Optimal. Leaf size=183 \[ \frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {b^2 (2 c d-b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{16 d^{5/2} (c d-b e)^{5/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {744, 734, 738,
212} \begin {gather*} -\frac {b^2 (2 c d-b e) \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 )}{16 d^{5/2} (c d-b e)^{5/2}}+\frac {\sqrt {b x+c x^2} (2 c d-b e) (x (2 c d-b e)+b d)}{8 d^2 (d+e x)^2 (c d-b e)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (d+e x)^3 (c d-b e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 734
Rule 738
Rule 744
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx &=-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {(2 c d-b e) \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx}{2 d (c d-b e)}\\ &=\frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {\left (b^2 (2 c d-b e)\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{16 d^2 (c d-b e)^2}\\ &=\frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {\left (b^2 (2 c d-b e)\right ) \text {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 )}{8 d^2 (c d-b e)^2}\\ &=\frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {b^2 (2 c d-b e) \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 )}{16 d^{5/2} (c d-b e)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 10.24, size = 191, normalized size = 1.04 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {e x^{3/2} (b+c x)}{(d+e x)^3}-\frac {3 (2 c d-b e) \left (\sqrt {d} \sqrt {c d-b e} \sqrt {x} \sqrt {b+c x} (2 c d x+b (d-e x))-b^2 (d+e x)^2 \tanh ^{-1}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )}{8 d^{3/2} (c d-b e)^{3/2} \sqrt {b+c x} (d+e x)^2}\right )}{3 d (-c d+b e) \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1084\) vs.
\(2(161)=322\).
time = 0.48, size = 1085, normalized size = 5.93
method | result | size |
default | \(\frac {\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{3 d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {e \left (b e -2 c d \right ) \left (\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{2 d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {e \left (b e -2 c d \right ) \left (\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {e \left (b e -2 c d \right ) \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}-\frac {2 c \,e^{2} \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{d \left (b e -c d \right )}\right )}{4 d \left (b e -c d \right )}-\frac {c \,e^{2} \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}\right )}{2 d \left (b e -c d \right )}}{e^{4}}\) | \(1085\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 473 vs.
\(2 (172) = 344\).
time = 1.19, size = 958, normalized size = 5.23 \begin {gather*} \left [-\frac {3 \, {\left (2 \, b^{2} c d^{4} - b^{3} x^{3} e^{4} + {\left (2 \, b^{2} c d x^{3} - 3 \, b^{3} d x^{2}\right )} e^{3} + 3 \, {\left (2 \, b^{2} c d^{2} x^{2} - b^{3} d^{2} x\right )} e^{2} + {\left (6 \, b^{2} c d^{3} x - b^{3} d^{3}\right )} e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {2 \, c d x - b x e + b d + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{x e + d}\right ) - 2 \, {\left (12 \, c^{3} d^{5} x + 6 \, b c^{2} d^{5} - 3 \, b^{3} d x^{2} e^{4} + {\left (7 \, b^{2} c d^{2} x^{2} - 8 \, b^{3} d^{2} x\right )} e^{3} - {\left (8 \, b c^{2} d^{3} x^{2} - 22 \, b^{2} c d^{3} x - 3 \, b^{3} d^{3}\right )} e^{2} + {\left (4 \, c^{3} d^{4} x^{2} - 26 \, b c^{2} d^{4} x - 9 \, b^{2} c d^{4}\right )} e\right )} \sqrt {c x^{2} + b x}}{48 \, {\left (c^{3} d^{9} - b^{3} d^{3} x^{3} e^{6} + 3 \, {\left (b^{2} c d^{4} x^{3} - b^{3} d^{4} x^{2}\right )} e^{5} - 3 \, {\left (b c^{2} d^{5} x^{3} - 3 \, b^{2} c d^{5} x^{2} + b^{3} d^{5} x\right )} e^{4} + {\left (c^{3} d^{6} x^{3} - 9 \, b c^{2} d^{6} x^{2} + 9 \, b^{2} c d^{6} x - b^{3} d^{6}\right )} e^{3} + 3 \, {\left (c^{3} d^{7} x^{2} - 3 \, b c^{2} d^{7} x + b^{2} c d^{7}\right )} e^{2} + 3 \, {\left (c^{3} d^{8} x - b c^{2} d^{8}\right )} e\right )}}, -\frac {3 \, {\left (2 \, b^{2} c d^{4} - b^{3} x^{3} e^{4} + {\left (2 \, b^{2} c d x^{3} - 3 \, b^{3} d x^{2}\right )} e^{3} + 3 \, {\left (2 \, b^{2} c d^{2} x^{2} - b^{3} d^{2} x\right )} e^{2} + {\left (6 \, b^{2} c d^{3} x - b^{3} d^{3}\right )} e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) - {\left (12 \, c^{3} d^{5} x + 6 \, b c^{2} d^{5} - 3 \, b^{3} d x^{2} e^{4} + {\left (7 \, b^{2} c d^{2} x^{2} - 8 \, b^{3} d^{2} x\right )} e^{3} - {\left (8 \, b c^{2} d^{3} x^{2} - 22 \, b^{2} c d^{3} x - 3 \, b^{3} d^{3}\right )} e^{2} + {\left (4 \, c^{3} d^{4} x^{2} - 26 \, b c^{2} d^{4} x - 9 \, b^{2} c d^{4}\right )} e\right )} \sqrt {c x^{2} + b x}}{24 \, {\left (c^{3} d^{9} - b^{3} d^{3} x^{3} e^{6} + 3 \, {\left (b^{2} c d^{4} x^{3} - b^{3} d^{4} x^{2}\right )} e^{5} - 3 \, {\left (b c^{2} d^{5} x^{3} - 3 \, b^{2} c d^{5} x^{2} + b^{3} d^{5} x\right )} e^{4} + {\left (c^{3} d^{6} x^{3} - 9 \, b c^{2} d^{6} x^{2} + 9 \, b^{2} c d^{6} x - b^{3} d^{6}\right )} e^{3} + 3 \, {\left (c^{3} d^{7} x^{2} - 3 \, b c^{2} d^{7} x + b^{2} c d^{7}\right )} e^{2} + 3 \, {\left (c^{3} d^{8} x - b c^{2} d^{8}\right )} e\right )}}\right ] \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 {\sqrt {x \left (b + c x\right )}}{\left (d + e x\right )^{4}}\, dx \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. 825 vs.
\(2 (172) = 344\).
time = 3.23, size = 825, normalized size = 4.51 \begin {gather*} \frac {{\left (2 \, b^{2} c d - b^{3} e\right )} \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 )}{8 \, {\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 {48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} c^{\frac {7}{2}} d^{4} e + 32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{4} d^{5} + 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c^{3} d^{4} e + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c^{\frac {7}{2}} d^{5} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b c^{\frac {5}{2}} d^{3} e^{2} - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} c^{\frac {5}{2}} d^{4} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c^{3} d^{5} - 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} c^{2} d^{3} e^{2} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} c^{2} d^{4} e + 4 \, b^{3} c^{\frac {5}{2}} d^{5} + 78 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b^{2} c^{\frac {3}{2}} d^{2} e^{3} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{3} c^{\frac {3}{2}} d^{3} e^{2} - 4 \, b^{4} c^{\frac {3}{2}} d^{4} e + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} b^{2} c d e^{4} + 74 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{3} c d^{2} e^{3} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{4} c d^{3} e^{2} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b^{3} \sqrt {c} d e^{4} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{4} \sqrt {c} d^{2} e^{3} + 3 \, b^{5} \sqrt {c} d^{3} e^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} b^{3} e^{5} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{4} d e^{4} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{5} d^{2} e^{3}}{24 \, {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{3}} \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 = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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