Optimal. Leaf size=143 \[ \frac {2 \sqrt {d+e x} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^5}-\frac {2 d^2 (c d-b e)^2}{3 e^5 (d+e x)^{3/2}}-\frac {4 c (d+e x)^{3/2} (2 c d-b e)}{3 e^5}+\frac {4 d (c d-b e) (2 c d-b e)}{e^5 \sqrt {d+e x}}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5} \]
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Rubi [A] time = 0.06, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {698} \begin {gather*} \frac {2 \sqrt {d+e x} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^5}-\frac {2 d^2 (c d-b e)^2}{3 e^5 (d+e x)^{3/2}}-\frac {4 c (d+e x)^{3/2} (2 c d-b e)}{3 e^5}+\frac {4 d (c d-b e) (2 c d-b e)}{e^5 \sqrt {d+e x}}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx &=\int \left (\frac {d^2 (c d-b e)^2}{e^4 (d+e x)^{5/2}}+\frac {2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^{3/2}}+\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 \sqrt {d+e x}}-\frac {2 c (2 c d-b e) \sqrt {d+e x}}{e^4}+\frac {c^2 (d+e x)^{3/2}}{e^4}\right ) \, dx\\ &=-\frac {2 d^2 (c d-b e)^2}{3 e^5 (d+e x)^{3/2}}+\frac {4 d (c d-b e) (2 c d-b e)}{e^5 \sqrt {d+e x}}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \sqrt {d+e x}}{e^5}-\frac {4 c (2 c d-b e) (d+e x)^{3/2}}{3 e^5}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 123, normalized size = 0.86 \begin {gather*} \frac {2 \left (5 b^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+10 b c e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+c^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 164, normalized size = 1.15 \begin {gather*} \frac {2 \left (-5 b^2 d^2 e^2+30 b^2 d e^2 (d+e x)+15 b^2 e^2 (d+e x)^2+10 b c d^3 e-90 b c d^2 e (d+e x)-90 b c d e (d+e x)^2+10 b c e (d+e x)^3-5 c^2 d^4+60 c^2 d^3 (d+e x)+90 c^2 d^2 (d+e x)^2-20 c^2 d (d+e x)^3+3 c^2 (d+e x)^4\right )}{15 e^5 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 159, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} e^{4} x^{4} + 128 \, c^{2} d^{4} - 160 \, b c d^{3} e + 40 \, b^{2} d^{2} e^{2} - 2 \, {\left (4 \, c^{2} d e^{3} - 5 \, b c e^{4}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{2} e^{2} - 20 \, b c d e^{3} + 5 \, b^{2} e^{4}\right )} x^{2} + 12 \, {\left (16 \, c^{2} d^{3} e - 20 \, b c d^{2} e^{2} + 5 \, b^{2} d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 182, normalized size = 1.27 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} e^{20} - 20 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d e^{20} + 90 \, \sqrt {x e + d} c^{2} d^{2} e^{20} + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} b c e^{21} - 90 \, \sqrt {x e + d} b c d e^{21} + 15 \, \sqrt {x e + d} b^{2} e^{22}\right )} e^{\left (-25\right )} + \frac {2 \, {\left (12 \, {\left (x e + d\right )} c^{2} d^{3} - c^{2} d^{4} - 18 \, {\left (x e + d\right )} b c d^{2} e + 2 \, b c d^{3} e + 6 \, {\left (x e + d\right )} b^{2} d e^{2} - b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\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 = 141, normalized size = 0.99 \begin {gather*} \frac {\frac {2}{5} c^{2} x^{4} e^{4}+\frac {4}{3} b c \,e^{4} x^{3}-\frac {16}{15} c^{2} d \,e^{3} x^{3}+2 b^{2} e^{4} x^{2}-8 b c d \,e^{3} x^{2}+\frac {32}{5} c^{2} d^{2} e^{2} x^{2}+8 b^{2} d \,e^{3} x -32 b c \,d^{2} e^{2} x +\frac {128}{5} c^{2} d^{3} e x +\frac {16}{3} b^{2} d^{2} e^{2}-\frac {64}{3} b c \,d^{3} e +\frac {256}{15} c^{2} d^{4}}{\left (e x +d \right )^{\frac {3}{2}} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 145, normalized size = 1.01 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} - 10 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \sqrt {e x + d}}{e^{4}} - \frac {5 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} - 6 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{4}}\right )}}{15 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 145, normalized size = 1.01 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {\sqrt {d+e\,x}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {\left (d+e\,x\right )\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )-\frac {2\,c^2\,d^4}{3}-\frac {2\,b^2\,d^2\,e^2}{3}+\frac {4\,b\,c\,d^3\,e}{3}}{e^5\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 34.66, size = 139, normalized size = 0.97 \begin {gather*} \frac {2 c^{2} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{5}} - \frac {2 d^{2} \left (b e - c d\right )^{2}}{3 e^{5} \left (d + e x\right )^{\frac {3}{2}}} + \frac {4 d \left (b e - 2 c d\right ) \left (b e - c d\right )}{e^{5} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (4 b c e - 8 c^{2} d\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (2 b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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