Optimal. Leaf size=78 \[ -\frac {2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (b d+(2 c d-b e) x)}{3 b^4 \sqrt {b x+c x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {742, 650}
\begin {gather*} \frac {8 (2 c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 742
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {(4 (2 c d-b e)) \int \frac {d+e x}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac {2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (b d+(2 c d-b e) x)}{3 b^4 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 95, normalized size = 1.22 \begin {gather*} \frac {32 c^3 d^2 x^3+16 b c^2 d x^2 (3 d-2 e x)-2 b^3 \left (d^2+6 d e x-3 e^2 x^2\right )+4 b^2 c x \left (3 d^2-12 d e x+e^2 x^2\right )}{3 b^4 (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. \(221\) vs.
\(2(70)=140\).
time = 0.47, size = 222, normalized size = 2.85
method | result | size |
risch | \(-\frac {2 d \left (c x +b \right ) \left (6 b e x -8 c d x +b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}+\frac {2 x \left (2 b e x c -8 c^{2} d x +3 b^{2} e -9 b c d \right ) \left (b e -c d \right )}{3 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right ) b^{4}}\) | \(94\) |
gosper | \(-\frac {2 x \left (c x +b \right ) \left (-2 b^{2} c \,e^{2} x^{3}+16 b \,c^{2} d e \,x^{3}-16 c^{3} d^{2} x^{3}-3 b^{3} e^{2} x^{2}+24 b^{2} c d e \,x^{2}-24 b \,c^{2} d^{2} x^{2}+6 b^{3} d e x -6 b^{2} c \,d^{2} x +d^{2} b^{3}\right )}{3 b^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\) | \(117\) |
trager | \(-\frac {2 \left (-2 b^{2} c \,e^{2} x^{3}+16 b \,c^{2} d e \,x^{3}-16 c^{3} d^{2} x^{3}-3 b^{3} e^{2} x^{2}+24 b^{2} c d e \,x^{2}-24 b \,c^{2} d^{2} x^{2}+6 b^{3} d e x -6 b^{2} c \,d^{2} x +d^{2} b^{3}\right ) \sqrt {c \,x^{2}+b x}}{3 b^{4} x^{2} \left (c x +b \right )^{2}}\) | \(121\) |
default | \(e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )+2 d e \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )+d^{2} \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )\) | \(222\) |
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. 203 vs.
\(2 (73) = 146\).
time = 0.28, size = 203, normalized size = 2.60 \begin {gather*} -\frac {4 \, c d^{2} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, c^{2} d^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {4 \, d x e}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {32 \, c d x e}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {2 \, d^{2}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, c d^{2}}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {4 \, x e^{2}}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, x e^{2}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {16 \, d e}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, e^{2}}{3 \, \sqrt {c x^{2} + b x} b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.62, size = 137, normalized size = 1.76 \begin {gather*} \frac {2 \, {\left (16 \, c^{3} d^{2} x^{3} + 24 \, b c^{2} d^{2} x^{2} + 6 \, b^{2} c d^{2} x - b^{3} d^{2} + {\left (2 \, b^{2} c x^{3} + 3 \, b^{3} x^{2}\right )} e^{2} - 2 \, {\left (8 \, b c^{2} d x^{3} + 12 \, b^{2} c d x^{2} + 3 \, b^{3} d x\right )} e\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\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 )^{2}}{\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 = 1.29, size = 111, normalized size = 1.42 \begin {gather*} \frac {2 \, {\left ({\left (x {\left (\frac {2 \, {\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2}\right )} x}{b^{4}} + \frac {3 \, {\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2}\right )}}{b^{4}}\right )} + \frac {6 \, {\left (b^{2} c d^{2} - b^{3} d e\right )}}{b^{4}}\right )} x - \frac {d^{2}}{b}\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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Mupad [B]
time = 0.36, size = 111, normalized size = 1.42 \begin {gather*} \frac {2\,\left (-b^3\,d^2-6\,b^3\,d\,e\,x+3\,b^3\,e^2\,x^2+6\,b^2\,c\,d^2\,x-24\,b^2\,c\,d\,e\,x^2+2\,b^2\,c\,e^2\,x^3+24\,b\,c^2\,d^2\,x^2-16\,b\,c^2\,d\,e\,x^3+16\,c^3\,d^2\,x^3\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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