3.2011 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{14}} \, dx\)

Optimal. Leaf size=360 \[ -\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^{12}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^7 (a+b x) (d+e x)^8}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^7 (a+b x) (d+e x)^9}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^{10}} \]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x)^13) +
(b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^12)
- (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*
x)^11) + (2*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d +
e*x)^10) - (5*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)
*(d + e*x)^9) + (3*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*
x)*(d + e*x)^8) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)
^7)

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Rubi [A]  time = 0.598714, antiderivative size = 360, 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 \[ -\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^{12}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^7 (a+b x) (d+e x)^8}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^7 (a+b x) (d+e x)^9}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^14,x]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x)^13) +
(b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^12)
- (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*
x)^11) + (2*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d +
e*x)^10) - (5*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)
*(d + e*x)^9) + (3*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*
x)*(d + e*x)^8) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)
^7)

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Rubi in Sympy [A]  time = 111.202, size = 279, normalized size = 0.78 \[ - \frac{b^{6} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{12012 \left (d + e x\right )^{7} \left (a e - b d\right )^{7}} + \frac{b^{5} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{1716 \left (d + e x\right )^{8} \left (a e - b d\right )^{6}} - \frac{b^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{429 \left (d + e x\right )^{9} \left (a e - b d\right )^{5}} + \frac{b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{143 \left (d + e x\right )^{10} \left (a e - b d\right )^{4}} - \frac{5 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{286 \left (d + e x\right )^{11} \left (a e - b d\right )^{3}} + \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{26 \left (d + e x\right )^{12} \left (a e - b d\right )^{2}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{13 \left (d + e x\right )^{13} \left (a e - b d\right )} \]

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)**14,x)

[Out]

-b**6*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(12012*(d + e*x)**7*(a*e - b*d)**7) +
b**5*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(1716*(d + e*x)**8*(a*e - b*d)**6) - b*
*4*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(429*(d + e*x)**9*(a*e - b*d)**5) + b**3*
(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(143*(d + e*x)**10*(a*e - b*d)**4) - 5*b**2*
(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(286*(d + e*x)**11*(a*e - b*d)**3) + b*(a**2
+ 2*a*b*x + b**2*x**2)**(7/2)/(26*(d + e*x)**12*(a*e - b*d)**2) - (a**2 + 2*a*b
*x + b**2*x**2)**(7/2)/(13*(d + e*x)**13*(a*e - b*d))

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Mathematica [A]  time = 0.341251, size = 295, normalized size = 0.82 \[ -\frac{\sqrt{(a+b x)^2} \left (924 a^6 e^6+462 a^5 b e^5 (d+13 e x)+210 a^4 b^2 e^4 \left (d^2+13 d e x+78 e^2 x^2\right )+84 a^3 b^3 e^3 \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+28 a^2 b^4 e^2 \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+7 a b^5 e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+b^6 \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )\right )}{12012 e^7 (a+b x) (d+e x)^{13}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^14,x]

[Out]

-(Sqrt[(a + b*x)^2]*(924*a^6*e^6 + 462*a^5*b*e^5*(d + 13*e*x) + 210*a^4*b^2*e^4*
(d^2 + 13*d*e*x + 78*e^2*x^2) + 84*a^3*b^3*e^3*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2
+ 286*e^3*x^3) + 28*a^2*b^4*e^2*(d^4 + 13*d^3*e*x + 78*d^2*e^2*x^2 + 286*d*e^3*x
^3 + 715*e^4*x^4) + 7*a*b^5*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x
^3 + 715*d*e^4*x^4 + 1287*e^5*x^5) + b^6*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 28
6*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5*x^5 + 1716*e^6*x^6)))/(12012*e^7*(a
+ b*x)*(d + e*x)^13)

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Maple [A]  time = 0.017, size = 392, normalized size = 1.1 \[ -{\frac{1716\,{x}^{6}{b}^{6}{e}^{6}+9009\,{x}^{5}a{b}^{5}{e}^{6}+1287\,{x}^{5}{b}^{6}d{e}^{5}+20020\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+5005\,{x}^{4}a{b}^{5}d{e}^{5}+715\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+24024\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+8008\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+2002\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+286\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+16380\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+6552\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+2184\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+546\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+78\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+6006\,x{a}^{5}b{e}^{6}+2730\,x{a}^{4}{b}^{2}d{e}^{5}+1092\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+364\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+91\,xa{b}^{5}{d}^{4}{e}^{2}+13\,x{b}^{6}{d}^{5}e+924\,{a}^{6}{e}^{6}+462\,{a}^{5}bd{e}^{5}+210\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+84\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+28\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+7\,{d}^{5}a{b}^{5}e+{b}^{6}{d}^{6}}{12012\,{e}^{7} \left ( ex+d \right ) ^{13} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^14,x)

[Out]

-1/12012/e^7*(1716*b^6*e^6*x^6+9009*a*b^5*e^6*x^5+1287*b^6*d*e^5*x^5+20020*a^2*b
^4*e^6*x^4+5005*a*b^5*d*e^5*x^4+715*b^6*d^2*e^4*x^4+24024*a^3*b^3*e^6*x^3+8008*a
^2*b^4*d*e^5*x^3+2002*a*b^5*d^2*e^4*x^3+286*b^6*d^3*e^3*x^3+16380*a^4*b^2*e^6*x^
2+6552*a^3*b^3*d*e^5*x^2+2184*a^2*b^4*d^2*e^4*x^2+546*a*b^5*d^3*e^3*x^2+78*b^6*d
^4*e^2*x^2+6006*a^5*b*e^6*x+2730*a^4*b^2*d*e^5*x+1092*a^3*b^3*d^2*e^4*x+364*a^2*
b^4*d^3*e^3*x+91*a*b^5*d^4*e^2*x+13*b^6*d^5*e*x+924*a^6*e^6+462*a^5*b*d*e^5+210*
a^4*b^2*d^2*e^4+84*a^3*b^3*d^3*e^3+28*a^2*b^4*d^4*e^2+7*a*b^5*d^5*e+b^6*d^6)*((b
*x+a)^2)^(5/2)/(e*x+d)^13/(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)^14,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287076, size = 655, normalized size = 1.82 \[ -\frac{1716 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 7 \, a b^{5} d^{5} e + 28 \, a^{2} b^{4} d^{4} e^{2} + 84 \, a^{3} b^{3} d^{3} e^{3} + 210 \, a^{4} b^{2} d^{2} e^{4} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6} + 1287 \,{\left (b^{6} d e^{5} + 7 \, a b^{5} e^{6}\right )} x^{5} + 715 \,{\left (b^{6} d^{2} e^{4} + 7 \, a b^{5} d e^{5} + 28 \, a^{2} b^{4} e^{6}\right )} x^{4} + 286 \,{\left (b^{6} d^{3} e^{3} + 7 \, a b^{5} d^{2} e^{4} + 28 \, a^{2} b^{4} d e^{5} + 84 \, a^{3} b^{3} e^{6}\right )} x^{3} + 78 \,{\left (b^{6} d^{4} e^{2} + 7 \, a b^{5} d^{3} e^{3} + 28 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} + 210 \, a^{4} b^{2} e^{6}\right )} x^{2} + 13 \,{\left (b^{6} d^{5} e + 7 \, a b^{5} d^{4} e^{2} + 28 \, a^{2} b^{4} d^{3} e^{3} + 84 \, a^{3} b^{3} d^{2} e^{4} + 210 \, a^{4} b^{2} d e^{5} + 462 \, a^{5} b e^{6}\right )} x}{12012 \,{\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} e^{7}\right )}} \]

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)^14,x, algorithm="fricas")

[Out]

-1/12012*(1716*b^6*e^6*x^6 + b^6*d^6 + 7*a*b^5*d^5*e + 28*a^2*b^4*d^4*e^2 + 84*a
^3*b^3*d^3*e^3 + 210*a^4*b^2*d^2*e^4 + 462*a^5*b*d*e^5 + 924*a^6*e^6 + 1287*(b^6
*d*e^5 + 7*a*b^5*e^6)*x^5 + 715*(b^6*d^2*e^4 + 7*a*b^5*d*e^5 + 28*a^2*b^4*e^6)*x
^4 + 286*(b^6*d^3*e^3 + 7*a*b^5*d^2*e^4 + 28*a^2*b^4*d*e^5 + 84*a^3*b^3*e^6)*x^3
+ 78*(b^6*d^4*e^2 + 7*a*b^5*d^3*e^3 + 28*a^2*b^4*d^2*e^4 + 84*a^3*b^3*d*e^5 + 2
10*a^4*b^2*e^6)*x^2 + 13*(b^6*d^5*e + 7*a*b^5*d^4*e^2 + 28*a^2*b^4*d^3*e^3 + 84*
a^3*b^3*d^2*e^4 + 210*a^4*b^2*d*e^5 + 462*a^5*b*e^6)*x)/(e^20*x^13 + 13*d*e^19*x
^12 + 78*d^2*e^18*x^11 + 286*d^3*e^17*x^10 + 715*d^4*e^16*x^9 + 1287*d^5*e^15*x^
8 + 1716*d^6*e^14*x^7 + 1716*d^7*e^13*x^6 + 1287*d^8*e^12*x^5 + 715*d^9*e^11*x^4
+ 286*d^10*e^10*x^3 + 78*d^11*e^9*x^2 + 13*d^12*e^8*x + d^13*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)**14,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.304039, size = 702, normalized size = 1.95 \[ -\frac{{\left (1716 \, b^{6} x^{6} e^{6}{\rm sign}\left (b x + a\right ) + 1287 \, b^{6} d x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 715 \, b^{6} d^{2} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 286 \, b^{6} d^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 78 \, b^{6} d^{4} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 13 \, b^{6} d^{5} x e{\rm sign}\left (b x + a\right ) + b^{6} d^{6}{\rm sign}\left (b x + a\right ) + 9009 \, a b^{5} x^{5} e^{6}{\rm sign}\left (b x + a\right ) + 5005 \, a b^{5} d x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 2002 \, a b^{5} d^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 546 \, a b^{5} d^{3} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 91 \, a b^{5} d^{4} x e^{2}{\rm sign}\left (b x + a\right ) + 7 \, a b^{5} d^{5} e{\rm sign}\left (b x + a\right ) + 20020 \, a^{2} b^{4} x^{4} e^{6}{\rm sign}\left (b x + a\right ) + 8008 \, a^{2} b^{4} d x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 2184 \, a^{2} b^{4} d^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 364 \, a^{2} b^{4} d^{3} x e^{3}{\rm sign}\left (b x + a\right ) + 28 \, a^{2} b^{4} d^{4} e^{2}{\rm sign}\left (b x + a\right ) + 24024 \, a^{3} b^{3} x^{3} e^{6}{\rm sign}\left (b x + a\right ) + 6552 \, a^{3} b^{3} d x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 1092 \, a^{3} b^{3} d^{2} x e^{4}{\rm sign}\left (b x + a\right ) + 84 \, a^{3} b^{3} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 16380 \, a^{4} b^{2} x^{2} e^{6}{\rm sign}\left (b x + a\right ) + 2730 \, a^{4} b^{2} d x e^{5}{\rm sign}\left (b x + a\right ) + 210 \, a^{4} b^{2} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 6006 \, a^{5} b x e^{6}{\rm sign}\left (b x + a\right ) + 462 \, a^{5} b d e^{5}{\rm sign}\left (b x + a\right ) + 924 \, a^{6} e^{6}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}}{12012 \,{\left (x e + d\right )}^{13}} \]

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)^14,x, algorithm="giac")

[Out]

-1/12012*(1716*b^6*x^6*e^6*sign(b*x + a) + 1287*b^6*d*x^5*e^5*sign(b*x + a) + 71
5*b^6*d^2*x^4*e^4*sign(b*x + a) + 286*b^6*d^3*x^3*e^3*sign(b*x + a) + 78*b^6*d^4
*x^2*e^2*sign(b*x + a) + 13*b^6*d^5*x*e*sign(b*x + a) + b^6*d^6*sign(b*x + a) +
9009*a*b^5*x^5*e^6*sign(b*x + a) + 5005*a*b^5*d*x^4*e^5*sign(b*x + a) + 2002*a*b
^5*d^2*x^3*e^4*sign(b*x + a) + 546*a*b^5*d^3*x^2*e^3*sign(b*x + a) + 91*a*b^5*d^
4*x*e^2*sign(b*x + a) + 7*a*b^5*d^5*e*sign(b*x + a) + 20020*a^2*b^4*x^4*e^6*sign
(b*x + a) + 8008*a^2*b^4*d*x^3*e^5*sign(b*x + a) + 2184*a^2*b^4*d^2*x^2*e^4*sign
(b*x + a) + 364*a^2*b^4*d^3*x*e^3*sign(b*x + a) + 28*a^2*b^4*d^4*e^2*sign(b*x +
a) + 24024*a^3*b^3*x^3*e^6*sign(b*x + a) + 6552*a^3*b^3*d*x^2*e^5*sign(b*x + a)
+ 1092*a^3*b^3*d^2*x*e^4*sign(b*x + a) + 84*a^3*b^3*d^3*e^3*sign(b*x + a) + 1638
0*a^4*b^2*x^2*e^6*sign(b*x + a) + 2730*a^4*b^2*d*x*e^5*sign(b*x + a) + 210*a^4*b
^2*d^2*e^4*sign(b*x + a) + 6006*a^5*b*x*e^6*sign(b*x + a) + 462*a^5*b*d*e^5*sign
(b*x + a) + 924*a^6*e^6*sign(b*x + a))*e^(-7)/(x*e + d)^13