Optimal. Leaf size=189 \[ \frac {B c^2 x}{e^5}+\frac {(B d-A e) \left (c d^2+a e^2\right )^2}{4 e^6 (d+e x)^4}-\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{3 e^6 (d+e x)^3}+\frac {c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^6 (d+e x)^2}-\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right )}{e^6 (d+e x)}-\frac {c^2 (5 B d-A e) \log (d+e x)}{e^6} \]
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Rubi [A]
time = 0.12, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {786}
\begin {gather*} -\frac {2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6 (d+e x)^4}+\frac {c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (d+e x)^2}-\frac {c^2 (5 B d-A e) \log (d+e x)}{e^6}+\frac {B c^2 x}{e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 786
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^5} \, dx &=\int \left (\frac {B c^2}{e^5}+\frac {(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^5}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^4}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^3}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^2}+\frac {c^2 (-5 B d+A e)}{e^5 (d+e x)}\right ) \, dx\\ &=\frac {B c^2 x}{e^5}+\frac {(B d-A e) \left (c d^2+a e^2\right )^2}{4 e^6 (d+e x)^4}-\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{3 e^6 (d+e x)^3}+\frac {c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^6 (d+e x)^2}-\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right )}{e^6 (d+e x)}-\frac {c^2 (5 B d-A e) \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 221, normalized size = 1.17 \begin {gather*} \frac {A e \left (-3 a^2 e^4-2 a c e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+c^2 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )-B \left (a^2 e^4 (d+4 e x)+6 a c e^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+c^2 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )-12 c^2 (5 B d-A e) (d+e x)^4 \log (d+e x)}{12 e^6 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.63, size = 240, normalized size = 1.27
method | result | size |
default | \(\frac {B \,c^{2} x}{e^{5}}+\frac {2 c \left (2 A c d e -B \,e^{2} a -5 B c \,d^{2}\right )}{e^{6} \left (e x +d \right )}-\frac {c \left (A a \,e^{3}+3 A c \,d^{2} e -3 a B d \,e^{2}-5 B c \,d^{3}\right )}{e^{6} \left (e x +d \right )^{2}}+\frac {c^{2} \left (A e -5 B d \right ) \ln \left (e x +d \right )}{e^{6}}-\frac {A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}}{4 e^{6} \left (e x +d \right )^{4}}-\frac {-4 A a c d \,e^{3}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}}{3 e^{6} \left (e x +d \right )^{3}}\) | \(240\) |
norman | \(\frac {\frac {B \,c^{2} x^{5}}{e}-\frac {3 A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}-25 A \,c^{2} d^{4} e +B \,a^{2} d \,e^{4}+6 B a c \,d^{3} e^{2}+125 B \,c^{2} d^{5}}{12 e^{6}}+\frac {2 \left (2 A \,c^{2} d e -B \,e^{2} a c -10 B \,c^{2} d^{2}\right ) x^{3}}{e^{3}}-\frac {\left (A a c \,e^{3}-9 A \,c^{2} d^{2} e +3 B a c d \,e^{2}+45 B \,c^{2} d^{3}\right ) x^{2}}{e^{4}}-\frac {\left (2 A a c d \,e^{3}-22 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+110 B \,c^{2} d^{4}\right ) x}{3 e^{5}}}{\left (e x +d \right )^{4}}+\frac {c^{2} \left (A e -5 B d \right ) \ln \left (e x +d \right )}{e^{6}}\) | \(240\) |
risch | \(\frac {B \,c^{2} x}{e^{5}}+\frac {\left (4 A \,c^{2} d \,e^{3}-2 B \,e^{4} a c -10 B \,c^{2} d^{2} e^{2}\right ) x^{3}-e c \left (A a \,e^{3}-9 A c \,d^{2} e +3 a B d \,e^{2}+25 B c \,d^{3}\right ) x^{2}+\left (-\frac {2}{3} A a c d \,e^{3}+\frac {22}{3} A \,c^{2} d^{3} e -\frac {1}{3} B \,e^{4} a^{2}-2 B a c \,d^{2} e^{2}-\frac {65}{3} B \,c^{2} d^{4}\right ) x -\frac {3 A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}-25 A \,c^{2} d^{4} e +B \,a^{2} d \,e^{4}+6 B a c \,d^{3} e^{2}+77 B \,c^{2} d^{5}}{12 e}}{e^{5} \left (e x +d \right )^{4}}+\frac {c^{2} \ln \left (e x +d \right ) A}{e^{5}}-\frac {5 c^{2} \ln \left (e x +d \right ) B d}{e^{6}}\) | \(241\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 261, normalized size = 1.38 \begin {gather*} B c^{2} x e^{\left (-5\right )} - {\left (5 \, B c^{2} d - A c^{2} e\right )} e^{\left (-6\right )} \log \left (x e + d\right ) - \frac {77 \, B c^{2} d^{5} - 25 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} + 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + 24 \, {\left (5 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} + 3 \, A a^{2} e^{5} + 12 \, {\left (25 \, B c^{2} d^{3} e^{2} - 9 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + A a c e^{5}\right )} x^{2} + 4 \, {\left (65 \, B c^{2} d^{4} e - 22 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} + 2 \, A a c d e^{4} + B a^{2} e^{5}\right )} x}{12 \, {\left (x^{4} e^{10} + 4 \, d x^{3} e^{9} + 6 \, d^{2} x^{2} e^{8} + 4 \, d^{3} x e^{7} + d^{4} e^{6}\right )}} \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. 380 vs.
\(2 (183) = 366\).
time = 4.18, size = 380, normalized size = 2.01 \begin {gather*} -\frac {77 \, B c^{2} d^{5} - {\left (12 \, B c^{2} x^{5} - 24 \, B a c x^{3} - 12 \, A a c x^{2} - 4 \, B a^{2} x - 3 \, A a^{2}\right )} e^{5} - {\left (48 \, B c^{2} d x^{4} + 48 \, A c^{2} d x^{3} - 36 \, B a c d x^{2} - 8 \, A a c d x - B a^{2} d\right )} e^{4} + 2 \, {\left (24 \, B c^{2} d^{2} x^{3} - 54 \, A c^{2} d^{2} x^{2} + 12 \, B a c d^{2} x + A a c d^{2}\right )} e^{3} + 2 \, {\left (126 \, B c^{2} d^{3} x^{2} - 44 \, A c^{2} d^{3} x + 3 \, B a c d^{3}\right )} e^{2} + {\left (248 \, B c^{2} d^{4} x - 25 \, A c^{2} d^{4}\right )} e + 12 \, {\left (5 \, B c^{2} d^{5} - A c^{2} x^{4} e^{5} + {\left (5 \, B c^{2} d x^{4} - 4 \, A c^{2} d x^{3}\right )} e^{4} + 2 \, {\left (10 \, B c^{2} d^{2} x^{3} - 3 \, A c^{2} d^{2} x^{2}\right )} e^{3} + 2 \, {\left (15 \, B c^{2} d^{3} x^{2} - 2 \, A c^{2} d^{3} x\right )} e^{2} + {\left (20 \, B c^{2} d^{4} x - A c^{2} d^{4}\right )} e\right )} \log \left (x e + d\right )}{12 \, {\left (x^{4} e^{10} + 4 \, d x^{3} e^{9} + 6 \, d^{2} x^{2} e^{8} + 4 \, d^{3} x e^{7} + d^{4} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 79.02, size = 304, normalized size = 1.61 \begin {gather*} \frac {B c^{2} x}{e^{5}} - \frac {c^{2} \left (- A e + 5 B d\right ) \log {\left (d + e x \right )}}{e^{6}} + \frac {- 3 A a^{2} e^{5} - 2 A a c d^{2} e^{3} + 25 A c^{2} d^{4} e - B a^{2} d e^{4} - 6 B a c d^{3} e^{2} - 77 B c^{2} d^{5} + x^{3} \cdot \left (48 A c^{2} d e^{4} - 24 B a c e^{5} - 120 B c^{2} d^{2} e^{3}\right ) + x^{2} \left (- 12 A a c e^{5} + 108 A c^{2} d^{2} e^{3} - 36 B a c d e^{4} - 300 B c^{2} d^{3} e^{2}\right ) + x \left (- 8 A a c d e^{4} + 88 A c^{2} d^{3} e^{2} - 4 B a^{2} e^{5} - 24 B a c d^{2} e^{3} - 260 B c^{2} d^{4} e\right )}{12 d^{4} e^{6} + 48 d^{3} e^{7} x + 72 d^{2} e^{8} x^{2} + 48 d e^{9} x^{3} + 12 e^{10} x^{4}} \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. 372 vs.
\(2 (183) = 366\).
time = 1.17, size = 372, normalized size = 1.97 \begin {gather*} {\left (x e + d\right )} B c^{2} e^{\left (-6\right )} + {\left (5 \, B c^{2} d - A c^{2} e\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {1}{12} \, {\left (\frac {120 \, B c^{2} d^{2} e^{22}}{x e + d} - \frac {60 \, B c^{2} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac {20 \, B c^{2} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B c^{2} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac {48 \, A c^{2} d e^{23}}{x e + d} + \frac {36 \, A c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac {16 \, A c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac {3 \, A c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac {24 \, B a c e^{24}}{x e + d} - \frac {36 \, B a c d e^{24}}{{\left (x e + d\right )}^{2}} + \frac {24 \, B a c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac {6 \, B a c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac {12 \, A a c e^{25}}{{\left (x e + d\right )}^{2}} - \frac {16 \, A a c d e^{25}}{{\left (x e + d\right )}^{3}} + \frac {6 \, A a c d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a^{2} e^{26}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a^{2} d e^{26}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a^{2} e^{27}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.84, size = 277, normalized size = 1.47 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,c^2\,e-5\,B\,c^2\,d\right )}{e^6}-\frac {x^3\,\left (10\,B\,c^2\,d^2\,e^2-4\,A\,c^2\,d\,e^3+2\,B\,a\,c\,e^4\right )+x\,\left (\frac {B\,a^2\,e^4}{3}+2\,B\,a\,c\,d^2\,e^2+\frac {2\,A\,a\,c\,d\,e^3}{3}+\frac {65\,B\,c^2\,d^4}{3}-\frac {22\,A\,c^2\,d^3\,e}{3}\right )+\frac {B\,a^2\,d\,e^4+3\,A\,a^2\,e^5+6\,B\,a\,c\,d^3\,e^2+2\,A\,a\,c\,d^2\,e^3+77\,B\,c^2\,d^5-25\,A\,c^2\,d^4\,e}{12\,e}+x^2\,\left (25\,B\,c^2\,d^3\,e-9\,A\,c^2\,d^2\,e^2+3\,B\,a\,c\,d\,e^3+A\,a\,c\,e^4\right )}{d^4\,e^5+4\,d^3\,e^6\,x+6\,d^2\,e^7\,x^2+4\,d\,e^8\,x^3+e^9\,x^4}+\frac {B\,c^2\,x}{e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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