Optimal. Leaf size=368 \[ -\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^{3/2}}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^7 (a+b x)}-\frac{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^7 (a+b x)}+\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) \sqrt{d+e x}} \]
[Out]
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Rubi [A] time = 0.397082, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ -\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^{3/2}}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^7 (a+b x)}-\frac{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^7 (a+b x)}+\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 52.857, size = 308, normalized size = 0.84 \[ \frac{256 b^{4} \left (3 a + 3 b x\right ) \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{5}} + \frac{1024 b^{4} \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{6}} + \frac{2048 b^{4} \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{7} \left (a + b x\right )} - \frac{128 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7 e^{4} \sqrt{d + e x}} - \frac{16 b^{2} \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{35 e^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{24 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{35 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{7 e \left (d + e x\right )^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(9/2),x)
[Out]
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Mathematica [A] time = 0.330782, size = 173, normalized size = 0.47 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (7 b^4 \left (75 a^2 e^2-140 a b d e+66 b^2 d^2\right )-14 b^5 e x (4 b d-5 a e)+\frac{700 b^3 (b d-a e)^3}{d+e x}-\frac{175 b^2 (b d-a e)^4}{(d+e x)^2}+\frac{42 b (b d-a e)^5}{(d+e x)^3}-\frac{5 (b d-a e)^6}{(d+e x)^4}+7 b^6 e^2 x^2\right )}{35 e^7 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(9/2),x]
[Out]
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Maple [A] time = 0.013, size = 393, normalized size = 1.1 \[ -{\frac{-14\,{x}^{6}{b}^{6}{e}^{6}-140\,{x}^{5}a{b}^{5}{e}^{6}+56\,{x}^{5}{b}^{6}d{e}^{5}-1050\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+1400\,{x}^{4}a{b}^{5}d{e}^{5}-560\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+1400\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-8400\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+11200\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-4480\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+350\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+2800\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-16800\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+22400\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-8960\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+84\,x{a}^{5}b{e}^{6}+280\,x{a}^{4}{b}^{2}d{e}^{5}+2240\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-13440\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+17920\,xa{b}^{5}{d}^{4}{e}^{2}-7168\,x{b}^{6}{d}^{5}e+10\,{a}^{6}{e}^{6}+24\,{a}^{5}bd{e}^{5}+80\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+640\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-3840\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+5120\,{d}^{5}a{b}^{5}e-2048\,{b}^{6}{d}^{6}}{35\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(9/2),x)
[Out]
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Maxima [A] time = 0.752028, size = 902, normalized size = 2.45 \[ \frac{2 \,{\left (7 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e - 96 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 6 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 35 \,{\left (2 \, b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} - 70 \,{\left (8 \, b^{5} d^{2} e^{3} - 12 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 70 \,{\left (16 \, b^{5} d^{3} e^{2} - 24 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} - 7 \,{\left (128 \, b^{5} d^{4} e - 192 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 8 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x\right )} a}{21 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (21 \, b^{5} e^{6} x^{6} + 3072 \, b^{5} d^{6} - 6400 \, a b^{4} d^{5} e + 3840 \, a^{2} b^{3} d^{4} e^{2} - 480 \, a^{3} b^{2} d^{3} e^{3} - 40 \, a^{4} b d^{2} e^{4} - 6 \, a^{5} d e^{5} - 7 \,{\left (12 \, b^{5} d e^{5} - 25 \, a b^{4} e^{6}\right )} x^{5} + 70 \,{\left (12 \, b^{5} d^{2} e^{4} - 25 \, a b^{4} d e^{5} + 15 \, a^{2} b^{3} e^{6}\right )} x^{4} + 70 \,{\left (96 \, b^{5} d^{3} e^{3} - 200 \, a b^{4} d^{2} e^{4} + 120 \, a^{2} b^{3} d e^{5} - 15 \, a^{3} b^{2} e^{6}\right )} x^{3} + 35 \,{\left (384 \, b^{5} d^{4} e^{2} - 800 \, a b^{4} d^{3} e^{3} + 480 \, a^{2} b^{3} d^{2} e^{4} - 60 \, a^{3} b^{2} d e^{5} - 5 \, a^{4} b e^{6}\right )} x^{2} + 7 \,{\left (1536 \, b^{5} d^{5} e - 3200 \, a b^{4} d^{4} e^{2} + 1920 \, a^{2} b^{3} d^{3} e^{3} - 240 \, a^{3} b^{2} d^{2} e^{4} - 20 \, a^{4} b d e^{5} - 3 \, a^{5} e^{6}\right )} x\right )} b}{105 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(9/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.295607, size = 524, normalized size = 1.42 \[ \frac{2 \,{\left (7 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 2560 \, a b^{5} d^{5} e + 1920 \, a^{2} b^{4} d^{4} e^{2} - 320 \, a^{3} b^{3} d^{3} e^{3} - 40 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 5 \, a^{6} e^{6} - 14 \,{\left (2 \, b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 35 \,{\left (8 \, b^{6} d^{2} e^{4} - 20 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 140 \,{\left (16 \, b^{6} d^{3} e^{3} - 40 \, a b^{5} d^{2} e^{4} + 30 \, a^{2} b^{4} d e^{5} - 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 35 \,{\left (128 \, b^{6} d^{4} e^{2} - 320 \, a b^{5} d^{3} e^{3} + 240 \, a^{2} b^{4} d^{2} e^{4} - 40 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 14 \,{\left (256 \, b^{6} d^{5} e - 640 \, a b^{5} d^{4} e^{2} + 480 \, a^{2} b^{4} d^{3} e^{3} - 80 \, a^{3} b^{3} d^{2} e^{4} - 10 \, a^{4} b^{2} d e^{5} - 3 \, a^{5} b e^{6}\right )} x\right )}}{35 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(9/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(9/2),x)
[Out]
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GIAC/XCAS [A] time = 0.319695, size = 844, normalized size = 2.29 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(9/2),x, algorithm="giac")
[Out]