Optimal. Leaf size=75 \[ \frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}{7 e^3}-\frac{2 (d+e x)^{9/2} (2 c d-b e)}{9 e^3}+\frac{2 c (d+e x)^{11/2}}{11 e^3} \]
[Out]
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Rubi [A] time = 0.102478, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}{7 e^3}-\frac{2 (d+e x)^{9/2} (2 c d-b e)}{9 e^3}+\frac{2 c (d+e x)^{11/2}}{11 e^3} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(5/2)*(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 14.9424, size = 70, normalized size = 0.93 \[ \frac{2 c \left (d + e x\right )^{\frac{11}{2}}}{11 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right )}{9 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - b d e + c d^{2}\right )}{7 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(5/2)*(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.0957415, size = 55, normalized size = 0.73 \[ \frac{2 (d+e x)^{7/2} \left (11 e (9 a e-2 b d+7 b e x)+c \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(5/2)*(a + b*x + c*x^2),x]
[Out]
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Maple [A] time = 0.006, size = 53, normalized size = 0.7 \[{\frac{126\,c{e}^{2}{x}^{2}+154\,b{e}^{2}x-56\,cdex+198\,a{e}^{2}-44\,bde+16\,c{d}^{2}}{693\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(5/2)*(c*x^2+b*x+a),x)
[Out]
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Maxima [A] time = 0.676716, size = 80, normalized size = 1.07 \[ \frac{2 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} c - 77 \,{\left (2 \, c d - b e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 99 \,{\left (c d^{2} - b d e + a e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{693 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20465, size = 201, normalized size = 2.68 \[ \frac{2 \,{\left (63 \, c e^{5} x^{5} + 8 \, c d^{5} - 22 \, b d^{4} e + 99 \, a d^{3} e^{2} + 7 \,{\left (23 \, c d e^{4} + 11 \, b e^{5}\right )} x^{4} +{\left (113 \, c d^{2} e^{3} + 209 \, b d e^{4} + 99 \, a e^{5}\right )} x^{3} + 3 \,{\left (c d^{3} e^{2} + 55 \, b d^{2} e^{3} + 99 \, a d e^{4}\right )} x^{2} -{\left (4 \, c d^{4} e - 11 \, b d^{3} e^{2} - 297 \, a d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.35318, size = 326, normalized size = 4.35 \[ \begin{cases} \frac{2 a d^{3} \sqrt{d + e x}}{7 e} + \frac{6 a d^{2} x \sqrt{d + e x}}{7} + \frac{6 a d e x^{2} \sqrt{d + e x}}{7} + \frac{2 a e^{2} x^{3} \sqrt{d + e x}}{7} - \frac{4 b d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{2 b d^{3} x \sqrt{d + e x}}{63 e} + \frac{10 b d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{38 b d e x^{3} \sqrt{d + e x}}{63} + \frac{2 b e^{2} x^{4} \sqrt{d + e x}}{9} + \frac{16 c d^{5} \sqrt{d + e x}}{693 e^{3}} - \frac{8 c d^{4} x \sqrt{d + e x}}{693 e^{2}} + \frac{2 c d^{3} x^{2} \sqrt{d + e x}}{231 e} + \frac{226 c d^{2} x^{3} \sqrt{d + e x}}{693} + \frac{46 c d e x^{4} \sqrt{d + e x}}{99} + \frac{2 c e^{2} x^{5} \sqrt{d + e x}}{11} & \text{for}\: e \neq 0 \\d^{\frac{5}{2}} \left (a x + \frac{b x^{2}}{2} + \frac{c x^{3}}{3}\right ) & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(5/2)*(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.21437, size = 560, normalized size = 7.47 \[ \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b d^{2} e^{\left (-1\right )} + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} c d^{2} e^{\left (-14\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a d^{2} + 66 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} b d e^{\left (-13\right )} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} c d e^{\left (-26\right )} + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a d + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a e^{\left (-12\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} b e^{\left (-25\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} c e^{\left (-42\right )}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)^(5/2),x, algorithm="giac")
[Out]