Optimal. Leaf size=227 \[ -\frac {B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x}{4 b^3 d^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2}{8 b^2 d^2}-\frac {B (b c-a d) g^3 x^3}{12 b d}-\frac {B (b f-a g)^4 \log (a+b x)}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 g}+\frac {B (d f-c g)^4 \log (c+d x)}{4 d^4 g} \]
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Rubi [A]
time = 0.21, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2548, 84}
\begin {gather*} -\frac {B g x (b c-a d) \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )}{4 b^3 d^3}+\frac {(f+g x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 g}-\frac {B (b f-a g)^4 \log (a+b x)}{4 b^4 g}-\frac {B g^2 x^2 (b c-a d) (-a d g-b c g+4 b d f)}{8 b^2 d^2}-\frac {B g^3 x^3 (b c-a d)}{12 b d}+\frac {B (d f-c g)^4 \log (c+d x)}{4 d^4 g} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rule 2548
Rubi steps
\begin {align*} \int (f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx &=\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 g}-\frac {B \int \frac {(b c-a d) (f+g x)^4}{(a+b x) (c+d x)} \, dx}{4 g}\\ &=\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 g}-\frac {(B (b c-a d)) \int \frac {(f+g x)^4}{(a+b x) (c+d x)} \, dx}{4 g}\\ &=\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 g}-\frac {(B (b c-a d)) \int \left (\frac {g^2 \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right )}{b^3 d^3}+\frac {g^3 (4 b d f-b c g-a d g) x}{b^2 d^2}+\frac {g^4 x^2}{b d}+\frac {(b f-a g)^4}{b^3 (b c-a d) (a+b x)}+\frac {(d f-c g)^4}{d^3 (-b c+a d) (c+d x)}\right ) \, dx}{4 g}\\ &=-\frac {B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x}{4 b^3 d^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2}{8 b^2 d^2}-\frac {B (b c-a d) g^3 x^3}{12 b d}-\frac {B (b f-a g)^4 \log (a+b x)}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 g}+\frac {B (d f-c g)^4 \log (c+d x)}{4 d^4 g}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 215, normalized size = 0.95 \begin {gather*} \frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-\frac {B \left (6 b d (b c-a d) g^2 \left (a^2 d^2 g^2+a b d g (-4 d f+c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x+3 b^2 d^2 (b c-a d) g^3 (4 b d f-b c g-a d g) x^2+2 b^3 d^3 (b c-a d) g^4 x^3+6 d^4 (b f-a g)^4 \log (a+b x)-6 b^4 (d f-c g)^4 \log (c+d x)\right )}{6 b^4 d^4}}{4 g} \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. \(7243\) vs.
\(2(215)=430\).
time = 0.42, size = 7244, normalized size = 31.91
method | result | size |
risch | \(\frac {\left (g x +f \right )^{4} B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{4 g}-\frac {B \ln \left (b x +a \right ) f^{4}}{4 g}+\frac {B \ln \left (-d x -c \right ) f^{4}}{4 g}+g^{2} A f \,x^{3}+\frac {g^{2} B a f \,x^{2}}{2 b}-\frac {g^{2} B c f \,x^{2}}{2 d}+\frac {g^{3} A \,x^{4}}{4}+\frac {B \ln \left (b x +a \right ) a \,f^{3}}{b}-\frac {B \ln \left (-d x -c \right ) c \,f^{3}}{d}-\frac {g^{3} B \ln \left (b x +a \right ) a^{4}}{4 b^{4}}+\frac {g^{3} B \ln \left (-d x -c \right ) c^{4}}{4 d^{4}}-\frac {g^{2} B \,a^{2} f x}{b^{2}}+\frac {3 g B a \,f^{2} x}{2 b}+\frac {g^{2} B \,c^{2} f x}{d^{2}}-\frac {3 g B c \,f^{2} x}{2 d}+\frac {g^{3} B a \,x^{3}}{12 b}-\frac {g^{3} B c \,x^{3}}{12 d}+\frac {3 g A \,f^{2} x^{2}}{2}-\frac {g^{3} B \,a^{2} x^{2}}{8 b^{2}}+\frac {g^{3} B \,c^{2} x^{2}}{8 d^{2}}+A \,f^{3} x +\frac {g^{3} B \,a^{3} x}{4 b^{3}}-\frac {g^{3} B \,c^{3} x}{4 d^{3}}+\frac {g^{2} B \ln \left (b x +a \right ) a^{3} f}{b^{3}}-\frac {3 g B \ln \left (b x +a \right ) a^{2} f^{2}}{2 b^{2}}-\frac {g^{2} B \ln \left (-d x -c \right ) c^{3} f}{d^{3}}+\frac {3 g B \ln \left (-d x -c \right ) c^{2} f^{2}}{2 d^{2}}\) | \(412\) |
derivativedivides | \(\text {Expression too large to display}\) | \(7244\) |
default | \(\text {Expression too large to display}\) | \(7244\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 423, normalized size = 1.86 \begin {gather*} \frac {1}{4} \, A g^{3} x^{4} + A f g^{2} x^{3} + \frac {3}{2} \, A f^{2} g x^{2} + {\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 f^{3} + \frac {3}{2} \, {\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 f^{2} g + \frac {1}{2} \, {\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 f g^{2} + \frac {1}{24} \, {\left (6 \, x^{4} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \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 )} B g^{3} + A f^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 444 vs.
\(2 (216) = 432\).
time = 0.48, size = 444, normalized size = 1.96 \begin {gather*} \frac {6 \, A b^{4} d^{4} g^{3} x^{4} + 2 \, {\left (12 \, A b^{4} d^{4} f g^{2} - {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g^{3}\right )} x^{3} + 3 \, {\left (12 \, A b^{4} d^{4} f^{2} g - 4 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} f g^{2} + {\left (B b^{4} c^{2} d^{2} - B a^{2} b^{2} d^{4}\right )} g^{3}\right )} x^{2} + 6 \, {\left (4 \, A b^{4} d^{4} f^{3} - 6 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} f^{2} g + 4 \, {\left (B b^{4} c^{2} d^{2} - B a^{2} b^{2} d^{4}\right )} f g^{2} - {\left (B b^{4} c^{3} d - B a^{3} b d^{4}\right )} g^{3}\right )} x + 6 \, {\left (4 \, B a b^{3} d^{4} f^{3} - 6 \, B a^{2} b^{2} d^{4} f^{2} g + 4 \, B a^{3} b d^{4} f g^{2} - B a^{4} d^{4} g^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left (4 \, B b^{4} c d^{3} f^{3} - 6 \, B b^{4} c^{2} d^{2} f^{2} g + 4 \, B b^{4} c^{3} d f g^{2} - B b^{4} c^{4} g^{3}\right )} \log \left (d x + c\right ) + 6 \, {\left (B b^{4} d^{4} g^{3} x^{4} + 4 \, B b^{4} d^{4} f g^{2} x^{3} + 6 \, B b^{4} d^{4} f^{2} g x^{2} + 4 \, B b^{4} d^{4} f^{3} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{24 \, b^{4} d^{4}} \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. 998 vs.
\(2 (207) = 414\).
time = 15.12, size = 998, normalized size = 4.40 \begin {gather*} \frac {A g^{3} x^{4}}{4} - \frac {B a \left (a g - 2 b f\right ) \left (a^{2} g^{2} - 2 a b f g + 2 b^{2} f^{2}\right ) \log {\left (x + \frac {B a^{4} c d^{3} g^{3} - 4 B a^{3} b c d^{3} f g^{2} + 6 B a^{2} b^{2} c d^{3} f^{2} g + \frac {B a^{2} d^{4} \left (a g - 2 b f\right ) \left (a^{2} g^{2} - 2 a b f g + 2 b^{2} f^{2}\right )}{b} + B a b^{3} c^{4} g^{3} - 4 B a b^{3} c^{3} d f g^{2} + 6 B a b^{3} c^{2} d^{2} f^{2} g - 8 B a b^{3} c d^{3} f^{3} - B a c d^{3} \left (a g - 2 b f\right ) \left (a^{2} g^{2} - 2 a b f g + 2 b^{2} f^{2}\right )}{B a^{4} d^{4} g^{3} - 4 B a^{3} b d^{4} f g^{2} + 6 B a^{2} b^{2} d^{4} f^{2} g - 4 B a b^{3} d^{4} f^{3} + B b^{4} c^{4} g^{3} - 4 B b^{4} c^{3} d f g^{2} + 6 B b^{4} c^{2} d^{2} f^{2} g - 4 B b^{4} c d^{3} f^{3}} \right )}}{4 b^{4}} + \frac {B c \left (c g - 2 d f\right ) \left (c^{2} g^{2} - 2 c d f g + 2 d^{2} f^{2}\right ) \log {\left (x + \frac {B a^{4} c d^{3} g^{3} - 4 B a^{3} b c d^{3} f g^{2} + 6 B a^{2} b^{2} c d^{3} f^{2} g + B a b^{3} c^{4} g^{3} - 4 B a b^{3} c^{3} d f g^{2} + 6 B a b^{3} c^{2} d^{2} f^{2} g - 8 B a b^{3} c d^{3} f^{3} - B a b^{3} c \left (c g - 2 d f\right ) \left (c^{2} g^{2} - 2 c d f g + 2 d^{2} f^{2}\right ) + \frac {B b^{4} c^{2} \left (c g - 2 d f\right ) \left (c^{2} g^{2} - 2 c d f g + 2 d^{2} f^{2}\right )}{d}}{B a^{4} d^{4} g^{3} - 4 B a^{3} b d^{4} f g^{2} + 6 B a^{2} b^{2} d^{4} f^{2} g - 4 B a b^{3} d^{4} f^{3} + B b^{4} c^{4} g^{3} - 4 B b^{4} c^{3} d f g^{2} + 6 B b^{4} c^{2} d^{2} f^{2} g - 4 B b^{4} c d^{3} f^{3}} \right )}}{4 d^{4}} + x^{3} \left (A f g^{2} + \frac {B a g^{3}}{12 b} - \frac {B c g^{3}}{12 d}\right ) + x^{2} \cdot \left (\frac {3 A f^{2} g}{2} - \frac {B a^{2} g^{3}}{8 b^{2}} + \frac {B a f g^{2}}{2 b} + \frac {B c^{2} g^{3}}{8 d^{2}} - \frac {B c f g^{2}}{2 d}\right ) + x \left (A f^{3} + \frac {B a^{3} g^{3}}{4 b^{3}} - \frac {B a^{2} f g^{2}}{b^{2}} + \frac {3 B a f^{2} g}{2 b} - \frac {B c^{3} g^{3}}{4 d^{3}} + \frac {B c^{2} f g^{2}}{d^{2}} - \frac {3 B c f^{2} g}{2 d}\right ) + \left (B f^{3} x + \frac {3 B f^{2} g x^{2}}{2} + B f g^{2} x^{3} + \frac {B g^{3} x^{4}}{4}\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] Leaf count of result is larger than twice the leaf count of optimal. 11299 vs.
\(2 (216) = 432\).
time = 5.80, size = 11299, normalized size = 49.78 \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 = 4.69, size = 741, normalized size = 3.26 \begin {gather*} x\,\left (\frac {4\,A\,b\,d\,f^3+12\,A\,a\,c\,f\,g^2+12\,A\,a\,d\,f^2\,g+12\,A\,b\,c\,f^2\,g+6\,B\,a\,d\,f^2\,g-6\,B\,b\,c\,f^2\,g}{4\,b\,d}+\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (\frac {4\,A\,a\,d\,g^3+4\,A\,b\,c\,g^3+B\,a\,d\,g^3-B\,b\,c\,g^3+12\,A\,b\,d\,f\,g^2}{4\,b\,d}-\frac {A\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}-\frac {4\,A\,a\,c\,g^3+12\,A\,a\,d\,f\,g^2+12\,A\,b\,c\,f\,g^2+12\,A\,b\,d\,f^2\,g+4\,B\,a\,d\,f\,g^2-4\,B\,b\,c\,f\,g^2}{4\,b\,d}+\frac {A\,a\,c\,g^3}{b\,d}\right )}{4\,b\,d}-\frac {a\,c\,\left (\frac {4\,A\,a\,d\,g^3+4\,A\,b\,c\,g^3+B\,a\,d\,g^3-B\,b\,c\,g^3+12\,A\,b\,d\,f\,g^2}{4\,b\,d}-\frac {A\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}\right )}{b\,d}\right )-x^2\,\left (\frac {\left (\frac {4\,A\,a\,d\,g^3+4\,A\,b\,c\,g^3+B\,a\,d\,g^3-B\,b\,c\,g^3+12\,A\,b\,d\,f\,g^2}{4\,b\,d}-\frac {A\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{8\,b\,d}-\frac {4\,A\,a\,c\,g^3+12\,A\,a\,d\,f\,g^2+12\,A\,b\,c\,f\,g^2+12\,A\,b\,d\,f^2\,g+4\,B\,a\,d\,f\,g^2-4\,B\,b\,c\,f\,g^2}{8\,b\,d}+\frac {A\,a\,c\,g^3}{2\,b\,d}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,f^3\,x+\frac {3\,B\,f^2\,g\,x^2}{2}+B\,f\,g^2\,x^3+\frac {B\,g^3\,x^4}{4}\right )+x^3\,\left (\frac {4\,A\,a\,d\,g^3+4\,A\,b\,c\,g^3+B\,a\,d\,g^3-B\,b\,c\,g^3+12\,A\,b\,d\,f\,g^2}{12\,b\,d}-\frac {A\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{12\,b\,d}\right )+\frac {A\,g^3\,x^4}{4}-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^4\,g^3-4\,B\,a^3\,b\,f\,g^2+6\,B\,a^2\,b^2\,f^2\,g-4\,B\,a\,b^3\,f^3\right )}{4\,b^4}+\frac {\ln \left (c+d\,x\right )\,\left (B\,c^4\,g^3-4\,B\,c^3\,d\,f\,g^2+6\,B\,c^2\,d^2\,f^2\,g-4\,B\,c\,d^3\,f^3\right )}{4\,d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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