Optimal. Leaf size=78 \[ \frac {8 (2 c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.03, 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 = {728, 636} \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 636
Rule 728
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 = 0.04, size = 95, normalized size = 1.22 \begin {gather*} \frac {-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 )+16 b c^2 d x^2 (3 d-2 e x)+32 c^3 d^2 x^3}{3 b^4 (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 125, normalized size = 1.60 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (b^3 \left (-d^2\right )-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 x^2 (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 129, normalized size = 1.65 \begin {gather*} -\frac {2 \, {\left (b^{3} d^{2} - 2 \, {\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2}\right )} x^{3} - 3 \, {\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2}\right )} x^{2} - 6 \, {\left (b^{2} c d^{2} - b^{3} d e\right )} x\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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giac [A] time = 0.27, 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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maple [A] time = 0.05, size = 117, normalized size = 1.50 \begin {gather*} -\frac {2 \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 ) x}{3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.33, 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 e x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {32 \, c d e x}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {4 \, e^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, e^{2} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \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 {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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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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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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