Optimal. Leaf size=230 \[ -\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac {2 \left (c x (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (c d-b e)^2}+\frac {e^4 \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {740, 822, 12, 724, 206} \begin {gather*} \frac {2 \left (c x (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (c d-b e)^2}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac {e^4 \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 740
Rule 822
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\begin {align*} \int \frac {1}{(d+e x) \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-4 b c d e-3 b^2 e^2\right )+2 c e (2 c d-b e) x}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-4 b c d e-3 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {4 \int \frac {3 b^4 e^4}{4 (d+e x) \sqrt {b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-4 b c d e-3 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {e^4 \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{d^2 (c d-b e)^2}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-4 b c d e-3 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {\left (2 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{d^2 (c d-b e)^2}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-4 b c d e-3 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {e^4 \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{d^{5/2} (c d-b e)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 238, normalized size = 1.03 \begin {gather*} \frac {6 b^4 e^4 x^{3/2} (b+c x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )+2 \sqrt {d} \sqrt {c d-b e} \left (b^5 e^2 (3 e x-d)+2 b^4 c e \left (d^2+3 e^2 x^2\right )-b^3 c^2 \left (d^3+9 d^2 e x-3 d e^2 x^2-3 e^3 x^3\right )+2 b^2 c^3 d x \left (3 d^2-18 d e x+e^2 x^2\right )+24 b c^4 d^2 x^2 (d-e x)+16 c^5 d^3 x^3\right )}{3 b^4 d^{5/2} (x (b+c x))^{3/2} (c d-b e)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.83, size = 319, normalized size = 1.39 \begin {gather*} \frac {2 e^4 \tanh ^{-1}\left (-\frac {e \sqrt {b x+c x^2}}{\sqrt {d} \sqrt {c d-b e}}+\frac {\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}+\frac {\sqrt {c} \sqrt {d}}{\sqrt {c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}}-\frac {2 \sqrt {b x+c x^2} \left (b^5 d e^2-3 b^5 e^3 x-2 b^4 c d^2 e-6 b^4 c e^3 x^2+b^3 c^2 d^3+9 b^3 c^2 d^2 e x-3 b^3 c^2 d e^2 x^2-3 b^3 c^2 e^3 x^3-6 b^2 c^3 d^3 x+36 b^2 c^3 d^2 e x^2-2 b^2 c^3 d e^2 x^3-24 b c^4 d^3 x^2+24 b c^4 d^2 e x^3-16 c^5 d^3 x^3\right )}{3 b^4 d^2 x^2 (b+c x)^2 (b e-c d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 1017, normalized size = 4.42 \begin {gather*} \left [\frac {3 \, {\left (b^{4} c^{2} e^{4} x^{4} + 2 \, b^{5} c e^{4} x^{3} + b^{6} e^{4} x^{2}\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) - 2 \, {\left (b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - {\left (16 \, c^{6} d^{5} - 40 \, b c^{5} d^{4} e + 26 \, b^{2} c^{4} d^{3} e^{2} + b^{3} c^{3} d^{2} e^{3} - 3 \, b^{4} c^{2} d e^{4}\right )} x^{3} - 3 \, {\left (8 \, b c^{5} d^{5} - 20 \, b^{2} c^{4} d^{4} e + 13 \, b^{3} c^{3} d^{3} e^{2} + b^{4} c^{2} d^{2} e^{3} - 2 \, b^{5} c d e^{4}\right )} x^{2} - 3 \, {\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + b^{5} c d^{2} e^{3} - b^{6} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left ({\left (b^{4} c^{5} d^{6} - 3 \, b^{5} c^{4} d^{5} e + 3 \, b^{6} c^{3} d^{4} e^{2} - b^{7} c^{2} d^{3} e^{3}\right )} x^{4} + 2 \, {\left (b^{5} c^{4} d^{6} - 3 \, b^{6} c^{3} d^{5} e + 3 \, b^{7} c^{2} d^{4} e^{2} - b^{8} c d^{3} e^{3}\right )} x^{3} + {\left (b^{6} c^{3} d^{6} - 3 \, b^{7} c^{2} d^{5} e + 3 \, b^{8} c d^{4} e^{2} - b^{9} d^{3} e^{3}\right )} x^{2}\right )}}, \frac {2 \, {\left (3 \, {\left (b^{4} c^{2} e^{4} x^{4} + 2 \, b^{5} c e^{4} x^{3} + b^{6} e^{4} x^{2}\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) - {\left (b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - {\left (16 \, c^{6} d^{5} - 40 \, b c^{5} d^{4} e + 26 \, b^{2} c^{4} d^{3} e^{2} + b^{3} c^{3} d^{2} e^{3} - 3 \, b^{4} c^{2} d e^{4}\right )} x^{3} - 3 \, {\left (8 \, b c^{5} d^{5} - 20 \, b^{2} c^{4} d^{4} e + 13 \, b^{3} c^{3} d^{3} e^{2} + b^{4} c^{2} d^{2} e^{3} - 2 \, b^{5} c d e^{4}\right )} x^{2} - 3 \, {\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + b^{5} c d^{2} e^{3} - b^{6} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}\right )}}{3 \, {\left ({\left (b^{4} c^{5} d^{6} - 3 \, b^{5} c^{4} d^{5} e + 3 \, b^{6} c^{3} d^{4} e^{2} - b^{7} c^{2} d^{3} e^{3}\right )} x^{4} + 2 \, {\left (b^{5} c^{4} d^{6} - 3 \, b^{6} c^{3} d^{5} e + 3 \, b^{7} c^{2} d^{4} e^{2} - b^{8} c d^{3} e^{3}\right )} x^{3} + {\left (b^{6} c^{3} d^{6} - 3 \, b^{7} c^{2} d^{5} e + 3 \, b^{8} c d^{4} e^{2} - b^{9} d^{3} e^{3}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 628, normalized size = 2.73 \begin {gather*} -\frac {2 \, \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right ) e^{4}}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e}} + \frac {2 \, {\left ({\left ({\left (\frac {{\left (16 \, c^{7} d^{10} - 56 \, b c^{6} d^{9} e + 66 \, b^{2} c^{5} d^{8} e^{2} - 25 \, b^{3} c^{4} d^{7} e^{3} - 4 \, b^{4} c^{3} d^{6} e^{4} + 3 \, b^{5} c^{2} d^{5} e^{5}\right )} x}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}} + \frac {3 \, {\left (8 \, b c^{6} d^{10} - 28 \, b^{2} c^{5} d^{9} e + 33 \, b^{3} c^{4} d^{8} e^{2} - 12 \, b^{4} c^{3} d^{7} e^{3} - 3 \, b^{5} c^{2} d^{6} e^{4} + 2 \, b^{6} c d^{5} e^{5}\right )}}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c^{5} d^{10} - 7 \, b^{3} c^{4} d^{9} e + 8 \, b^{4} c^{3} d^{8} e^{2} - 2 \, b^{5} c^{2} d^{7} e^{3} - 2 \, b^{6} c d^{6} e^{4} + b^{7} d^{5} e^{5}\right )}}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}}\right )} x - \frac {b^{3} c^{4} d^{10} - 4 \, b^{4} c^{3} d^{9} e + 6 \, b^{5} c^{2} d^{8} e^{2} - 4 \, b^{6} c d^{7} e^{3} + b^{7} d^{6} e^{4}}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}}\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 950, normalized size = 4.13 \begin {gather*} \frac {2 c \,e^{3} x}{\left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b \,d^{2}}-\frac {4 c^{2} e^{2} x}{\left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b^{2} d}-\frac {e^{3} \ln \left (\frac {-\frac {2 \left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}}{x +\frac {d}{e}}\right )}{\left (b e -c d \right )^{2} \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, d^{2}}-\frac {2 c \,e^{2}}{\left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b d}-\frac {2 c e x}{3 \left (b e -c d \right ) \left (\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}\right )^{\frac {3}{2}} b d}+\frac {4 c^{2} x}{3 \left (b e -c d \right ) \left (\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}\right )^{\frac {3}{2}} b^{2}}+\frac {2 e^{3}}{\left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, d^{2}}+\frac {2 c}{3 \left (b e -c d \right ) \left (\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}\right )^{\frac {3}{2}} b}+\frac {16 c^{2} e x}{3 \left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b^{3} d}-\frac {32 c^{3} x}{3 \left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b^{4}}-\frac {2 e}{3 \left (b e -c d \right ) \left (\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}\right )^{\frac {3}{2}} d}+\frac {8 c e}{3 \left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b^{2} d}-\frac {16 c^{2}}{3 \left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b^{3}} \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 {1}{{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (d+e\,x\right )} \,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 {1}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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