Optimal. Leaf size=248 \[ \frac{6 c (d+e x)^{13/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{11 e^7}+\frac{2 d (d+e x)^{9/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac{2 d^3 (d+e x)^{5/2} (c d-b e)^3}{5 e^7}-\frac{6 d^2 (d+e x)^{7/2} (c d-b e)^2 (2 c d-b e)}{7 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7} \]
[Out]
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Rubi [A] time = 0.308744, antiderivative size = 248, 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 \[ \frac{6 c (d+e x)^{13/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{11 e^7}+\frac{2 d (d+e x)^{9/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac{2 d^3 (d+e x)^{5/2} (c d-b e)^3}{5 e^7}-\frac{6 d^2 (d+e x)^{7/2} (c d-b e)^2 (2 c d-b e)}{7 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)*(b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 58.0707, size = 243, normalized size = 0.98 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{7}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{15}{2}} \left (b e - 2 c d\right )}{5 e^{7}} + \frac{6 c \left (d + e x\right )^{\frac{13}{2}} \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{13 e^{7}} - \frac{2 d^{3} \left (d + e x\right )^{\frac{5}{2}} \left (b e - c d\right )^{3}}{5 e^{7}} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{7 e^{7}} - \frac{2 d \left (d + e x\right )^{\frac{9}{2}} \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{3 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{11 e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)*(c*x**2+b*x)**3,x)
[Out]
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Mathematica [A] time = 0.204397, size = 231, normalized size = 0.93 \[ \frac{2 (d+e x)^{5/2} \left (221 b^3 e^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+51 b^2 c e^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+17 b c^2 e \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )+c^3 \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )}{255255 e^7} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)*(b*x + c*x^2)^3,x]
[Out]
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Maple [A] time = 0.011, size = 286, normalized size = 1.2 \[ -{\frac{-30030\,{c}^{3}{x}^{6}{e}^{6}-102102\,b{c}^{2}{e}^{6}{x}^{5}+24024\,{c}^{3}d{e}^{5}{x}^{5}-117810\,{b}^{2}c{e}^{6}{x}^{4}+78540\,b{c}^{2}d{e}^{5}{x}^{4}-18480\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-46410\,{b}^{3}{e}^{6}{x}^{3}+85680\,{b}^{2}cd{e}^{5}{x}^{3}-57120\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+13440\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+30940\,{b}^{3}d{e}^{5}{x}^{2}-57120\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+38080\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-8960\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-17680\,{b}^{3}{d}^{2}{e}^{4}x+32640\,{b}^{2}c{d}^{3}{e}^{3}x-21760\,b{c}^{2}{d}^{4}{e}^{2}x+5120\,{c}^{3}{d}^{5}ex+7072\,{b}^{3}{d}^{3}{e}^{3}-13056\,{b}^{2}c{d}^{4}{e}^{2}+8704\,b{c}^{2}{d}^{5}e-2048\,{c}^{3}{d}^{6}}{255255\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)*(c*x^2+b*x)^3,x)
[Out]
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Maxima [A] time = 0.706785, size = 366, normalized size = 1.48 \[ \frac{2 \,{\left (15015 \,{\left (e x + d\right )}^{\frac{17}{2}} c^{3} - 51051 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 58905 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 23205 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 85085 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 109395 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 51051 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{255255 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3*(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232089, size = 504, normalized size = 2.03 \[ \frac{2 \,{\left (15015 \, c^{3} e^{8} x^{8} + 1024 \, c^{3} d^{8} - 4352 \, b c^{2} d^{7} e + 6528 \, b^{2} c d^{6} e^{2} - 3536 \, b^{3} d^{5} e^{3} + 3003 \,{\left (6 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \,{\left (c^{3} d^{2} e^{6} + 272 \, b c^{2} d e^{7} + 255 \, b^{2} c e^{8}\right )} x^{6} - 21 \,{\left (12 \, c^{3} d^{3} e^{5} - 51 \, b c^{2} d^{2} e^{6} - 3570 \, b^{2} c d e^{7} - 1105 \, b^{3} e^{8}\right )} x^{5} + 35 \,{\left (8 \, c^{3} d^{4} e^{4} - 34 \, b c^{2} d^{3} e^{5} + 51 \, b^{2} c d^{2} e^{6} + 884 \, b^{3} d e^{7}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{5} e^{3} - 272 \, b c^{2} d^{4} e^{4} + 408 \, b^{2} c d^{3} e^{5} - 221 \, b^{3} d^{2} e^{6}\right )} x^{3} + 6 \,{\left (64 \, c^{3} d^{6} e^{2} - 272 \, b c^{2} d^{5} e^{3} + 408 \, b^{2} c d^{4} e^{4} - 221 \, b^{3} d^{3} e^{5}\right )} x^{2} - 8 \,{\left (64 \, c^{3} d^{7} e - 272 \, b c^{2} d^{6} e^{2} + 408 \, b^{2} c d^{5} e^{3} - 221 \, b^{3} d^{4} e^{4}\right )} x\right )} \sqrt{e x + d}}{255255 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3*(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.50949, size = 738, normalized size = 2.98 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(3/2)*(c*x**2+b*x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.218541, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3*(e*x + d)^(3/2),x, algorithm="giac")
[Out]