3.2009 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=359 \[ -\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^7 (a+b x) (d+e x)^9}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{10}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)^6}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^7 (a+b x) (d+e x)^7}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^7 (a+b x) (d+e x)^8} \]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^11) +
(3*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^10
) - (5*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*
x)^9) + (5*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d
+ e*x)^8) - (15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x
)*(d + e*x)^7) + (b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*
(d + e*x)^6) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^5)

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Rubi [A]  time = 0.604596, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^7 (a+b x) (d+e x)^9}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{10}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)^6}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^7 (a+b x) (d+e x)^7}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^7 (a+b x) (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^12,x]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^11) +
(3*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^10
) - (5*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*
x)^9) + (5*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d
+ e*x)^8) - (15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x
)*(d + e*x)^7) + (b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*
(d + e*x)^6) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^5)

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Rubi in Sympy [A]  time = 69.5832, size = 199, normalized size = 0.55 \[ - \frac{b^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{2310 \left (d + e x\right )^{7} \left (a e - b d\right )^{5}} + \frac{b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{330 \left (d + e x\right )^{8} \left (a e - b d\right )^{4}} - \frac{2 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{165 \left (d + e x\right )^{9} \left (a e - b d\right )^{3}} + \frac{2 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{55 \left (d + e x\right )^{10} \left (a e - b d\right )^{2}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{11 \left (d + e x\right )^{11} \left (a e - b d\right )} \]

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)**12,x)

[Out]

-b**4*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(2310*(d + e*x)**7*(a*e - b*d)**5) + b
**3*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(330*(d + e*x)**8*(a*e - b*d)**4) - 2*b*
*2*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(165*(d + e*x)**9*(a*e - b*d)**3) + 2*b*(
a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(55*(d + e*x)**10*(a*e - b*d)**2) - (a**2 + 2
*a*b*x + b**2*x**2)**(7/2)/(11*(d + e*x)**11*(a*e - b*d))

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Mathematica [A]  time = 0.232801, size = 295, normalized size = 0.82 \[ -\frac{\sqrt{(a+b x)^2} \left (210 a^6 e^6+126 a^5 b e^5 (d+11 e x)+70 a^4 b^2 e^4 \left (d^2+11 d e x+55 e^2 x^2\right )+35 a^3 b^3 e^3 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+15 a^2 b^4 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b^5 e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+b^6 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )}{2310 e^7 (a+b x) (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^12,x]

[Out]

-(Sqrt[(a + b*x)^2]*(210*a^6*e^6 + 126*a^5*b*e^5*(d + 11*e*x) + 70*a^4*b^2*e^4*(
d^2 + 11*d*e*x + 55*e^2*x^2) + 35*a^3*b^3*e^3*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 +
165*e^3*x^3) + 15*a^2*b^4*e^2*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^
3 + 330*e^4*x^4) + 5*a*b^5*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^
3 + 330*d*e^4*x^4 + 462*e^5*x^5) + b^6*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*
d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6)))/(2310*e^7*(a + b*
x)*(d + e*x)^11)

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Maple [A]  time = 0.017, size = 392, normalized size = 1.1 \[ -{\frac{462\,{x}^{6}{b}^{6}{e}^{6}+2310\,{x}^{5}a{b}^{5}{e}^{6}+462\,{x}^{5}{b}^{6}d{e}^{5}+4950\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+1650\,{x}^{4}a{b}^{5}d{e}^{5}+330\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+5775\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+2475\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+825\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+165\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+3850\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+1925\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+825\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+275\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+55\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+1386\,x{a}^{5}b{e}^{6}+770\,x{a}^{4}{b}^{2}d{e}^{5}+385\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+165\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+55\,xa{b}^{5}{d}^{4}{e}^{2}+11\,x{b}^{6}{d}^{5}e+210\,{a}^{6}{e}^{6}+126\,{a}^{5}bd{e}^{5}+70\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+35\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+15\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+5\,{d}^{5}a{b}^{5}e+{b}^{6}{d}^{6}}{2310\,{e}^{7} \left ( ex+d \right ) ^{11} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^12,x)

[Out]

-1/2310/e^7*(462*b^6*e^6*x^6+2310*a*b^5*e^6*x^5+462*b^6*d*e^5*x^5+4950*a^2*b^4*e
^6*x^4+1650*a*b^5*d*e^5*x^4+330*b^6*d^2*e^4*x^4+5775*a^3*b^3*e^6*x^3+2475*a^2*b^
4*d*e^5*x^3+825*a*b^5*d^2*e^4*x^3+165*b^6*d^3*e^3*x^3+3850*a^4*b^2*e^6*x^2+1925*
a^3*b^3*d*e^5*x^2+825*a^2*b^4*d^2*e^4*x^2+275*a*b^5*d^3*e^3*x^2+55*b^6*d^4*e^2*x
^2+1386*a^5*b*e^6*x+770*a^4*b^2*d*e^5*x+385*a^3*b^3*d^2*e^4*x+165*a^2*b^4*d^3*e^
3*x+55*a*b^5*d^4*e^2*x+11*b^6*d^5*e*x+210*a^6*e^6+126*a^5*b*d*e^5+70*a^4*b^2*d^2
*e^4+35*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2+5*a*b^5*d^5*e+b^6*d^6)*((b*x+a)^2)^(5
/2)/(e*x+d)^11/(b*x+a)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288136, size = 625, normalized size = 1.74 \[ -\frac{462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \,{\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \,{\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \,{\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \,{\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \,{\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \,{\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \]

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)^12,x, algorithm="fricas")

[Out]

-1/2310*(462*b^6*e^6*x^6 + b^6*d^6 + 5*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 35*a^3
*b^3*d^3*e^3 + 70*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 210*a^6*e^6 + 462*(b^6*d*e
^5 + 5*a*b^5*e^6)*x^5 + 330*(b^6*d^2*e^4 + 5*a*b^5*d*e^5 + 15*a^2*b^4*e^6)*x^4 +
165*(b^6*d^3*e^3 + 5*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 + 35*a^3*b^3*e^6)*x^3 + 5
5*(b^6*d^4*e^2 + 5*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2*e^4 + 35*a^3*b^3*d*e^5 + 70*a^
4*b^2*e^6)*x^2 + 11*(b^6*d^5*e + 5*a*b^5*d^4*e^2 + 15*a^2*b^4*d^3*e^3 + 35*a^3*b
^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 126*a^5*b*e^6)*x)/(e^18*x^11 + 11*d*e^17*x^10 +
55*d^2*e^16*x^9 + 165*d^3*e^15*x^8 + 330*d^4*e^14*x^7 + 462*d^5*e^13*x^6 + 462*d
^6*e^12*x^5 + 330*d^7*e^11*x^4 + 165*d^8*e^10*x^3 + 55*d^9*e^9*x^2 + 11*d^10*e^8
*x + d^11*e^7)

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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)**12,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296383, size = 702, normalized size = 1.96 \[ -\frac{{\left (462 \, b^{6} x^{6} e^{6}{\rm sign}\left (b x + a\right ) + 462 \, b^{6} d x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 330 \, b^{6} d^{2} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 165 \, b^{6} d^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 55 \, b^{6} d^{4} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 11 \, b^{6} d^{5} x e{\rm sign}\left (b x + a\right ) + b^{6} d^{6}{\rm sign}\left (b x + a\right ) + 2310 \, a b^{5} x^{5} e^{6}{\rm sign}\left (b x + a\right ) + 1650 \, a b^{5} d x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 825 \, a b^{5} d^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 275 \, a b^{5} d^{3} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 55 \, a b^{5} d^{4} x e^{2}{\rm sign}\left (b x + a\right ) + 5 \, a b^{5} d^{5} e{\rm sign}\left (b x + a\right ) + 4950 \, a^{2} b^{4} x^{4} e^{6}{\rm sign}\left (b x + a\right ) + 2475 \, a^{2} b^{4} d x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 825 \, a^{2} b^{4} d^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 165 \, a^{2} b^{4} d^{3} x e^{3}{\rm sign}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2}{\rm sign}\left (b x + a\right ) + 5775 \, a^{3} b^{3} x^{3} e^{6}{\rm sign}\left (b x + a\right ) + 1925 \, a^{3} b^{3} d x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 385 \, a^{3} b^{3} d^{2} x e^{4}{\rm sign}\left (b x + a\right ) + 35 \, a^{3} b^{3} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 3850 \, a^{4} b^{2} x^{2} e^{6}{\rm sign}\left (b x + a\right ) + 770 \, a^{4} b^{2} d x e^{5}{\rm sign}\left (b x + a\right ) + 70 \, a^{4} b^{2} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 1386 \, a^{5} b x e^{6}{\rm sign}\left (b x + a\right ) + 126 \, a^{5} b d e^{5}{\rm sign}\left (b x + a\right ) + 210 \, a^{6} e^{6}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}}{2310 \,{\left (x e + d\right )}^{11}} \]

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)^12,x, algorithm="giac")

[Out]

-1/2310*(462*b^6*x^6*e^6*sign(b*x + a) + 462*b^6*d*x^5*e^5*sign(b*x + a) + 330*b
^6*d^2*x^4*e^4*sign(b*x + a) + 165*b^6*d^3*x^3*e^3*sign(b*x + a) + 55*b^6*d^4*x^
2*e^2*sign(b*x + a) + 11*b^6*d^5*x*e*sign(b*x + a) + b^6*d^6*sign(b*x + a) + 231
0*a*b^5*x^5*e^6*sign(b*x + a) + 1650*a*b^5*d*x^4*e^5*sign(b*x + a) + 825*a*b^5*d
^2*x^3*e^4*sign(b*x + a) + 275*a*b^5*d^3*x^2*e^3*sign(b*x + a) + 55*a*b^5*d^4*x*
e^2*sign(b*x + a) + 5*a*b^5*d^5*e*sign(b*x + a) + 4950*a^2*b^4*x^4*e^6*sign(b*x
+ a) + 2475*a^2*b^4*d*x^3*e^5*sign(b*x + a) + 825*a^2*b^4*d^2*x^2*e^4*sign(b*x +
a) + 165*a^2*b^4*d^3*x*e^3*sign(b*x + a) + 15*a^2*b^4*d^4*e^2*sign(b*x + a) + 5
775*a^3*b^3*x^3*e^6*sign(b*x + a) + 1925*a^3*b^3*d*x^2*e^5*sign(b*x + a) + 385*a
^3*b^3*d^2*x*e^4*sign(b*x + a) + 35*a^3*b^3*d^3*e^3*sign(b*x + a) + 3850*a^4*b^2
*x^2*e^6*sign(b*x + a) + 770*a^4*b^2*d*x*e^5*sign(b*x + a) + 70*a^4*b^2*d^2*e^4*
sign(b*x + a) + 1386*a^5*b*x*e^6*sign(b*x + a) + 126*a^5*b*d*e^5*sign(b*x + a) +
210*a^6*e^6*sign(b*x + a))*e^(-7)/(x*e + d)^11