Optimal. Leaf size=129 \[ -\frac{8 b^3 (d+e x)^{13/2} (b d-a e)}{13 e^5}+\frac{12 b^2 (d+e x)^{11/2} (b d-a e)^2}{11 e^5}-\frac{8 b (d+e x)^{9/2} (b d-a e)^3}{9 e^5}+\frac{2 (d+e x)^{7/2} (b d-a e)^4}{7 e^5}+\frac{2 b^4 (d+e x)^{15/2}}{15 e^5} \]
[Out]
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Rubi [A] time = 0.146484, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{8 b^3 (d+e x)^{13/2} (b d-a e)}{13 e^5}+\frac{12 b^2 (d+e x)^{11/2} (b d-a e)^2}{11 e^5}-\frac{8 b (d+e x)^{9/2} (b d-a e)^3}{9 e^5}+\frac{2 (d+e x)^{7/2} (b d-a e)^4}{7 e^5}+\frac{2 b^4 (d+e x)^{15/2}}{15 e^5} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 56.5049, size = 119, normalized size = 0.92 \[ \frac{2 b^{4} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{5}} + \frac{8 b^{3} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )}{13 e^{5}} + \frac{12 b^{2} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{2}}{11 e^{5}} + \frac{8 b \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{3}}{9 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4}}{7 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.184956, size = 154, normalized size = 1.19 \[ \frac{2 (d+e x)^{7/2} \left (6435 a^4 e^4+2860 a^3 b e^3 (7 e x-2 d)+390 a^2 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+60 a b^3 e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^4 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Maple [A] time = 0.013, size = 186, normalized size = 1.4 \[{\frac{6006\,{x}^{4}{b}^{4}{e}^{4}+27720\,{x}^{3}a{b}^{3}{e}^{4}-3696\,{x}^{3}{b}^{4}d{e}^{3}+49140\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-15120\,{x}^{2}a{b}^{3}d{e}^{3}+2016\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+40040\,x{a}^{3}b{e}^{4}-21840\,x{a}^{2}{b}^{2}d{e}^{3}+6720\,xa{b}^{3}{d}^{2}{e}^{2}-896\,x{b}^{4}{d}^{3}e+12870\,{a}^{4}{e}^{4}-11440\,{a}^{3}bd{e}^{3}+6240\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-1920\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
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Maxima [A] time = 0.727806, size = 244, normalized size = 1.89 \[ \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} b^{4} - 13860 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 24570 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 20020 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{45045 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.207395, size = 509, normalized size = 3.95 \[ \frac{2 \,{\left (3003 \, b^{4} e^{7} x^{7} + 128 \, b^{4} d^{7} - 960 \, a b^{3} d^{6} e + 3120 \, a^{2} b^{2} d^{5} e^{2} - 5720 \, a^{3} b d^{4} e^{3} + 6435 \, a^{4} d^{3} e^{4} + 231 \,{\left (31 \, b^{4} d e^{6} + 60 \, a b^{3} e^{7}\right )} x^{6} + 63 \,{\left (71 \, b^{4} d^{2} e^{5} + 540 \, a b^{3} d e^{6} + 390 \, a^{2} b^{2} e^{7}\right )} x^{5} + 35 \,{\left (b^{4} d^{3} e^{4} + 636 \, a b^{3} d^{2} e^{5} + 1794 \, a^{2} b^{2} d e^{6} + 572 \, a^{3} b e^{7}\right )} x^{4} - 5 \,{\left (8 \, b^{4} d^{4} e^{3} - 60 \, a b^{3} d^{3} e^{4} - 8814 \, a^{2} b^{2} d^{2} e^{5} - 10868 \, a^{3} b d e^{6} - 1287 \, a^{4} e^{7}\right )} x^{3} + 3 \,{\left (16 \, b^{4} d^{5} e^{2} - 120 \, a b^{3} d^{4} e^{3} + 390 \, a^{2} b^{2} d^{3} e^{4} + 14300 \, a^{3} b d^{2} e^{5} + 6435 \, a^{4} d e^{6}\right )} x^{2} -{\left (64 \, b^{4} d^{6} e - 480 \, a b^{3} d^{5} e^{2} + 1560 \, a^{2} b^{2} d^{4} e^{3} - 2860 \, a^{3} b d^{3} e^{4} - 19305 \, a^{4} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.3666, size = 960, normalized size = 7.44 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.233968, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(5/2),x, algorithm="giac")
[Out]