Optimal. Leaf size=291 \[ \frac {(-5 b e g+8 c d g+2 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{7/2} e^2}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+8 c d g+2 c e f)}{c^3 e^2 (2 c d-b e)}-\frac {2 (d+e x)^2 (-5 b e g+8 c d g+2 c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^4 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.43, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {788, 668, 640, 621, 204} \begin {gather*} -\frac {2 (d+e x)^2 (-5 b e g+8 c d g+2 c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+8 c d g+2 c e f)}{c^3 e^2 (2 c d-b e)}+\frac {(-5 b e g+8 c d g+2 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{7/2} e^2}+\frac {2 (d+e x)^4 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 621
Rule 640
Rule 668
Rule 788
Rubi steps
\begin {align*} \int \frac {(d+e x)^4 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^4}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(2 c e f+8 c d g-5 b e g) \int \frac {(d+e x)^3}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^4}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (2 c e f+8 c d g-5 b e g) (d+e x)^2}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(2 c e f+8 c d g-5 b e g) \int \frac {d+e x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c^2 e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^4}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (2 c e f+8 c d g-5 b e g) (d+e x)^2}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(2 c e f+8 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^2 (2 c d-b e)}+\frac {(2 c e f+8 c d g-5 b e g) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 c^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^4}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (2 c e f+8 c d g-5 b e g) (d+e x)^2}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(2 c e f+8 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^2 (2 c d-b e)}+\frac {(2 c e f+8 c d g-5 b e g) \operatorname {Subst}\left (\int \frac {1}{-4 c e^2-x^2} \, dx,x,\frac {-b e^2-2 c e^2 x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{c^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^4}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (2 c e f+8 c d g-5 b e g) (d+e x)^2}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(2 c e f+8 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^2 (2 c d-b e)}+\frac {(2 c e f+8 c d g-5 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{7/2} e^2}\\ \end {align*}
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Mathematica [C] time = 0.25, size = 139, normalized size = 0.48 \begin {gather*} \frac {2 (d+e x)^4 \left (\left (\frac {b e-c d+c e x}{b e-2 c d}\right )^{3/2} (-5 b e g+8 c d g+2 c e f) \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};\frac {c (d+e x)}{2 c d-b e}\right )-5 (-b e g+c d g+c e f)\right )}{15 c e^2 (b e-2 c d) ((d+e x) (c (d-e x)-b e))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 39.87, size = 24206, normalized size = 83.18 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.55, size = 881, normalized size = 3.03 \begin {gather*} \left [\frac {3 \, {\left ({\left (2 \, c^{3} e^{3} f + {\left (8 \, c^{3} d e^{2} - 5 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 2 \, {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + {\left (8 \, c^{3} d^{3} - 21 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g - 2 \, {\left (2 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} f + {\left (8 \, c^{3} d^{2} e - 13 \, b c^{2} d e^{2} + 5 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} + 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (3 \, c^{3} e^{2} g x^{2} + 2 \, {\left (2 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f + {\left (19 \, c^{3} d^{2} - 34 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g - 2 \, {\left (4 \, c^{3} e^{2} f + {\left (13 \, c^{3} d e - 10 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{12 \, {\left (c^{6} e^{4} x^{2} + c^{6} d^{2} e^{2} - 2 \, b c^{5} d e^{3} + b^{2} c^{4} e^{4} - 2 \, {\left (c^{6} d e^{3} - b c^{5} e^{4}\right )} x\right )}}, -\frac {3 \, {\left ({\left (2 \, c^{3} e^{3} f + {\left (8 \, c^{3} d e^{2} - 5 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 2 \, {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + {\left (8 \, c^{3} d^{3} - 21 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g - 2 \, {\left (2 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} f + {\left (8 \, c^{3} d^{2} e - 13 \, b c^{2} d e^{2} + 5 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \, {\left (3 \, c^{3} e^{2} g x^{2} + 2 \, {\left (2 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f + {\left (19 \, c^{3} d^{2} - 34 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g - 2 \, {\left (4 \, c^{3} e^{2} f + {\left (13 \, c^{3} d e - 10 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{6 \, {\left (c^{6} e^{4} x^{2} + c^{6} d^{2} e^{2} - 2 \, b c^{5} d e^{3} + b^{2} c^{4} e^{4} - 2 \, {\left (c^{6} d e^{3} - b c^{5} e^{4}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.87, size = 1116, normalized size = 3.84 \begin {gather*} -\frac {\sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left ({\left ({\left ({\left (\frac {3 \, {\left (16 \, c^{6} d^{4} g e^{8} - 32 \, b c^{5} d^{3} g e^{9} + 24 \, b^{2} c^{4} d^{2} g e^{10} - 8 \, b^{3} c^{3} d g e^{11} + b^{4} c^{2} g e^{12}\right )} x}{16 \, c^{7} d^{4} e^{6} - 32 \, b c^{6} d^{3} e^{7} + 24 \, b^{2} c^{5} d^{2} e^{8} - 8 \, b^{3} c^{4} d e^{9} + b^{4} c^{3} e^{10}} - \frac {4 \, {\left (80 \, c^{6} d^{5} g e^{7} + 32 \, c^{6} d^{4} f e^{8} - 240 \, b c^{5} d^{4} g e^{8} - 64 \, b c^{5} d^{3} f e^{9} + 280 \, b^{2} c^{4} d^{3} g e^{9} + 48 \, b^{2} c^{4} d^{2} f e^{10} - 160 \, b^{3} c^{3} d^{2} g e^{10} - 16 \, b^{3} c^{3} d f e^{11} + 45 \, b^{4} c^{2} d g e^{11} + 2 \, b^{4} c^{2} f e^{12} - 5 \, b^{5} c g e^{12}\right )}}{16 \, c^{7} d^{4} e^{6} - 32 \, b c^{6} d^{3} e^{7} + 24 \, b^{2} c^{5} d^{2} e^{8} - 8 \, b^{3} c^{4} d e^{9} + b^{4} c^{3} e^{10}}\right )} x - \frac {3 \, {\left (160 \, c^{6} d^{6} g e^{6} + 64 \, c^{6} d^{5} f e^{7} - 352 \, b c^{5} d^{5} g e^{7} - 96 \, b c^{5} d^{4} f e^{8} + 224 \, b^{2} c^{4} d^{4} g e^{8} + 32 \, b^{2} c^{4} d^{3} f e^{9} + 32 \, b^{3} c^{3} d^{3} g e^{9} + 16 \, b^{3} c^{3} d^{2} f e^{10} - 94 \, b^{4} c^{2} d^{2} g e^{10} - 12 \, b^{4} c^{2} d f e^{11} + 38 \, b^{5} c d g e^{11} + 2 \, b^{5} c f e^{12} - 5 \, b^{6} g e^{12}\right )}}{16 \, c^{7} d^{4} e^{6} - 32 \, b c^{6} d^{3} e^{7} + 24 \, b^{2} c^{5} d^{2} e^{8} - 8 \, b^{3} c^{4} d e^{9} + b^{4} c^{3} e^{10}}\right )} x + \frac {6 \, {\left (32 \, c^{6} d^{7} g e^{5} - 192 \, b c^{5} d^{6} g e^{6} - 32 \, b c^{5} d^{5} f e^{7} + 384 \, b^{2} c^{4} d^{5} g e^{7} + 64 \, b^{2} c^{4} d^{4} f e^{8} - 368 \, b^{3} c^{3} d^{4} g e^{8} - 48 \, b^{3} c^{3} d^{3} f e^{9} + 186 \, b^{4} c^{2} d^{3} g e^{9} + 16 \, b^{4} c^{2} d^{2} f e^{10} - 48 \, b^{5} c d^{2} g e^{10} - 2 \, b^{5} c d f e^{11} + 5 \, b^{6} d g e^{11}\right )}}{16 \, c^{7} d^{4} e^{6} - 32 \, b c^{6} d^{3} e^{7} + 24 \, b^{2} c^{5} d^{2} e^{8} - 8 \, b^{3} c^{4} d e^{9} + b^{4} c^{3} e^{10}}\right )} x + \frac {304 \, c^{6} d^{8} g e^{4} + 64 \, c^{6} d^{7} f e^{5} - 1152 \, b c^{5} d^{7} g e^{5} - 224 \, b c^{5} d^{6} f e^{6} + 1784 \, b^{2} c^{4} d^{6} g e^{6} + 288 \, b^{2} c^{4} d^{5} f e^{7} - 1448 \, b^{3} c^{3} d^{5} g e^{7} - 176 \, b^{3} c^{3} d^{4} f e^{8} + 651 \, b^{4} c^{2} d^{4} g e^{8} + 52 \, b^{4} c^{2} d^{3} f e^{9} - 154 \, b^{5} c d^{3} g e^{9} - 6 \, b^{5} c d^{2} f e^{10} + 15 \, b^{6} d^{2} g e^{10}}{16 \, c^{7} d^{4} e^{6} - 32 \, b c^{6} d^{3} e^{7} + 24 \, b^{2} c^{5} d^{2} e^{8} - 8 \, b^{3} c^{4} d e^{9} + b^{4} c^{3} e^{10}}\right )}}{3 \, {\left (c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e\right )}^{2}} + \frac {{\left (8 \, c d g + 2 \, c f e - 5 \, b g e\right )} \sqrt {-c e^{2}} e^{\left (-3\right )} \log \left ({\left | -2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c - \sqrt {-c e^{2}} b \right |}\right )}{2 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 5032, normalized size = 17.29 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^4}{{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{4} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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