 b d g x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \, dx\\ &=(3 a c g) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx+(3 b d g) \int x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx+(3 (b c+a d) g) \int x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx\\ &=3 a A c g x+\frac {3}{2} (b c+a d) g x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+b d g x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+(3 a B c g) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx-(b B d g) \int \frac {(b c-a d) x^3}{(a+b x) (c+d x)} \, dx-\frac {1}{2} (3 B (b c+a d) g) \int \frac {(b c-a d) x^2}{(a+b x) (c+d x)} \, dx\\ &=3 a A c g x+\frac {3 a B c g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b}+\frac {3}{2} (b c+a d) g x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+b d g x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-\frac {(3 a B c (b c-a d) g) \int \frac {1}{c+d x} \, dx}{b}-(b B d (b c-a d) g) \int \frac {x^3}{(a+b x) (c+d x)} \, dx-\frac {1}{2} (3 B (b c-a d) (b c+a d) g) \int \frac {x^2}{(a+b x) (c+d x)} \, dx\\ &=3 a A c g x+\frac {3 a B c g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b}+\frac {3}{2} (b c+a d) g x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+b d g x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-\frac {3 a B c (b c-a d) g \log (c+d x)}{b d}-(b B d (b c-a d) g) \int \left (\frac {-b c-a d}{b^2 d^2}+\frac {x}{b d}-\frac {a^3}{b^2 (b c-a d) (a+b x)}-\frac {c^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx-\frac {1}{2} (3 B (b c-a d) (b c+a d) g) \int \left (\frac {1}{b d}+\frac {a^2}{b (b c-a d) (a+b x)}+\frac {c^2}{d (-b c+a d) (c+d x)}\right ) \, dx\\ &=3 a A c g x-\frac {B (b c-a d) (b c+a d) g x}{2 b d}-\frac {1}{2} B (b c-a d) g x^2+\frac {a^3 B d g \log (a+b x)}{b^2}-\frac {3 a^2 B (b c+a d) g \log (a+b x)}{2 b^2}+\frac {3 a B c g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b}+\frac {3}{2} (b c+a d) g x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+b d g x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-\frac {b B c^3 g \log (c+d x)}{d^2}-\frac {3 a B c (b c-a d) g \log (c+d x)}{b d}+\frac {3 B c^2 (b c+a d) g \log (c+d x)}{2 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 181, normalized size = 1.29 \begin {gather*} \frac {g i \left (-a^2 B d^2 (3 b c+a d) \log (a+b x)+b \left (d x \left (a^2 B d^2-b^2 B c (c+d x)+A b^2 d x (3 c+2 d x)+a b d (6 A c+3 A d x+B d x)\right )+B d^2 \left (6 a^2 c+3 a b x (2 c+d x)+b^2 x^2 (3 c+2 d x)\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+B c \left (b^2 c^2-3 a b c d+6 a^2 d^2\right ) \log (c+d x)\right )\right )}{6 b^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2139\) vs.
\(2(132)=264\).
time = 0.47, size = 2140, normalized size = 15.29
method | result | size |
risch | \(\frac {g i B x \left (2 b d \,x^{2}+3 x a d +3 b c x +6 c a \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6}+\frac {i g b d A \,x^{3}}{3}+\frac {i g d A a \,x^{2}}{2}+\frac {i g b A c \,x^{2}}{2}+\frac {i g d B a \,x^{2}}{6}-\frac {i g b B c \,x^{2}}{6}+i g A a c x -\frac {i g d B \ln \left (b x +a \right ) a^{3}}{6 b^{2}}+\frac {i g B \ln \left (b x +a \right ) a^{2} c}{2 b}-\frac {i g B \ln \left (-d x -c \right ) a \,c^{2}}{2 d}+\frac {i g b B \ln \left (-d x -c \right ) c^{3}}{6 d^{2}}+\frac {i g d B \,a^{2} x}{6 b}-\frac {i g b B \,c^{2} x}{6 d}\) | \(206\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2140\) |
default | \(\text {Expression too large to display}\) | \(2140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 363 vs. \(2 (130) = 260\).
time = 0.29, size = 363, normalized size = 2.59 \begin {gather*} \frac {1}{3} i \, A b d g x^{3} + \frac {1}{2} i \, A b c g x^{2} + \frac {1}{2} i \, A a d g x^{2} + i \, {\left (x \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a c g + \frac {1}{2} i \, {\left (x^{2} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \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 b c g + \frac {1}{2} i \, {\left (x^{2} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \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 a d g + \frac {1}{6} i \, {\left (2 \, x^{3} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \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 )} B b d g + i \, A a c g x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 230, normalized size = 1.64 \begin {gather*} \frac {2 i \, A b^{3} d^{3} g x^{3} + {\left ({\left (3 i \, A - i \, B\right )} b^{3} c d^{2} + {\left (3 i \, A + i \, B\right )} a b^{2} d^{3}\right )} g x^{2} + {\left (-i \, B b^{3} c^{2} d + 6 i \, A a b^{2} c d^{2} + i \, B a^{2} b d^{3}\right )} g x + {\left (3 i \, B a^{2} b c d^{2} - i \, B a^{3} d^{3}\right )} g \log \left (\frac {b x + a}{b}\right ) + {\left (i \, B b^{3} c^{3} - 3 i \, B a b^{2} c^{2} d\right )} g \log \left (\frac {d x + c}{d}\right ) + {\left (2 i \, B b^{3} d^{3} g x^{3} + 6 i \, B a b^{2} c d^{2} g x - 3 \, {\left (-i \, B b^{3} c d^{2} - i \, B a b^{2} d^{3}\right )} g x^{2}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{6 \, b^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 498 vs.
\(2 (126) = 252\).
time = 1.66, size = 498, normalized size = 3.56 \begin {gather*} \frac {A b d g i x^{3}}{3} - \frac {B a^{2} g i \left (a d - 3 b c\right ) \log {\left (x + \frac {B a^{3} c d^{2} g i + \frac {B a^{3} d^{2} g i \left (a d - 3 b c\right )}{b} - 6 B a^{2} b c^{2} d g i - B a^{2} c d g i \left (a d - 3 b c\right ) + B a b^{2} c^{3} g i}{B a^{3} d^{3} g i - 3 B a^{2} b c d^{2} g i - 3 B a b^{2} c^{2} d g i + B b^{3} c^{3} g i} \right )}}{6 b^{2}} - \frac {B c^{2} g i \left (3 a d - b c\right ) \log {\left (x + \frac {B a^{3} c d^{2} g i - 6 B a^{2} b c^{2} d g i + B a b^{2} c^{3} g i + B a b c^{2} g i \left (3 a d - b c\right ) - \frac {B b^{2} c^{3} g i \left (3 a d - b c\right )}{d}}{B a^{3} d^{3} g i - 3 B a^{2} b c d^{2} g i - 3 B a b^{2} c^{2} d g i + B b^{3} c^{3} g i} \right )}}{6 d^{2}} + x^{2} \left (\frac {A a d g i}{2} + \frac {A b c g i}{2} + \frac {B a d g i}{6} - \frac {B b c g i}{6}\right ) + x \left (A a c g i + \frac {B a^{2} d g i}{6 b} - \frac {B b c^{2} g i}{6 d}\right ) + \left (B a c g i x + \frac {B a d g i x^{2}}{2} + \frac {B b c g i x^{2}}{2} + \frac {B b d g i x^{3}}{3}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 2362 vs. \(2 (130) = 260\).
time = 4.10, size = 2362, normalized size = 16.87 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.71, size = 282, normalized size = 2.01 \begin {gather*} x^2\,\left (\frac {g\,i\,\left (6\,A\,a\,d+6\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{6}-\frac {A\,g\,i\,\left (6\,a\,d+6\,b\,c\right )}{12}\right )-x\,\left (\frac {\left (\frac {g\,i\,\left (6\,A\,a\,d+6\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{3}-\frac {A\,g\,i\,\left (6\,a\,d+6\,b\,c\right )}{6}\right )\,\left (6\,a\,d+6\,b\,c\right )}{6\,b\,d}+A\,a\,c\,g\,i-\frac {g\,i\,\left (2\,A\,a^2\,d^2+2\,A\,b^2\,c^2+B\,a^2\,d^2-B\,b^2\,c^2+8\,A\,a\,b\,c\,d\right )}{2\,b\,d}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,b\,d\,g\,i\,x^3}{3}+\frac {B\,g\,i\,\left (a\,d+b\,c\right )\,x^2}{2}+B\,a\,c\,g\,i\,x\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^3\,d\,g\,i-3\,B\,a^2\,b\,c\,g\,i\right )}{6\,b^2}+\frac {\ln \left (c+d\,x\right )\,\left (B\,b\,c^3\,g\,i-3\,B\,a\,c^2\,d\,g\,i\right )}{6\,d^2}+\frac {A\,b\,d\,g\,i\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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