Optimal. Leaf size=181 \[ -\frac{12 b^5 (d+e x)^{7/2} (b d-a e)}{7 e^7}+\frac{6 b^4 (d+e x)^{5/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{3/2} (b d-a e)^3}{3 e^7}+\frac{30 b^2 \sqrt{d+e x} (b d-a e)^4}{e^7}+\frac{12 b (b d-a e)^5}{e^7 \sqrt{d+e x}}-\frac{2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac{2 b^6 (d+e x)^{9/2}}{9 e^7} \]
[Out]
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Rubi [A] time = 0.172807, antiderivative size = 181, 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 \[ -\frac{12 b^5 (d+e x)^{7/2} (b d-a e)}{7 e^7}+\frac{6 b^4 (d+e x)^{5/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{3/2} (b d-a e)^3}{3 e^7}+\frac{30 b^2 \sqrt{d+e x} (b d-a e)^4}{e^7}+\frac{12 b (b d-a e)^5}{e^7 \sqrt{d+e x}}-\frac{2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac{2 b^6 (d+e x)^{9/2}}{9 e^7} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 77.1802, size = 168, normalized size = 0.93 \[ \frac{2 b^{6} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{7}} + \frac{12 b^{5} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )}{7 e^{7}} + \frac{6 b^{4} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}}{e^{7}} + \frac{40 b^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}}{3 e^{7}} + \frac{30 b^{2} \sqrt{d + e x} \left (a e - b d\right )^{4}}{e^{7}} - \frac{12 b \left (a e - b d\right )^{5}}{e^{7} \sqrt{d + e x}} - \frac{2 \left (a e - b d\right )^{6}}{3 e^{7} \left (d + e x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(5/2),x)
[Out]
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Mathematica [A] time = 0.331966, size = 220, normalized size = 1.22 \[ \frac{2 \sqrt{d+e x} \left (3 b^4 e^2 x^2 \left (63 a^2 e^2-72 a b d e+23 b^2 d^2\right )+2 b^3 e x \left (210 a^3 e^3-441 a^2 b d e^2+333 a b^2 d^2 e-88 b^3 d^3\right )+b^2 \left (945 a^4 e^4-3360 a^3 b d e^3+4599 a^2 b^2 d^2 e^2-2844 a b^3 d^3 e+667 b^4 d^4\right )-2 b^5 e^3 x^3 (13 b d-27 a e)+\frac{378 b (b d-a e)^5}{d+e x}-\frac{21 (b d-a e)^6}{(d+e x)^2}+7 b^6 e^4 x^4\right )}{63 e^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^(5/2),x]
[Out]
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Maple [B] time = 0.013, size = 377, normalized size = 2.1 \[ -{\frac{-14\,{x}^{6}{b}^{6}{e}^{6}-108\,{x}^{5}a{b}^{5}{e}^{6}+24\,{x}^{5}{b}^{6}d{e}^{5}-378\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+216\,{x}^{4}a{b}^{5}d{e}^{5}-48\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-840\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+1008\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-576\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+128\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-1890\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+5040\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-6048\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+3456\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-768\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+756\,x{a}^{5}b{e}^{6}-7560\,x{a}^{4}{b}^{2}d{e}^{5}+20160\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-24192\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+13824\,xa{b}^{5}{d}^{4}{e}^{2}-3072\,x{b}^{6}{d}^{5}e+42\,{a}^{6}{e}^{6}+504\,{a}^{5}bd{e}^{5}-5040\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+13440\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-16128\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+9216\,{d}^{5}a{b}^{5}e-2048\,{b}^{6}{d}^{6}}{63\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(5/2),x)
[Out]
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Maxima [A] time = 0.733804, size = 481, normalized size = 2.66 \[ \frac{2 \,{\left (\frac{7 \,{\left (e x + d\right )}^{\frac{9}{2}} b^{6} - 54 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 420 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 945 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} \sqrt{e x + d}}{e^{6}} - \frac{21 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6} - 18 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{6}}\right )}}{63 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206181, size = 494, normalized size = 2.73 \[ \frac{2 \,{\left (7 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 4608 \, a b^{5} d^{5} e + 8064 \, a^{2} b^{4} d^{4} e^{2} - 6720 \, a^{3} b^{3} d^{3} e^{3} + 2520 \, a^{4} b^{2} d^{2} e^{4} - 252 \, a^{5} b d e^{5} - 21 \, a^{6} e^{6} - 6 \,{\left (2 \, b^{6} d e^{5} - 9 \, a b^{5} e^{6}\right )} x^{5} + 3 \,{\left (8 \, b^{6} d^{2} e^{4} - 36 \, a b^{5} d e^{5} + 63 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \,{\left (16 \, b^{6} d^{3} e^{3} - 72 \, a b^{5} d^{2} e^{4} + 126 \, a^{2} b^{4} d e^{5} - 105 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{4} e^{2} - 576 \, a b^{5} d^{3} e^{3} + 1008 \, a^{2} b^{4} d^{2} e^{4} - 840 \, a^{3} b^{3} d e^{5} + 315 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \,{\left (256 \, b^{6} d^{5} e - 1152 \, a b^{5} d^{4} e^{2} + 2016 \, a^{2} b^{4} d^{3} e^{3} - 1680 \, a^{3} b^{3} d^{2} e^{4} + 630 \, a^{4} b^{2} d e^{5} - 63 \, a^{5} b e^{6}\right )} x\right )}}{63 \,{\left (e^{8} x + d e^{7}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{6}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217535, size = 624, normalized size = 3.45 \[ \frac{2}{63} \,{\left (7 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{6} e^{56} - 54 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} d e^{56} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d^{2} e^{56} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{3} e^{56} + 945 \, \sqrt{x e + d} b^{6} d^{4} e^{56} + 54 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{5} e^{57} - 378 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} d e^{57} + 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d^{2} e^{57} - 3780 \, \sqrt{x e + d} a b^{5} d^{3} e^{57} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{4} e^{58} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} d e^{58} + 5670 \, \sqrt{x e + d} a^{2} b^{4} d^{2} e^{58} + 420 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{3} e^{59} - 3780 \, \sqrt{x e + d} a^{3} b^{3} d e^{59} + 945 \, \sqrt{x e + d} a^{4} b^{2} e^{60}\right )} e^{\left (-63\right )} + \frac{2 \,{\left (18 \,{\left (x e + d\right )} b^{6} d^{5} - b^{6} d^{6} - 90 \,{\left (x e + d\right )} a b^{5} d^{4} e + 6 \, a b^{5} d^{5} e + 180 \,{\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} - 15 \, a^{2} b^{4} d^{4} e^{2} - 180 \,{\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{3} d^{3} e^{3} + 90 \,{\left (x e + d\right )} a^{4} b^{2} d e^{4} - 15 \, a^{4} b^{2} d^{2} e^{4} - 18 \,{\left (x e + d\right )} a^{5} b e^{5} + 6 \, a^{5} b d e^{5} - a^{6} e^{6}\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^(5/2),x, algorithm="giac")
[Out]