Optimal. Leaf size=267 \[ -\frac{2 (d+e x)^{13/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{13 e^6}+\frac{2 (d+e x)^{11/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{11 e^6}-\frac{2 d^2 (d+e x)^{7/2} (B d-A e) (c d-b e)^2}{7 e^6}-\frac{2 c (d+e x)^{15/2} (-A c e-2 b B e+5 B c d)}{15 e^6}+\frac{2 d (d+e x)^{9/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{9 e^6}+\frac{2 B c^2 (d+e x)^{17/2}}{17 e^6} \]
[Out]
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Rubi [A] time = 0.479387, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 (d+e x)^{13/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{13 e^6}+\frac{2 (d+e x)^{11/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{11 e^6}-\frac{2 d^2 (d+e x)^{7/2} (B d-A e) (c d-b e)^2}{7 e^6}-\frac{2 c (d+e x)^{15/2} (-A c e-2 b B e+5 B c d)}{15 e^6}+\frac{2 d (d+e x)^{9/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{9 e^6}+\frac{2 B c^2 (d+e x)^{17/2}}{17 e^6} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^(5/2)*(b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 107.65, size = 292, normalized size = 1.09 \[ \frac{2 B c^{2} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{6}} + \frac{2 c \left (d + e x\right )^{\frac{15}{2}} \left (A c e + 2 B b e - 5 B c d\right )}{15 e^{6}} + \frac{2 d^{2} \left (d + e x\right )^{\frac{7}{2}} \left (A e - B d\right ) \left (b e - c d\right )^{2}}{7 e^{6}} - \frac{2 d \left (d + e x\right )^{\frac{9}{2}} \left (b e - c d\right ) \left (2 A b e^{2} - 4 A c d e - 3 B b d e + 5 B c d^{2}\right )}{9 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{13}{2}} \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{13 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{11 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(5/2)*(c*x**2+b*x)**2,x)
[Out]
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Mathematica [A] time = 0.478347, size = 273, normalized size = 1.02 \[ \frac{2 (d+e x)^{7/2} \left (17 A e \left (65 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+30 b c e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )+B \left (255 b^2 e^2 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+34 b c e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )-5 c^2 \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )\right )}{765765 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^(5/2)*(b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.011, size = 341, normalized size = 1.3 \[{\frac{90090\,B{c}^{2}{x}^{5}{e}^{5}+102102\,A{c}^{2}{e}^{5}{x}^{4}+204204\,Bbc{e}^{5}{x}^{4}-60060\,B{c}^{2}d{e}^{4}{x}^{4}+235620\,Abc{e}^{5}{x}^{3}-62832\,A{c}^{2}d{e}^{4}{x}^{3}+117810\,B{b}^{2}{e}^{5}{x}^{3}-125664\,Bbcd{e}^{4}{x}^{3}+36960\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+139230\,A{b}^{2}{e}^{5}{x}^{2}-128520\,Abcd{e}^{4}{x}^{2}+34272\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-64260\,B{b}^{2}d{e}^{4}{x}^{2}+68544\,Bbc{d}^{2}{e}^{3}{x}^{2}-20160\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-61880\,A{b}^{2}d{e}^{4}x+57120\,Abc{d}^{2}{e}^{3}x-15232\,A{c}^{2}{d}^{3}{e}^{2}x+28560\,B{b}^{2}{d}^{2}{e}^{3}x-30464\,Bbc{d}^{3}{e}^{2}x+8960\,B{c}^{2}{d}^{4}ex+17680\,A{b}^{2}{d}^{2}{e}^{3}-16320\,Abc{d}^{3}{e}^{2}+4352\,A{c}^{2}{d}^{4}e-8160\,B{b}^{2}{d}^{3}{e}^{2}+8704\,Bbc{d}^{4}e-2560\,B{c}^{2}{d}^{5}}{765765\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(5/2)*(c*x^2+b*x)^2,x)
[Out]
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Maxima [A] time = 0.695844, size = 393, normalized size = 1.47 \[ \frac{2 \,{\left (45045 \,{\left (e x + d\right )}^{\frac{17}{2}} B c^{2} - 51051 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 58905 \,{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 69615 \,{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 85085 \,{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 109395 \,{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{765765 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)*(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.273526, size = 667, normalized size = 2.5 \[ \frac{2 \,{\left (45045 \, B c^{2} e^{8} x^{8} - 1280 \, B c^{2} d^{8} + 8840 \, A b^{2} d^{5} e^{3} + 2176 \,{\left (2 \, B b c + A c^{2}\right )} d^{7} e - 4080 \,{\left (B b^{2} + 2 \, A b c\right )} d^{6} e^{2} + 3003 \,{\left (35 \, B c^{2} d e^{7} + 17 \,{\left (2 \, B b c + A c^{2}\right )} e^{8}\right )} x^{7} + 231 \,{\left (275 \, B c^{2} d^{2} e^{6} + 527 \,{\left (2 \, B b c + A c^{2}\right )} d e^{7} + 255 \,{\left (B b^{2} + 2 \, A b c\right )} e^{8}\right )} x^{6} + 63 \,{\left (5 \, B c^{2} d^{3} e^{5} + 1105 \, A b^{2} e^{8} + 1207 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{6} + 2295 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{7}\right )} x^{5} - 35 \,{\left (10 \, B c^{2} d^{4} e^{4} - 5083 \, A b^{2} d e^{7} - 17 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{5} - 2703 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{6}\right )} x^{4} + 5 \,{\left (80 \, B c^{2} d^{5} e^{3} + 24973 \, A b^{2} d^{2} e^{6} - 136 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e^{4} + 255 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{5}\right )} x^{3} - 3 \,{\left (160 \, B c^{2} d^{6} e^{2} - 1105 \, A b^{2} d^{3} e^{5} - 272 \,{\left (2 \, B b c + A c^{2}\right )} d^{5} e^{3} + 510 \,{\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{4}\right )} x^{2} + 4 \,{\left (160 \, B c^{2} d^{7} e - 1105 \, A b^{2} d^{4} e^{4} - 272 \,{\left (2 \, B b c + A c^{2}\right )} d^{6} e^{2} + 510 \,{\left (B b^{2} + 2 \, A b c\right )} d^{5} e^{3}\right )} x\right )} \sqrt{e x + d}}{765765 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)*(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.8542, size = 1556, normalized size = 5.83 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**(5/2)*(c*x**2+b*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.312511, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)*(e*x + d)^(5/2),x, algorithm="giac")
[Out]