Optimal. Leaf size=208 \[ -\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (b c d^2 (4 c d-5 b e)+(2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^4 c^2}+\frac {2 e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 208, 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 = {752, 832, 654,
634, 212} \begin {gather*} -\frac {2 (d+e x)^3 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 e \sqrt {b x+c x^2} (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{3 b^4 c^2}+\frac {4 (d+e x) \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{3 b^4 c \sqrt {b x+c x^2}}+\frac {2 e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 654
Rule 752
Rule 832
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {(d+e x)^2 (d (4 c d-5 b e)-e (2 c d-b e) x)}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (b c d^2 (4 c d-5 b e)+(2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {4 \int \frac {\frac {1}{2} b d e \left (8 c^2 d^2-12 b c d e+b^2 e^2\right )+\frac {1}{2} e (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c}\\ &=-\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (b c d^2 (4 c d-5 b e)+(2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^4 c^2}+\frac {e^4 \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{c^2}\\ &=-\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (b c d^2 (4 c d-5 b e)+(2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^4 c^2}+\frac {\left (2 e^4\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c^2}\\ &=-\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (b c d^2 (4 c d-5 b e)+(2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^4 c^2}+\frac {2 e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 195, normalized size = 0.94 \begin {gather*} -\frac {2 \left (\sqrt {c} \left (3 b^5 e^4 x^2-16 c^5 d^4 x^3+4 b^4 c e^4 x^3+8 b c^4 d^3 x^2 (-3 d+4 e x)-6 b^2 c^3 d^2 x \left (d^2-8 d e x+2 e^2 x^2\right )+b^3 c^2 d \left (d^3+12 d^2 e x-18 d e^2 x^2-4 e^3 x^3\right )\right )+3 b^4 e^4 x^{3/2} (b+c x)^{3/2} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{3 b^4 c^{5/2} (x (b+c x))^{3/2}} \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. \(598\) vs.
\(2(190)=380\).
time = 0.49, size = 599, normalized size = 2.88
method | result | size |
risch | \(-\frac {2 d^{3} \left (c x +b \right ) \left (12 b e x -8 c d x +b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}+\frac {e^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {5}{2}}}+\frac {2 b \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, e^{4}}{3 c^{4} \left (\frac {b}{c}+x \right )^{2}}-\frac {8 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d \,e^{3}}{3 c^{3} \left (\frac {b}{c}+x \right )^{2}}+\frac {4 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{2} e^{2}}{b \,c^{2} \left (\frac {b}{c}+x \right )^{2}}-\frac {8 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{3} e}{3 b^{2} c \left (\frac {b}{c}+x \right )^{2}}+\frac {2 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{4}}{3 b^{3} \left (\frac {b}{c}+x \right )^{2}}-\frac {8 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, e^{4}}{3 c^{3} \left (\frac {b}{c}+x \right )}+\frac {8 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d \,e^{3}}{3 b \,c^{2} \left (\frac {b}{c}+x \right )}+\frac {8 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{2} e^{2}}{b^{2} c \left (\frac {b}{c}+x \right )}-\frac {40 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{3} e}{3 b^{3} \left (\frac {b}{c}+x \right )}+\frac {16 c \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{4}}{3 b^{4} \left (\frac {b}{c}+x \right )}\) | \(506\) |
default | \(e^{4} \left (-\frac {x^{3}}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )}{2 c}+\frac {-\frac {x}{c \sqrt {c \,x^{2}+b x}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x}}+\frac {2 c x +b}{b c \sqrt {c \,x^{2}+b x}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {3}{2}}}}{c}\right )+4 d \,e^{3} \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )+6 d^{2} e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )+4 d^{3} e \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )+d^{4} \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )\) | \(599\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 450 vs.
\(2 (196) = 392\).
time = 0.29, size = 450, normalized size = 2.16 \begin {gather*} -\frac {1}{3} \, x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )} e^{4} - \frac {4 \, c d^{4} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, c^{2} d^{4} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {8 \, d^{3} x e}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {64 \, c d^{3} x e}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {2 \, d^{4}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, c d^{4}}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {4 \, d x^{2} e^{3}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {8 \, d^{2} x e^{2}}{\sqrt {c x^{2} + b x} b^{2}} - \frac {4 \, d^{2} x e^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {32 \, d^{3} e}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {4 \, b d x e^{3}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} + \frac {8 \, d x e^{3}}{3 \, \sqrt {c x^{2} + b x} b c} + \frac {4 \, d^{2} e^{2}}{\sqrt {c x^{2} + b x} b c} - \frac {4 \, x e^{4}}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {4 \, d e^{3}}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {e^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {5}{2}}} - \frac {2 \, \sqrt {c x^{2} + b x} e^{4}}{3 \, b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.71, size = 542, normalized size = 2.61 \begin {gather*} \left [\frac {3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )} \sqrt {c} e^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (16 \, c^{6} d^{4} x^{3} + 24 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + 4 \, b^{3} c^{3} d x^{3} e^{3} - b^{3} c^{3} d^{4} - {\left (4 \, b^{4} c^{2} x^{3} + 3 \, b^{5} c x^{2}\right )} e^{4} + 6 \, {\left (2 \, b^{2} c^{4} d^{2} x^{3} + 3 \, b^{3} c^{3} d^{2} x^{2}\right )} e^{2} - 4 \, {\left (8 \, b c^{5} d^{3} x^{3} + 12 \, b^{2} c^{4} d^{3} x^{2} + 3 \, b^{3} c^{3} d^{3} x\right )} e\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) e^{4} - {\left (16 \, c^{6} d^{4} x^{3} + 24 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + 4 \, b^{3} c^{3} d x^{3} e^{3} - b^{3} c^{3} d^{4} - {\left (4 \, b^{4} c^{2} x^{3} + 3 \, b^{5} c x^{2}\right )} e^{4} + 6 \, {\left (2 \, b^{2} c^{4} d^{2} x^{3} + 3 \, b^{3} c^{3} d^{2} x^{2}\right )} e^{2} - 4 \, {\left (8 \, b c^{5} d^{3} x^{3} + 12 \, b^{2} c^{4} d^{3} x^{2} + 3 \, b^{3} c^{3} d^{3} x\right )} e\right )} \sqrt {c x^{2} + b x}\right )}}{3 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\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 {\left (d + e x\right )^{4}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.36, size = 209, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (\frac {d^{4}}{b} - {\left (x {\left (\frac {4 \, {\left (4 \, c^{5} d^{4} - 8 \, b c^{4} d^{3} e + 3 \, b^{2} c^{3} d^{2} e^{2} + b^{3} c^{2} d e^{3} - b^{4} c e^{4}\right )} x}{b^{4} c^{2}} + \frac {3 \, {\left (8 \, b c^{4} d^{4} - 16 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - b^{5} e^{4}\right )}}{b^{4} c^{2}}\right )} + \frac {6 \, {\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e\right )}}{b^{4} c^{2}}\right )} x\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} - \frac {e^{4} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {5}{2}}} \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 (d+e\,x\right )}^4}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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