Optimal. Leaf size=145 \[ \frac {2 (d+e x)^{5/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{5 e^5}+\frac {2 d^2 \sqrt {d+e x} (c d-b e)^2}{e^5}-\frac {4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}-\frac {4 d (d+e x)^{3/2} (c d-b e) (2 c d-b e)}{3 e^5}+\frac {2 c^2 (d+e x)^{9/2}}{9 e^5} \]
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Rubi [A] time = 0.06, antiderivative size = 145, 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 (d+e x)^{5/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{5 e^5}+\frac {2 d^2 \sqrt {d+e x} (c d-b e)^2}{e^5}-\frac {4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}-\frac {4 d (d+e x)^{3/2} (c d-b e) (2 c d-b e)}{3 e^5}+\frac {2 c^2 (d+e x)^{9/2}}{9 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}{\sqrt {d+e x}} \, dx &=\int \left (\frac {d^2 (c d-b e)^2}{e^4 \sqrt {d+e x}}+\frac {2 d (c d-b e) (-2 c d+b e) \sqrt {d+e x}}{e^4}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{5/2}}{e^4}+\frac {c^2 (d+e x)^{7/2}}{e^4}\right ) \, dx\\ &=\frac {2 d^2 (c d-b e)^2 \sqrt {d+e x}}{e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{3/2}}{3 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{7/2}}{7 e^5}+\frac {2 c^2 (d+e x)^{9/2}}{9 e^5}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 124, normalized size = 0.86 \begin {gather*} \frac {2 \sqrt {d+e x} \left (21 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 b c e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 164, normalized size = 1.13 \begin {gather*} \frac {2 \sqrt {d+e x} \left (315 b^2 d^2 e^2-210 b^2 d e^2 (d+e x)+63 b^2 e^2 (d+e x)^2-630 b c d^3 e+630 b c d^2 e (d+e x)-378 b c d e (d+e x)^2+90 b c e (d+e x)^3+315 c^2 d^4-420 c^2 d^3 (d+e x)+378 c^2 d^2 (d+e x)^2-180 c^2 d (d+e x)^3+35 c^2 (d+e x)^4\right )}{315 e^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 138, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (35 \, c^{2} e^{4} x^{4} + 128 \, c^{2} d^{4} - 288 \, b c d^{3} e + 168 \, b^{2} d^{2} e^{2} - 10 \, {\left (4 \, c^{2} d e^{3} - 9 \, b c e^{4}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{2} e^{2} - 36 \, b c d e^{3} + 21 \, b^{2} e^{4}\right )} x^{2} - 4 \, {\left (16 \, c^{2} d^{3} e - 36 \, b c d^{2} e^{2} + 21 \, b^{2} d e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 168, normalized size = 1.16 \begin {gather*} \frac {2}{315} \, {\left (21 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} b^{2} e^{\left (-2\right )} + 18 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} b c e^{\left (-3\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c^{2} e^{\left (-4\right )}\right )} e^{\left (-1\right )} \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 = 0.97 \begin {gather*} \frac {2 \left (35 c^{2} x^{4} e^{4}+90 b c \,e^{4} x^{3}-40 c^{2} d \,e^{3} x^{3}+63 b^{2} e^{4} x^{2}-108 b c d \,e^{3} x^{2}+48 c^{2} d^{2} e^{2} x^{2}-84 b^{2} d \,e^{3} x +144 b c \,d^{2} e^{2} x -64 c^{2} d^{3} e x +168 b^{2} d^{2} e^{2}-288 b c \,d^{3} e +128 c^{2} d^{4}\right ) \sqrt {e x +d}}{315 e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 160, normalized size = 1.10 \begin {gather*} \frac {2 \, {\left (\frac {21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2}}{e^{2}} + \frac {18 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b c}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )}}{315 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 138, normalized size = 0.95 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{3\,e^5}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{5\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}}{e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 51.40, size = 418, normalized size = 2.88 \begin {gather*} \begin {cases} \frac {- \frac {2 b^{2} d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {2 b^{2} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {4 b c d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {4 b c \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {2 c^{2} d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{4}} - \frac {2 c^{2} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}}}{e} & \text {for}\: e \neq 0 \\\frac {\frac {b^{2} x^{3}}{3} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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