3.2012 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15}} \, dx\)

Optimal. Leaf size=362 \[ -\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^{12}}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{14 e^7 (a+b x) (d+e x)^{14}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^7 (a+b x) (d+e x)^9}-\frac{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{11 e^7 (a+b x) (d+e x)^{11}} \]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(14*e^7*(a + b*x)*(d + e*x)^14) +
(6*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x)^1
3) - (5*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)*(d + e
*x)^12) + (20*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)
*(d + e*x)^11) - (3*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a +
b*x)*(d + e*x)^10) + (2*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(
a + b*x)*(d + e*x)^9) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a + b*x)*(d
+ e*x)^8)

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Rubi [A]  time = 0.600484, antiderivative size = 362, 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 \[ -\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^{12}}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{14 e^7 (a+b x) (d+e x)^{14}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^7 (a+b x) (d+e x)^9}-\frac{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{11 e^7 (a+b x) (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^15,x]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(14*e^7*(a + b*x)*(d + e*x)^14) +
(6*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x)^1
3) - (5*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)*(d + e
*x)^12) + (20*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)
*(d + e*x)^11) - (3*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a +
b*x)*(d + e*x)^10) + (2*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(
a + b*x)*(d + e*x)^9) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a + b*x)*(d
+ e*x)^8)

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Rubi in Sympy [A]  time = 52.7027, size = 280, normalized size = 0.77 \[ - \frac{3 b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{8008 e^{6} \left (d + e x\right )^{9}} + \frac{b^{5} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{24024 e^{7} \left (a + b x\right ) \left (d + e x\right )^{9}} - \frac{b^{4} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2002 e^{5} \left (d + e x\right )^{10}} - \frac{5 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{1001 e^{4} \left (d + e x\right )^{11}} - \frac{b^{2} \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{364 e^{3} \left (d + e x\right )^{12}} - \frac{3 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{91 e^{2} \left (d + e x\right )^{13}} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{14 e \left (d + e x\right )^{14}} \]

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)**15,x)

[Out]

-3*b**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(8008*e**6*(d + e*x)**9) + b**5*(a*e -
b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(24024*e**7*(a + b*x)*(d + e*x)**9) - b**4
*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2002*e**5*(d + e*x)**10) - 5*b*
*3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(1001*e**4*(d + e*x)**11) - b**2*(5*a + 5
*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(364*e**3*(d + e*x)**12) - 3*b*(a**2 +
2*a*b*x + b**2*x**2)**(5/2)/(91*e**2*(d + e*x)**13) - (a + b*x)*(a**2 + 2*a*b*x
+ b**2*x**2)**(5/2)/(14*e*(d + e*x)**14)

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Mathematica [A]  time = 0.222448, size = 295, normalized size = 0.81 \[ -\frac{\sqrt{(a+b x)^2} \left (1716 a^6 e^6+792 a^5 b e^5 (d+14 e x)+330 a^4 b^2 e^4 \left (d^2+14 d e x+91 e^2 x^2\right )+120 a^3 b^3 e^3 \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )+36 a^2 b^4 e^2 \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )+8 a b^5 e \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )+b^6 \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )\right )}{24024 e^7 (a+b x) (d+e x)^{14}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^15,x]

[Out]

-(Sqrt[(a + b*x)^2]*(1716*a^6*e^6 + 792*a^5*b*e^5*(d + 14*e*x) + 330*a^4*b^2*e^4
*(d^2 + 14*d*e*x + 91*e^2*x^2) + 120*a^3*b^3*e^3*(d^3 + 14*d^2*e*x + 91*d*e^2*x^
2 + 364*e^3*x^3) + 36*a^2*b^4*e^2*(d^4 + 14*d^3*e*x + 91*d^2*e^2*x^2 + 364*d*e^3
*x^3 + 1001*e^4*x^4) + 8*a*b^5*e*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^
3*x^3 + 1001*d*e^4*x^4 + 2002*e^5*x^5) + b^6*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2
+ 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2002*d*e^5*x^5 + 3003*e^6*x^6)))/(24024*e
^7*(a + b*x)*(d + e*x)^14)

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Maple [A]  time = 0.018, size = 392, normalized size = 1.1 \[ -{\frac{3003\,{x}^{6}{b}^{6}{e}^{6}+16016\,{x}^{5}a{b}^{5}{e}^{6}+2002\,{x}^{5}{b}^{6}d{e}^{5}+36036\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+8008\,{x}^{4}a{b}^{5}d{e}^{5}+1001\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+43680\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+13104\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+2912\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+364\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+30030\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+10920\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+3276\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+728\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+91\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+11088\,x{a}^{5}b{e}^{6}+4620\,x{a}^{4}{b}^{2}d{e}^{5}+1680\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+504\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+112\,xa{b}^{5}{d}^{4}{e}^{2}+14\,x{b}^{6}{d}^{5}e+1716\,{a}^{6}{e}^{6}+792\,{a}^{5}bd{e}^{5}+330\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+120\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+36\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+8\,{d}^{5}a{b}^{5}e+{b}^{6}{d}^{6}}{24024\,{e}^{7} \left ( ex+d \right ) ^{14} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^15,x)

[Out]

-1/24024/e^7*(3003*b^6*e^6*x^6+16016*a*b^5*e^6*x^5+2002*b^6*d*e^5*x^5+36036*a^2*
b^4*e^6*x^4+8008*a*b^5*d*e^5*x^4+1001*b^6*d^2*e^4*x^4+43680*a^3*b^3*e^6*x^3+1310
4*a^2*b^4*d*e^5*x^3+2912*a*b^5*d^2*e^4*x^3+364*b^6*d^3*e^3*x^3+30030*a^4*b^2*e^6
*x^2+10920*a^3*b^3*d*e^5*x^2+3276*a^2*b^4*d^2*e^4*x^2+728*a*b^5*d^3*e^3*x^2+91*b
^6*d^4*e^2*x^2+11088*a^5*b*e^6*x+4620*a^4*b^2*d*e^5*x+1680*a^3*b^3*d^2*e^4*x+504
*a^2*b^4*d^3*e^3*x+112*a*b^5*d^4*e^2*x+14*b^6*d^5*e*x+1716*a^6*e^6+792*a^5*b*d*e
^5+330*a^4*b^2*d^2*e^4+120*a^3*b^3*d^3*e^3+36*a^2*b^4*d^4*e^2+8*a*b^5*d^5*e+b^6*
d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^14/(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)^15,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299707, size = 670, normalized size = 1.85 \[ -\frac{3003 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 8 \, a b^{5} d^{5} e + 36 \, a^{2} b^{4} d^{4} e^{2} + 120 \, a^{3} b^{3} d^{3} e^{3} + 330 \, a^{4} b^{2} d^{2} e^{4} + 792 \, a^{5} b d e^{5} + 1716 \, a^{6} e^{6} + 2002 \,{\left (b^{6} d e^{5} + 8 \, a b^{5} e^{6}\right )} x^{5} + 1001 \,{\left (b^{6} d^{2} e^{4} + 8 \, a b^{5} d e^{5} + 36 \, a^{2} b^{4} e^{6}\right )} x^{4} + 364 \,{\left (b^{6} d^{3} e^{3} + 8 \, a b^{5} d^{2} e^{4} + 36 \, a^{2} b^{4} d e^{5} + 120 \, a^{3} b^{3} e^{6}\right )} x^{3} + 91 \,{\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 36 \, a^{2} b^{4} d^{2} e^{4} + 120 \, a^{3} b^{3} d e^{5} + 330 \, a^{4} b^{2} e^{6}\right )} x^{2} + 14 \,{\left (b^{6} d^{5} e + 8 \, a b^{5} d^{4} e^{2} + 36 \, a^{2} b^{4} d^{3} e^{3} + 120 \, a^{3} b^{3} d^{2} e^{4} + 330 \, a^{4} b^{2} d e^{5} + 792 \, a^{5} b e^{6}\right )} x}{24024 \,{\left (e^{21} x^{14} + 14 \, d e^{20} x^{13} + 91 \, d^{2} e^{19} x^{12} + 364 \, d^{3} e^{18} x^{11} + 1001 \, d^{4} e^{17} x^{10} + 2002 \, d^{5} e^{16} x^{9} + 3003 \, d^{6} e^{15} x^{8} + 3432 \, d^{7} e^{14} x^{7} + 3003 \, d^{8} e^{13} x^{6} + 2002 \, d^{9} e^{12} x^{5} + 1001 \, d^{10} e^{11} x^{4} + 364 \, d^{11} e^{10} x^{3} + 91 \, d^{12} e^{9} x^{2} + 14 \, d^{13} e^{8} x + d^{14} e^{7}\right )}} \]

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)^15,x, algorithm="fricas")

[Out]

-1/24024*(3003*b^6*e^6*x^6 + b^6*d^6 + 8*a*b^5*d^5*e + 36*a^2*b^4*d^4*e^2 + 120*
a^3*b^3*d^3*e^3 + 330*a^4*b^2*d^2*e^4 + 792*a^5*b*d*e^5 + 1716*a^6*e^6 + 2002*(b
^6*d*e^5 + 8*a*b^5*e^6)*x^5 + 1001*(b^6*d^2*e^4 + 8*a*b^5*d*e^5 + 36*a^2*b^4*e^6
)*x^4 + 364*(b^6*d^3*e^3 + 8*a*b^5*d^2*e^4 + 36*a^2*b^4*d*e^5 + 120*a^3*b^3*e^6)
*x^3 + 91*(b^6*d^4*e^2 + 8*a*b^5*d^3*e^3 + 36*a^2*b^4*d^2*e^4 + 120*a^3*b^3*d*e^
5 + 330*a^4*b^2*e^6)*x^2 + 14*(b^6*d^5*e + 8*a*b^5*d^4*e^2 + 36*a^2*b^4*d^3*e^3
+ 120*a^3*b^3*d^2*e^4 + 330*a^4*b^2*d*e^5 + 792*a^5*b*e^6)*x)/(e^21*x^14 + 14*d*
e^20*x^13 + 91*d^2*e^19*x^12 + 364*d^3*e^18*x^11 + 1001*d^4*e^17*x^10 + 2002*d^5
*e^16*x^9 + 3003*d^6*e^15*x^8 + 3432*d^7*e^14*x^7 + 3003*d^8*e^13*x^6 + 2002*d^9
*e^12*x^5 + 1001*d^10*e^11*x^4 + 364*d^11*e^10*x^3 + 91*d^12*e^9*x^2 + 14*d^13*e
^8*x + d^14*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)**15,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.302951, size = 702, normalized size = 1.94 \[ -\frac{{\left (3003 \, b^{6} x^{6} e^{6}{\rm sign}\left (b x + a\right ) + 2002 \, b^{6} d x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 1001 \, b^{6} d^{2} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 364 \, b^{6} d^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 91 \, b^{6} d^{4} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 14 \, b^{6} d^{5} x e{\rm sign}\left (b x + a\right ) + b^{6} d^{6}{\rm sign}\left (b x + a\right ) + 16016 \, a b^{5} x^{5} e^{6}{\rm sign}\left (b x + a\right ) + 8008 \, a b^{5} d x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 2912 \, a b^{5} d^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 728 \, a b^{5} d^{3} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 112 \, a b^{5} d^{4} x e^{2}{\rm sign}\left (b x + a\right ) + 8 \, a b^{5} d^{5} e{\rm sign}\left (b x + a\right ) + 36036 \, a^{2} b^{4} x^{4} e^{6}{\rm sign}\left (b x + a\right ) + 13104 \, a^{2} b^{4} d x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 3276 \, a^{2} b^{4} d^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 504 \, a^{2} b^{4} d^{3} x e^{3}{\rm sign}\left (b x + a\right ) + 36 \, a^{2} b^{4} d^{4} e^{2}{\rm sign}\left (b x + a\right ) + 43680 \, a^{3} b^{3} x^{3} e^{6}{\rm sign}\left (b x + a\right ) + 10920 \, a^{3} b^{3} d x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 1680 \, a^{3} b^{3} d^{2} x e^{4}{\rm sign}\left (b x + a\right ) + 120 \, a^{3} b^{3} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 30030 \, a^{4} b^{2} x^{2} e^{6}{\rm sign}\left (b x + a\right ) + 4620 \, a^{4} b^{2} d x e^{5}{\rm sign}\left (b x + a\right ) + 330 \, a^{4} b^{2} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 11088 \, a^{5} b x e^{6}{\rm sign}\left (b x + a\right ) + 792 \, a^{5} b d e^{5}{\rm sign}\left (b x + a\right ) + 1716 \, a^{6} e^{6}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}}{24024 \,{\left (x e + d\right )}^{14}} \]

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)^15,x, algorithm="giac")

[Out]

-1/24024*(3003*b^6*x^6*e^6*sign(b*x + a) + 2002*b^6*d*x^5*e^5*sign(b*x + a) + 10
01*b^6*d^2*x^4*e^4*sign(b*x + a) + 364*b^6*d^3*x^3*e^3*sign(b*x + a) + 91*b^6*d^
4*x^2*e^2*sign(b*x + a) + 14*b^6*d^5*x*e*sign(b*x + a) + b^6*d^6*sign(b*x + a) +
16016*a*b^5*x^5*e^6*sign(b*x + a) + 8008*a*b^5*d*x^4*e^5*sign(b*x + a) + 2912*a
*b^5*d^2*x^3*e^4*sign(b*x + a) + 728*a*b^5*d^3*x^2*e^3*sign(b*x + a) + 112*a*b^5
*d^4*x*e^2*sign(b*x + a) + 8*a*b^5*d^5*e*sign(b*x + a) + 36036*a^2*b^4*x^4*e^6*s
ign(b*x + a) + 13104*a^2*b^4*d*x^3*e^5*sign(b*x + a) + 3276*a^2*b^4*d^2*x^2*e^4*
sign(b*x + a) + 504*a^2*b^4*d^3*x*e^3*sign(b*x + a) + 36*a^2*b^4*d^4*e^2*sign(b*
x + a) + 43680*a^3*b^3*x^3*e^6*sign(b*x + a) + 10920*a^3*b^3*d*x^2*e^5*sign(b*x
+ a) + 1680*a^3*b^3*d^2*x*e^4*sign(b*x + a) + 120*a^3*b^3*d^3*e^3*sign(b*x + a)
+ 30030*a^4*b^2*x^2*e^6*sign(b*x + a) + 4620*a^4*b^2*d*x*e^5*sign(b*x + a) + 330
*a^4*b^2*d^2*e^4*sign(b*x + a) + 11088*a^5*b*x*e^6*sign(b*x + a) + 792*a^5*b*d*e
^5*sign(b*x + a) + 1716*a^6*e^6*sign(b*x + a))*e^(-7)/(x*e + d)^14