Optimal. Leaf size=210 \[ \frac {(e f+d g)^2}{64 d^6 e^3 (d-e x)}-\frac {(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac {e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac {(3 e f-d g) (e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac {f (e f+d g)}{16 d^5 e^2 (d+e x)^2}-\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d+e x)}+\frac {(e f+d g) (3 e f+d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^7 e^3} \]
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Rubi [A]
time = 0.16, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {862, 90, 214}
\begin {gather*} \frac {(d g+e f) (d g+3 e f) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^7 e^3}+\frac {(d g+e f)^2}{64 d^6 e^3 (d-e x)}-\frac {(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d+e x)}-\frac {f (d g+e f)}{16 d^5 e^2 (d+e x)^2}-\frac {(3 e f-d g) (d g+e f)}{48 d^4 e^3 (d+e x)^3}-\frac {(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac {e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 214
Rule 862
Rubi steps
\begin {align*} \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(f+g x)^2}{(d-e x)^2 (d+e x)^6} \, dx\\ &=\int \left (\frac {(e f+d g)^2}{64 d^6 e^2 (d-e x)^2}+\frac {(-e f+d g)^2}{4 d^2 e^2 (d+e x)^6}+\frac {e^2 f^2-d^2 g^2}{4 d^3 e^2 (d+e x)^5}+\frac {(3 e f-d g) (e f+d g)}{16 d^4 e^2 (d+e x)^4}+\frac {f (e f+d g)}{8 d^5 e (d+e x)^3}+\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^2 (d+e x)^2}+\frac {(e f+d g) (3 e f+d g)}{32 d^6 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac {(e f+d g)^2}{64 d^6 e^3 (d-e x)}-\frac {(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac {e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac {(3 e f-d g) (e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac {f (e f+d g)}{16 d^5 e^2 (d+e x)^2}-\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d+e x)}+\frac {((e f+d g) (3 e f+d g)) \int \frac {1}{d^2-e^2 x^2} \, dx}{32 d^6 e^2}\\ &=\frac {(e f+d g)^2}{64 d^6 e^3 (d-e x)}-\frac {(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac {e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac {(3 e f-d g) (e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac {f (e f+d g)}{16 d^5 e^2 (d+e x)^2}-\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d+e x)}+\frac {(e f+d g) (3 e f+d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^7 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 229, normalized size = 1.09 \begin {gather*} \frac {\frac {15 d (e f+d g)^2}{d-e x}-\frac {48 d^5 (e f-d g)^2}{(d+e x)^5}+\frac {60 d^4 \left (-e^2 f^2+d^2 g^2\right )}{(d+e x)^4}+\frac {20 d^3 \left (-3 e^2 f^2-2 d e f g+d^2 g^2\right )}{(d+e x)^3}-\frac {60 d^2 e f (e f+d g)}{(d+e x)^2}-\frac {15 d \left (5 e^2 f^2+6 d e f g+d^2 g^2\right )}{d+e x}-15 \left (3 e^2 f^2+4 d e f g+d^2 g^2\right ) \log (d-e x)+15 \left (3 e^2 f^2+4 d e f g+d^2 g^2\right ) \log (d+e x)}{960 d^7 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 278, normalized size = 1.32
method | result | size |
default | \(\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{64 e^{3} d^{7}}-\frac {d^{2} g^{2}+6 d e f g +5 e^{2} f^{2}}{64 e^{3} d^{6} \left (e x +d \right )}-\frac {-d^{2} g^{2}+e^{2} f^{2}}{16 e^{3} d^{3} \left (e x +d \right )^{4}}-\frac {-d^{2} g^{2}+2 d e f g +3 e^{2} f^{2}}{48 e^{3} d^{4} \left (e x +d \right )^{3}}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{20 e^{3} d^{2} \left (e x +d \right )^{5}}-\frac {f \left (d g +e f \right )}{16 d^{5} e^{2} \left (e x +d \right )^{2}}+\frac {\left (-d^{2} g^{2}-4 d e f g -3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{64 e^{3} d^{7}}+\frac {d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{64 e^{3} d^{6} \left (-e x +d \right )}\) | \(278\) |
norman | \(\frac {\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) x^{3}}{6 d^{4}}-\frac {\left (d^{2} g^{2}-4 d e f g -13 e^{2} f^{2}\right ) x^{2}}{8 e \,d^{3}}+\frac {e \left (7 d^{2} g^{2}+4 d e f g -27 e^{2} f^{2}\right ) x^{4}}{24 d^{5}}+\frac {e^{2} \left (79 d^{2} g^{2}-68 d e f g -531 e^{2} f^{2}\right ) x^{5}}{480 d^{6}}+\frac {e^{3} \left (d^{2} g^{2}-2 d e f g -9 e^{2} f^{2}\right ) x^{6}}{30 d^{7}}-\frac {\left (d^{2} g^{2}+4 d e f g -29 e^{2} f^{2}\right ) x}{32 e^{2} d^{2}}}{\left (e x +d \right )^{5} \left (-e x +d \right )}-\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{64 e^{3} d^{7}}+\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{64 e^{3} d^{7}}\) | \(285\) |
risch | \(\frac {\frac {e^{2} \left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) x^{5}}{32 d^{6}}+\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) e \,x^{4}}{8 d^{5}}+\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) x^{3}}{6 d^{4}}+\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) x^{2}}{24 d^{3} e}+\frac {\left (49 d^{2} g^{2}-188 d e f g -141 e^{2} f^{2}\right ) x}{480 e^{2} d^{2}}+\frac {d^{2} g^{2}-2 d e f g -9 e^{2} f^{2}}{30 e^{3} d}}{\left (e x +d \right )^{4} \left (-e^{2} x^{2}+d^{2}\right )}-\frac {\ln \left (-e x +d \right ) g^{2}}{64 e^{3} d^{5}}-\frac {\ln \left (-e x +d \right ) f g}{16 e^{2} d^{6}}-\frac {3 \ln \left (-e x +d \right ) f^{2}}{64 e \,d^{7}}+\frac {\ln \left (e x +d \right ) g^{2}}{64 e^{3} d^{5}}+\frac {\ln \left (e x +d \right ) f g}{16 e^{2} d^{6}}+\frac {3 \ln \left (e x +d \right ) f^{2}}{64 e \,d^{7}}\) | \(317\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 323, normalized size = 1.54 \begin {gather*} -\frac {16 \, d^{7} g^{2} - 32 \, d^{6} f g e - 144 \, d^{5} f^{2} e^{2} + 15 \, {\left (d^{2} g^{2} e^{5} + 4 \, d f g e^{6} + 3 \, f^{2} e^{7}\right )} x^{5} + 60 \, {\left (d^{3} g^{2} e^{4} + 4 \, d^{2} f g e^{5} + 3 \, d f^{2} e^{6}\right )} x^{4} + 80 \, {\left (d^{4} g^{2} e^{3} + 4 \, d^{3} f g e^{4} + 3 \, d^{2} f^{2} e^{5}\right )} x^{3} + 20 \, {\left (d^{5} g^{2} e^{2} + 4 \, d^{4} f g e^{3} + 3 \, d^{3} f^{2} e^{4}\right )} x^{2} + {\left (49 \, d^{6} g^{2} e - 188 \, d^{5} f g e^{2} - 141 \, d^{4} f^{2} e^{3}\right )} x}{480 \, {\left (d^{6} x^{6} e^{9} + 4 \, d^{7} x^{5} e^{8} + 5 \, d^{8} x^{4} e^{7} - 5 \, d^{10} x^{2} e^{5} - 4 \, d^{11} x e^{4} - d^{12} e^{3}\right )}} + \frac {{\left (d^{2} g^{2} + 4 \, d f g e + 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e + d\right )}{64 \, d^{7}} - \frac {{\left (d^{2} g^{2} + 4 \, d f g e + 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e - d\right )}{64 \, d^{7}} \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. 685 vs.
\(2 (204) = 408\).
time = 4.15, size = 685, normalized size = 3.26 \begin {gather*} -\frac {32 \, d^{8} g^{2} + 90 \, d f^{2} x^{5} e^{7} + 120 \, {\left (d^{2} f g x^{5} + 3 \, d^{2} f^{2} x^{4}\right )} e^{6} + 30 \, {\left (d^{3} g^{2} x^{5} + 16 \, d^{3} f g x^{4} + 16 \, d^{3} f^{2} x^{3}\right )} e^{5} + 40 \, {\left (3 \, d^{4} g^{2} x^{4} + 16 \, d^{4} f g x^{3} + 3 \, d^{4} f^{2} x^{2}\right )} e^{4} + 2 \, {\left (80 \, d^{5} g^{2} x^{3} + 80 \, d^{5} f g x^{2} - 141 \, d^{5} f^{2} x\right )} e^{3} + 8 \, {\left (5 \, d^{6} g^{2} x^{2} - 47 \, d^{6} f g x - 36 \, d^{6} f^{2}\right )} e^{2} + 2 \, {\left (49 \, d^{7} g^{2} x - 32 \, d^{7} f g\right )} e + 15 \, {\left (d^{8} g^{2} - 3 \, f^{2} x^{6} e^{8} - 4 \, {\left (d f g x^{6} + 3 \, d f^{2} x^{5}\right )} e^{7} - {\left (d^{2} g^{2} x^{6} + 16 \, d^{2} f g x^{5} + 15 \, d^{2} f^{2} x^{4}\right )} e^{6} - 4 \, {\left (d^{3} g^{2} x^{5} + 5 \, d^{3} f g x^{4}\right )} e^{5} - 5 \, {\left (d^{4} g^{2} x^{4} - 3 \, d^{4} f^{2} x^{2}\right )} e^{4} + 4 \, {\left (5 \, d^{5} f g x^{2} + 3 \, d^{5} f^{2} x\right )} e^{3} + {\left (5 \, d^{6} g^{2} x^{2} + 16 \, d^{6} f g x + 3 \, d^{6} f^{2}\right )} e^{2} + 4 \, {\left (d^{7} g^{2} x + d^{7} f g\right )} e\right )} \log \left (x e + d\right ) - 15 \, {\left (d^{8} g^{2} - 3 \, f^{2} x^{6} e^{8} - 4 \, {\left (d f g x^{6} + 3 \, d f^{2} x^{5}\right )} e^{7} - {\left (d^{2} g^{2} x^{6} + 16 \, d^{2} f g x^{5} + 15 \, d^{2} f^{2} x^{4}\right )} e^{6} - 4 \, {\left (d^{3} g^{2} x^{5} + 5 \, d^{3} f g x^{4}\right )} e^{5} - 5 \, {\left (d^{4} g^{2} x^{4} - 3 \, d^{4} f^{2} x^{2}\right )} e^{4} + 4 \, {\left (5 \, d^{5} f g x^{2} + 3 \, d^{5} f^{2} x\right )} e^{3} + {\left (5 \, d^{6} g^{2} x^{2} + 16 \, d^{6} f g x + 3 \, d^{6} f^{2}\right )} e^{2} + 4 \, {\left (d^{7} g^{2} x + d^{7} f g\right )} e\right )} \log \left (x e - d\right )}{960 \, {\left (d^{7} x^{6} e^{9} + 4 \, d^{8} x^{5} e^{8} + 5 \, d^{9} x^{4} e^{7} - 5 \, d^{11} x^{2} e^{5} - 4 \, d^{12} x e^{4} - d^{13} e^{3}\right )}} \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. 427 vs.
\(2 (192) = 384\).
time = 1.06, size = 427, normalized size = 2.03 \begin {gather*} \frac {- 16 d^{7} g^{2} + 32 d^{6} e f g + 144 d^{5} e^{2} f^{2} + x^{5} \left (- 15 d^{2} e^{5} g^{2} - 60 d e^{6} f g - 45 e^{7} f^{2}\right ) + x^{4} \left (- 60 d^{3} e^{4} g^{2} - 240 d^{2} e^{5} f g - 180 d e^{6} f^{2}\right ) + x^{3} \left (- 80 d^{4} e^{3} g^{2} - 320 d^{3} e^{4} f g - 240 d^{2} e^{5} f^{2}\right ) + x^{2} \left (- 20 d^{5} e^{2} g^{2} - 80 d^{4} e^{3} f g - 60 d^{3} e^{4} f^{2}\right ) + x \left (- 49 d^{6} e g^{2} + 188 d^{5} e^{2} f g + 141 d^{4} e^{3} f^{2}\right )}{- 480 d^{12} e^{3} - 1920 d^{11} e^{4} x - 2400 d^{10} e^{5} x^{2} + 2400 d^{8} e^{7} x^{4} + 1920 d^{7} e^{8} x^{5} + 480 d^{6} e^{9} x^{6}} - \frac {\left (d g + e f\right ) \left (d g + 3 e f\right ) \log {\left (- \frac {d \left (d g + e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 4 d e f g + 3 e^{2} f^{2}\right )} + x \right )}}{64 d^{7} e^{3}} + \frac {\left (d g + e f\right ) \left (d g + 3 e f\right ) \log {\left (\frac {d \left (d g + e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 4 d e f g + 3 e^{2} f^{2}\right )} + x \right )}}{64 d^{7} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.00, size = 296, normalized size = 1.41 \begin {gather*} \frac {{\left (d^{2} g^{2} + 4 \, d f g e + 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right )}{64 \, d^{7}} - \frac {{\left (d^{2} g^{2} + 4 \, d f g e + 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e - d \right |}\right )}{64 \, d^{7}} - \frac {{\left (16 \, d^{8} g^{2} - 32 \, d^{7} f g e - 144 \, d^{6} f^{2} e^{2} + 15 \, {\left (d^{3} g^{2} e^{5} + 4 \, d^{2} f g e^{6} + 3 \, d f^{2} e^{7}\right )} x^{5} + 60 \, {\left (d^{4} g^{2} e^{4} + 4 \, d^{3} f g e^{5} + 3 \, d^{2} f^{2} e^{6}\right )} x^{4} + 80 \, {\left (d^{5} g^{2} e^{3} + 4 \, d^{4} f g e^{4} + 3 \, d^{3} f^{2} e^{5}\right )} x^{3} + 20 \, {\left (d^{6} g^{2} e^{2} + 4 \, d^{5} f g e^{3} + 3 \, d^{4} f^{2} e^{4}\right )} x^{2} + {\left (49 \, d^{7} g^{2} e - 188 \, d^{6} f g e^{2} - 141 \, d^{5} f^{2} e^{3}\right )} x\right )} e^{\left (-3\right )}}{480 \, {\left (x e + d\right )}^{5} {\left (x e - d\right )} d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.72, size = 314, normalized size = 1.50 \begin {gather*} \frac {\frac {x^3\,\left (d^2\,g^2+4\,d\,e\,f\,g+3\,e^2\,f^2\right )}{6\,d^4}-\frac {-d^2\,g^2+2\,d\,e\,f\,g+9\,e^2\,f^2}{30\,d\,e^3}+\frac {e\,x^4\,\left (d^2\,g^2+4\,d\,e\,f\,g+3\,e^2\,f^2\right )}{8\,d^5}-\frac {x\,\left (-49\,d^2\,g^2+188\,d\,e\,f\,g+141\,e^2\,f^2\right )}{480\,d^2\,e^2}+\frac {x^2\,\left (d^2\,g^2+4\,d\,e\,f\,g+3\,e^2\,f^2\right )}{24\,d^3\,e}+\frac {e^2\,x^5\,\left (d^2\,g^2+4\,d\,e\,f\,g+3\,e^2\,f^2\right )}{32\,d^6}}{d^6+4\,d^5\,e\,x+5\,d^4\,e^2\,x^2-5\,d^2\,e^4\,x^4-4\,d\,e^5\,x^5-e^6\,x^6}+\frac {\mathrm {atanh}\left (\frac {e\,x\,\left (d\,g+e\,f\right )\,\left (d\,g+3\,e\,f\right )}{d\,\left (d^2\,g^2+4\,d\,e\,f\,g+3\,e^2\,f^2\right )}\right )\,\left (d\,g+e\,f\right )\,\left (d\,g+3\,e\,f\right )}{32\,d^7\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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