Optimal. Leaf size=182 \[ \frac {2 B (b c-a d)^4 g^4 x}{5 d^4}-\frac {B (b c-a d)^3 g^4 (a+b x)^2}{5 b d^3}+\frac {2 B (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b}-\frac {2 B (b c-a d)^5 g^4 \log (c+d x)}{5 b d^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2548, 21, 45}
\begin {gather*} \frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b}-\frac {2 B g^4 (b c-a d)^5 \log (c+d x)}{5 b d^5}+\frac {2 B g^4 x (b c-a d)^4}{5 d^4}-\frac {B g^4 (a+b x)^2 (b c-a d)^3}{5 b d^3}+\frac {2 B g^4 (a+b x)^3 (b c-a d)^2}{15 b d^2}-\frac {B g^4 (a+b x)^4 (b c-a d)}{10 b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 45
Rule 2548
Rubi steps
\begin {align*} \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx &=\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b}-\frac {B \int \frac {2 (b c-a d) g^5 (a+b x)^4}{c+d x} \, dx}{5 b g}\\ &=\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b}-\frac {\left (2 B (b c-a d) g^4\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{5 b}\\ &=\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b}-\frac {\left (2 B (b c-a d) g^4\right ) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{5 b}\\ &=\frac {2 B (b c-a d)^4 g^4 x}{5 d^4}-\frac {B (b c-a d)^3 g^4 (a+b x)^2}{5 b d^3}+\frac {2 B (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b}-\frac {2 B (b c-a d)^5 g^4 \log (c+d x)}{5 b d^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 144, normalized size = 0.79 \begin {gather*} \frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+\frac {B (b c-a d) \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )}{6 d^5}\right )}{5 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1957\) vs.
\(2(170)=340\).
time = 0.34, size = 1958, normalized size = 10.76
method | result | size |
risch | \(\frac {8 g^{4} B \,a^{4} x}{5}+\frac {2 g^{4} b^{4} B \,c^{4} x}{5 d^{4}}-\frac {2 g^{4} b^{4} B \ln \left (d x +c \right ) c^{5}}{5 d^{5}}-\frac {2 g^{4} B \ln \left (d x +c \right ) a^{4} c}{d}+\frac {g^{4} b^{3} B a \,x^{4}}{10}+g^{4} b^{3} A a \,x^{4}-\frac {g^{4} b^{4} B c \,x^{4}}{10 d}+2 g^{4} b^{2} A \,a^{2} x^{3}-\frac {4 g^{4} b B \,a^{3} c x}{d}+\frac {4 g^{4} b^{2} B \,a^{2} c^{2} x}{d^{2}}+\frac {4 g^{4} b B \ln \left (d x +c \right ) a^{3} c^{2}}{d^{2}}-\frac {4 g^{4} b^{2} B \ln \left (d x +c \right ) a^{2} c^{3}}{d^{3}}+\frac {2 g^{4} b^{3} B \ln \left (d x +c \right ) a \,c^{4}}{d^{4}}+\frac {2 g^{4} B \ln \left (d x +c \right ) a^{5}}{5 b}+\frac {\left (b x +a \right )^{5} g^{4} B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{5 b}-\frac {2 g^{4} b^{3} B a \,c^{3} x}{d^{3}}+\frac {8 g^{4} b^{2} B \,a^{2} x^{3}}{15}+\frac {2 g^{4} b^{4} B \,c^{2} x^{3}}{15 d^{2}}+2 g^{4} b A \,a^{3} x^{2}+\frac {6 g^{4} b B \,a^{3} x^{2}}{5}-\frac {g^{4} b^{4} B \,c^{3} x^{2}}{5 d^{3}}+g^{4} A \,a^{4} x -\frac {2 g^{4} b^{3} B a c \,x^{3}}{3 d}-\frac {2 g^{4} b^{2} B \,a^{2} c \,x^{2}}{d}+\frac {g^{4} b^{3} B a \,c^{2} x^{2}}{d^{2}}+\frac {g^{4} b^{4} A \,x^{5}}{5}\) | \(446\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1958\) |
default | \(\text {Expression too large to display}\) | \(1958\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 900 vs.
\(2 (171) = 342\).
time = 0.33, size = 900, normalized size = 4.95 \begin {gather*} \frac {1}{5} \, A b^{4} g^{4} x^{5} + A a b^{3} g^{4} x^{4} + 2 \, A a^{2} b^{2} g^{4} x^{3} + 2 \, A a^{3} b g^{4} x^{2} + {\left (x \log \left (\frac {b^{2} x^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b x e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {2 \, a \log \left (b x + a\right )}{b} - \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B a^{4} g^{4} + 2 \, {\left (x^{2} \log \left (\frac {b^{2} x^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b x e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B a^{3} b g^{4} + 2 \, {\left (x^{3} \log \left (\frac {b^{2} x^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b x e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\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 a^{2} b^{2} g^{4} + \frac {1}{3} \, {\left (3 \, x^{4} \log \left (\frac {b^{2} x^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b x e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\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 a b^{3} g^{4} + \frac {1}{30} \, {\left (6 \, x^{5} \log \left (\frac {b^{2} x^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b x e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} B b^{4} g^{4} + A a^{4} g^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 452 vs.
\(2 (171) = 342\).
time = 0.43, size = 452, normalized size = 2.48 \begin {gather*} \frac {6 \, A b^{5} d^{5} g^{4} x^{5} + 12 \, B a^{5} d^{5} g^{4} \log \left (b x + a\right ) - 3 \, {\left (B b^{5} c d^{4} - {\left (10 \, A + B\right )} a b^{4} d^{5}\right )} g^{4} x^{4} + 4 \, {\left (B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} + {\left (15 \, A + 4 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{4} x^{3} - 6 \, {\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 10 \, B a^{2} b^{3} c d^{4} - 2 \, {\left (5 \, A + 3 \, B\right )} a^{3} b^{2} d^{5}\right )} g^{4} x^{2} + 6 \, {\left (2 \, B b^{5} c^{4} d - 10 \, B a b^{4} c^{3} d^{2} + 20 \, B a^{2} b^{3} c^{2} d^{3} - 20 \, B a^{3} b^{2} c d^{4} + {\left (5 \, A + 8 \, B\right )} a^{4} b d^{5}\right )} g^{4} x - 12 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4}\right )} g^{4} \log \left (d x + c\right ) + 6 \, {\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B a b^{4} d^{5} g^{4} x^{4} + 10 \, B a^{2} b^{3} d^{5} g^{4} x^{3} + 10 \, B a^{3} b^{2} d^{5} g^{4} x^{2} + 5 \, B a^{4} b d^{5} g^{4} x\right )} \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{30 \, b d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 998 vs.
\(2 (163) = 326\).
time = 4.44, size = 998, normalized size = 5.48 \begin {gather*} \frac {A b^{4} g^{4} x^{5}}{5} + \frac {2 B a^{5} g^{4} \log {\left (x + \frac {\frac {2 B a^{6} d^{5} g^{4}}{b} + 10 B a^{5} c d^{4} g^{4} - 20 B a^{4} b c^{2} d^{3} g^{4} + 20 B a^{3} b^{2} c^{3} d^{2} g^{4} - 10 B a^{2} b^{3} c^{4} d g^{4} + 2 B a b^{4} c^{5} g^{4}}{2 B a^{5} d^{5} g^{4} + 10 B a^{4} b c d^{4} g^{4} - 20 B a^{3} b^{2} c^{2} d^{3} g^{4} + 20 B a^{2} b^{3} c^{3} d^{2} g^{4} - 10 B a b^{4} c^{4} d g^{4} + 2 B b^{5} c^{5} g^{4}} \right )}}{5 b} - \frac {2 B c g^{4} \cdot \left (5 a^{4} d^{4} - 10 a^{3} b c d^{3} + 10 a^{2} b^{2} c^{2} d^{2} - 5 a b^{3} c^{3} d + b^{4} c^{4}\right ) \log {\left (x + \frac {12 B a^{5} c d^{4} g^{4} - 20 B a^{4} b c^{2} d^{3} g^{4} + 20 B a^{3} b^{2} c^{3} d^{2} g^{4} - 10 B a^{2} b^{3} c^{4} d g^{4} + 2 B a b^{4} c^{5} g^{4} - 2 B a c g^{4} \cdot \left (5 a^{4} d^{4} - 10 a^{3} b c d^{3} + 10 a^{2} b^{2} c^{2} d^{2} - 5 a b^{3} c^{3} d + b^{4} c^{4}\right ) + \frac {2 B b c^{2} g^{4} \cdot \left (5 a^{4} d^{4} - 10 a^{3} b c d^{3} + 10 a^{2} b^{2} c^{2} d^{2} - 5 a b^{3} c^{3} d + b^{4} c^{4}\right )}{d}}{2 B a^{5} d^{5} g^{4} + 10 B a^{4} b c d^{4} g^{4} - 20 B a^{3} b^{2} c^{2} d^{3} g^{4} + 20 B a^{2} b^{3} c^{3} d^{2} g^{4} - 10 B a b^{4} c^{4} d g^{4} + 2 B b^{5} c^{5} g^{4}} \right )}}{5 d^{5}} + x^{4} \left (A a b^{3} g^{4} + \frac {B a b^{3} g^{4}}{10} - \frac {B b^{4} c g^{4}}{10 d}\right ) + x^{3} \cdot \left (2 A a^{2} b^{2} g^{4} + \frac {8 B a^{2} b^{2} g^{4}}{15} - \frac {2 B a b^{3} c g^{4}}{3 d} + \frac {2 B b^{4} c^{2} g^{4}}{15 d^{2}}\right ) + x^{2} \cdot \left (2 A a^{3} b g^{4} + \frac {6 B a^{3} b g^{4}}{5} - \frac {2 B a^{2} b^{2} c g^{4}}{d} + \frac {B a b^{3} c^{2} g^{4}}{d^{2}} - \frac {B b^{4} c^{3} g^{4}}{5 d^{3}}\right ) + x \left (A a^{4} g^{4} + \frac {8 B a^{4} g^{4}}{5} - \frac {4 B a^{3} b c g^{4}}{d} + \frac {4 B a^{2} b^{2} c^{2} g^{4}}{d^{2}} - \frac {2 B a b^{3} c^{3} g^{4}}{d^{3}} + \frac {2 B b^{4} c^{4} g^{4}}{5 d^{4}}\right ) + \left (B a^{4} g^{4} x + 2 B a^{3} b g^{4} x^{2} + 2 B a^{2} b^{2} g^{4} x^{3} + B a b^{3} g^{4} x^{4} + \frac {B b^{4} g^{4} x^{5}}{5}\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 496 vs.
\(2 (171) = 342\).
time = 255.27, size = 496, normalized size = 2.73 \begin {gather*} \frac {2 \, B a^{5} g^{4} \log \left (b x + a\right )}{5 \, b} + \frac {1}{5} \, {\left (A b^{4} g^{4} + B b^{4} g^{4}\right )} x^{5} - \frac {{\left (B b^{4} c g^{4} - 10 \, A a b^{3} d g^{4} - 11 \, B a b^{3} d g^{4}\right )} x^{4}}{10 \, d} + \frac {2 \, {\left (B b^{4} c^{2} g^{4} - 5 \, B a b^{3} c d g^{4} + 15 \, A a^{2} b^{2} d^{2} g^{4} + 19 \, B a^{2} b^{2} d^{2} g^{4}\right )} x^{3}}{15 \, d^{2}} + \frac {1}{5} \, {\left (B b^{4} g^{4} x^{5} + 5 \, B a b^{3} g^{4} x^{4} + 10 \, B a^{2} b^{2} g^{4} x^{3} + 10 \, B a^{3} b g^{4} x^{2} + 5 \, B a^{4} g^{4} x\right )} \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {{\left (B b^{4} c^{3} g^{4} - 5 \, B a b^{3} c^{2} d g^{4} + 10 \, B a^{2} b^{2} c d^{2} g^{4} - 10 \, A a^{3} b d^{3} g^{4} - 16 \, B a^{3} b d^{3} g^{4}\right )} x^{2}}{5 \, d^{3}} + \frac {{\left (2 \, B b^{4} c^{4} g^{4} - 10 \, B a b^{3} c^{3} d g^{4} + 20 \, B a^{2} b^{2} c^{2} d^{2} g^{4} - 20 \, B a^{3} b c d^{3} g^{4} + 5 \, A a^{4} d^{4} g^{4} + 13 \, B a^{4} d^{4} g^{4}\right )} x}{5 \, d^{4}} - \frac {2 \, {\left (B b^{4} c^{5} g^{4} - 5 \, B a b^{3} c^{4} d g^{4} + 10 \, B a^{2} b^{2} c^{3} d^{2} g^{4} - 10 \, B a^{3} b c^{2} d^{3} g^{4} + 5 \, B a^{4} c d^{4} g^{4}\right )} \log \left (-d x - c\right )}{5 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.99, size = 1025, normalized size = 5.63 \begin {gather*} x^2\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{10\,b\,d}+\frac {a^2\,b\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}-\frac {a\,c\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{2\,b\,d}\right )-x^3\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{15\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{3\,d}+\frac {A\,a\,b^3\,c\,g^4}{3\,d}\right )+x\,\left (\frac {a^3\,g^4\,\left (5\,A\,a\,d+10\,A\,b\,c+4\,B\,a\,d-4\,B\,b\,c\right )}{d}-\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{5\,b\,d}+\frac {2\,a^2\,b\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}-\frac {a\,c\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{b\,d}\right )}{5\,b\,d}+\frac {a\,c\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{b\,d}\right )+\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (B\,a^4\,g^4\,x+2\,B\,a^3\,b\,g^4\,x^2+2\,B\,a^2\,b^2\,g^4\,x^3+B\,a\,b^3\,g^4\,x^4+\frac {B\,b^4\,g^4\,x^5}{5}\right )+x^4\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{20\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{20\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (10\,B\,a^4\,c\,d^4\,g^4-20\,B\,a^3\,b\,c^2\,d^3\,g^4+20\,B\,a^2\,b^2\,c^3\,d^2\,g^4-10\,B\,a\,b^3\,c^4\,d\,g^4+2\,B\,b^4\,c^5\,g^4\right )}{5\,d^5}+\frac {A\,b^4\,g^4\,x^5}{5}+\frac {2\,B\,a^5\,g^4\,\ln \left (a+b\,x\right )}{5\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________