Optimal. Leaf size=147 \[ \frac{2 (d+e x)^{13/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{13 e^5}+\frac{2 d^2 (d+e x)^{9/2} (c d-b e)^2}{9 e^5}-\frac{4 c (d+e x)^{15/2} (2 c d-b e)}{15 e^5}-\frac{4 d (d+e x)^{11/2} (c d-b e) (2 c d-b e)}{11 e^5}+\frac{2 c^2 (d+e x)^{17/2}}{17 e^5} \]
[Out]
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Rubi [A] time = 0.228194, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 (d+e x)^{13/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{13 e^5}+\frac{2 d^2 (d+e x)^{9/2} (c d-b e)^2}{9 e^5}-\frac{4 c (d+e x)^{15/2} (2 c d-b e)}{15 e^5}-\frac{4 d (d+e x)^{11/2} (c d-b e) (2 c d-b e)}{11 e^5}+\frac{2 c^2 (d+e x)^{17/2}}{17 e^5} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(7/2)*(b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 35.0533, size = 141, normalized size = 0.96 \[ \frac{2 c^{2} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{15}{2}} \left (b e - 2 c d\right )}{15 e^{5}} + \frac{2 d^{2} \left (d + e x\right )^{\frac{9}{2}} \left (b e - c d\right )^{2}}{9 e^{5}} - \frac{4 d \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )}{11 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{13}{2}} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{13 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(7/2)*(c*x**2+b*x)**2,x)
[Out]
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Mathematica [A] time = 0.154503, size = 124, normalized size = 0.84 \[ \frac{2 (d+e x)^{9/2} \left (85 b^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+34 b c e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+c^2 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )}{109395 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(7/2)*(b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.009, size = 141, normalized size = 1. \[{\frac{12870\,{c}^{2}{x}^{4}{e}^{4}+29172\,bc{e}^{4}{x}^{3}-6864\,{c}^{2}d{e}^{3}{x}^{3}+16830\,{b}^{2}{e}^{4}{x}^{2}-13464\,bcd{e}^{3}{x}^{2}+3168\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-6120\,{b}^{2}d{e}^{3}x+4896\,bc{d}^{2}{e}^{2}x-1152\,{c}^{2}{d}^{3}ex+1360\,{b}^{2}{d}^{2}{e}^{2}-1088\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{109395\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(7/2)*(c*x^2+b*x)^2,x)
[Out]
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Maxima [A] time = 0.693652, size = 188, normalized size = 1.28 \[ \frac{2 \,{\left (6435 \,{\left (e x + d\right )}^{\frac{17}{2}} c^{2} - 14586 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 8415 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 19890 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 12155 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{109395 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212804, size = 392, normalized size = 2.67 \[ \frac{2 \,{\left (6435 \, c^{2} e^{8} x^{8} + 128 \, c^{2} d^{8} - 544 \, b c d^{7} e + 680 \, b^{2} d^{6} e^{2} + 858 \,{\left (26 \, c^{2} d e^{7} + 17 \, b c e^{8}\right )} x^{7} + 33 \,{\left (802 \, c^{2} d^{2} e^{6} + 1564 \, b c d e^{7} + 255 \, b^{2} e^{8}\right )} x^{6} + 36 \,{\left (303 \, c^{2} d^{3} e^{5} + 1751 \, b c d^{2} e^{6} + 850 \, b^{2} d e^{7}\right )} x^{5} + 5 \,{\left (7 \, c^{2} d^{4} e^{4} + 5440 \, b c d^{3} e^{5} + 7786 \, b^{2} d^{2} e^{6}\right )} x^{4} - 10 \,{\left (4 \, c^{2} d^{5} e^{3} - 17 \, b c d^{4} e^{4} - 1802 \, b^{2} d^{3} e^{5}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{6} e^{2} - 68 \, b c d^{5} e^{3} + 85 \, b^{2} d^{4} e^{4}\right )} x^{2} - 4 \,{\left (16 \, c^{2} d^{7} e - 68 \, b c d^{6} e^{2} + 85 \, b^{2} d^{5} e^{3}\right )} x\right )} \sqrt{e x + d}}{109395 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(e*x + d)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 37.7433, size = 590, normalized size = 4.01 \[ \begin{cases} \frac{16 b^{2} d^{6} \sqrt{d + e x}}{1287 e^{3}} - \frac{8 b^{2} d^{5} x \sqrt{d + e x}}{1287 e^{2}} + \frac{2 b^{2} d^{4} x^{2} \sqrt{d + e x}}{429 e} + \frac{424 b^{2} d^{3} x^{3} \sqrt{d + e x}}{1287} + \frac{916 b^{2} d^{2} e x^{4} \sqrt{d + e x}}{1287} + \frac{80 b^{2} d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{2 b^{2} e^{3} x^{6} \sqrt{d + e x}}{13} - \frac{64 b c d^{7} \sqrt{d + e x}}{6435 e^{4}} + \frac{32 b c d^{6} x \sqrt{d + e x}}{6435 e^{3}} - \frac{8 b c d^{5} x^{2} \sqrt{d + e x}}{2145 e^{2}} + \frac{4 b c d^{4} x^{3} \sqrt{d + e x}}{1287 e} + \frac{640 b c d^{3} x^{4} \sqrt{d + e x}}{1287} + \frac{824 b c d^{2} e x^{5} \sqrt{d + e x}}{715} + \frac{184 b c d e^{2} x^{6} \sqrt{d + e x}}{195} + \frac{4 b c e^{3} x^{7} \sqrt{d + e x}}{15} + \frac{256 c^{2} d^{8} \sqrt{d + e x}}{109395 e^{5}} - \frac{128 c^{2} d^{7} x \sqrt{d + e x}}{109395 e^{4}} + \frac{32 c^{2} d^{6} x^{2} \sqrt{d + e x}}{36465 e^{3}} - \frac{16 c^{2} d^{5} x^{3} \sqrt{d + e x}}{21879 e^{2}} + \frac{14 c^{2} d^{4} x^{4} \sqrt{d + e x}}{21879 e} + \frac{2424 c^{2} d^{3} x^{5} \sqrt{d + e x}}{12155} + \frac{1604 c^{2} d^{2} e x^{6} \sqrt{d + e x}}{3315} + \frac{104 c^{2} d e^{2} x^{7} \sqrt{d + e x}}{255} + \frac{2 c^{2} e^{3} x^{8} \sqrt{d + e x}}{17} & \text{for}\: e \neq 0 \\d^{\frac{7}{2}} \left (\frac{b^{2} x^{3}}{3} + \frac{b c x^{4}}{2} + \frac{c^{2} x^{5}}{5}\right ) & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(7/2)*(c*x**2+b*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.224329, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(e*x + d)^(7/2),x, algorithm="giac")
[Out]