Optimal. Leaf size=302 \[ \frac{15 e^4 (a+b x) (b d-a e)^2 \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{20 e^3 (b d-a e)^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^5 x (a+b x) (6 b d-5 a e)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^6 x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.58308, antiderivative size = 302, 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 \[ \frac{15 e^4 (a+b x) (b d-a e)^2 \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{20 e^3 (b d-a e)^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^5 x (a+b x) (6 b d-5 a e)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^6 x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 53.929, size = 284, normalized size = 0.94 \[ - \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{6}}{8 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{e \left (d + e x\right )^{5}}{2 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 e^{2} \left (2 a + 2 b x\right ) \left (d + e x\right )^{4}}{8 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 e^{3} \left (d + e x\right )^{3}}{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{15 e^{4} \left (2 a + 2 b x\right ) \left (d + e x\right )^{2}}{4 b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{15 e^{5} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{7}} + \frac{15 e^{4} \left (a + b x\right ) \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{7} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**6/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.294074, size = 313, normalized size = 1.04 \[ \frac{57 a^6 e^6+14 a^5 b e^5 (12 e x-11 d)+a^4 b^2 e^4 \left (125 d^2-496 d e x+132 e^2 x^2\right )-4 a^3 b^3 e^3 \left (5 d^3-110 d^2 e x+126 d e^2 x^2+8 e^3 x^3\right )-a^2 b^4 e^2 \left (5 d^4+80 d^3 e x-540 d^2 e^2 x^2+96 d e^3 x^3+68 e^4 x^4\right )-2 a b^5 e \left (d^5+10 d^4 e x+60 d^3 e^2 x^2-120 d^2 e^3 x^3-48 d e^4 x^4+6 e^5 x^5\right )+60 e^4 (a+b x)^4 (b d-a e)^2 \log (a+b x)+b^6 \left (-\left (d^6+8 d^5 e x+30 d^4 e^2 x^2+80 d^3 e^3 x^3-24 d e^5 x^5-2 e^6 x^6\right )\right )}{4 b^7 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.027, size = 661, normalized size = 2.2 \[{\frac{ \left ( -480\,\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{4}d{e}^{5}+240\,\ln \left ( bx+a \right ){x}^{3}a{b}^{5}{d}^{2}{e}^{4}-720\,\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{3}d{e}^{5}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-480\,\ln \left ( bx+a \right ) x{a}^{4}{b}^{2}d{e}^{5}+240\,\ln \left ( bx+a \right ) x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+57\,{a}^{6}{e}^{6}-{b}^{6}{d}^{6}-12\,{x}^{5}a{b}^{5}{e}^{6}+24\,{x}^{5}{b}^{6}d{e}^{5}-80\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-32\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-68\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+132\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-30\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+168\,x{a}^{5}b{e}^{6}-8\,x{b}^{6}{d}^{5}e-496\,x{a}^{4}{b}^{2}d{e}^{5}+240\,\ln \left ( bx+a \right ){x}^{3}{a}^{3}{b}^{3}{e}^{6}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{4}{b}^{2}{e}^{6}+240\,\ln \left ( bx+a \right ) x{a}^{5}b{e}^{6}-120\,\ln \left ( bx+a \right ){a}^{5}bd{e}^{5}+60\,\ln \left ( bx+a \right ){a}^{4}{b}^{2}{d}^{2}{e}^{4}+540\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-120\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-2\,{d}^{5}a{b}^{5}e-20\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-5\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+60\,\ln \left ( bx+a \right ){a}^{6}{e}^{6}-120\,\ln \left ( bx+a \right ){x}^{4}a{b}^{5}d{e}^{5}+440\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-80\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-20\,xa{b}^{5}{d}^{4}{e}^{2}+240\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+60\,\ln \left ( bx+a \right ){x}^{4}{a}^{2}{b}^{4}{e}^{6}+60\,\ln \left ( bx+a \right ){x}^{4}{b}^{6}{d}^{2}{e}^{4}-504\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+96\,{x}^{4}a{b}^{5}d{e}^{5}-96\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+125\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-154\,{a}^{5}bd{e}^{5}+2\,{x}^{6}{b}^{6}{e}^{6} \right ) \left ( bx+a \right ) }{4\,{b}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.789447, size = 844, normalized size = 2.79 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.212642, size = 772, normalized size = 2.56 \[ \frac{2 \, b^{6} e^{6} x^{6} - b^{6} d^{6} - 2 \, a b^{5} d^{5} e - 5 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 125 \, a^{4} b^{2} d^{2} e^{4} - 154 \, a^{5} b d e^{5} + 57 \, a^{6} e^{6} + 12 \,{\left (2 \, b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 4 \,{\left (24 \, a b^{5} d e^{5} - 17 \, a^{2} b^{4} e^{6}\right )} x^{4} - 16 \,{\left (5 \, b^{6} d^{3} e^{3} - 15 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{3} - 6 \,{\left (5 \, b^{6} d^{4} e^{2} + 20 \, a b^{5} d^{3} e^{3} - 90 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} - 22 \, a^{4} b^{2} e^{6}\right )} x^{2} - 4 \,{\left (2 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 20 \, a^{2} b^{4} d^{3} e^{3} - 110 \, a^{3} b^{3} d^{2} e^{4} + 124 \, a^{4} b^{2} d e^{5} - 42 \, a^{5} b e^{6}\right )} x + 60 \,{\left (a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} + a^{6} e^{6} +{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \,{\left (a b^{5} d^{2} e^{4} - 2 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 6 \,{\left (a^{2} b^{4} d^{2} e^{4} - 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 4 \,{\left (a^{3} b^{3} d^{2} e^{4} - 2 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x\right )} \log \left (b x + a\right )}{4 \,{\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{6}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**6/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.585388, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")
[Out]