3.2013 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{16}} \, dx\)

Optimal. Leaf size=362 \[ -\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{13 e^7 (a+b x) (d+e x)^{13}}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{14}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}+\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{10}}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{12}} \]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x)*(d + e*x)^15) +
(3*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^14
) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d +
e*x)^13) + (5*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*
(d + e*x)^12) - (15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a
+ b*x)*(d + e*x)^11) + (3*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*
(a + b*x)*(d + e*x)^10) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)*(
d + e*x)^9)

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Rubi [A]  time = 0.609802, 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 \[ -\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{13 e^7 (a+b x) (d+e x)^{13}}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{14}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}+\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{10}}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{12}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^16,x]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x)*(d + e*x)^15) +
(3*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^14
) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d +
e*x)^13) + (5*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*
(d + e*x)^12) - (15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a
+ b*x)*(d + e*x)^11) + (3*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*
(a + b*x)*(d + e*x)^10) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)*(
d + e*x)^9)

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Rubi in Sympy [A]  time = 52.5837, size = 277, normalized size = 0.77 \[ - \frac{2 b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 e^{6} \left (d + e x\right )^{10}} + \frac{b^{5} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{45045 e^{7} \left (a + b x\right ) \left (d + e x\right )^{10}} - \frac{b^{4} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3003 e^{5} \left (d + e x\right )^{11}} - \frac{b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{273 e^{4} \left (d + e x\right )^{12}} - \frac{b^{2} \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{455 e^{3} \left (d + e x\right )^{13}} - \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{35 e^{2} \left (d + e x\right )^{14}} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{15 e \left (d + e x\right )^{15}} \]

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)**16,x)

[Out]

-2*b**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9009*e**6*(d + e*x)**10) + b**5*(a*e -
b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(45045*e**7*(a + b*x)*(d + e*x)**10) - b*
*4*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3003*e**5*(d + e*x)**11) - b*
*3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(273*e**4*(d + e*x)**12) - b**2*(5*a + 5*
b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(455*e**3*(d + e*x)**13) - b*(a**2 + 2*
a*b*x + b**2*x**2)**(5/2)/(35*e**2*(d + e*x)**14) - (a + b*x)*(a**2 + 2*a*b*x +
b**2*x**2)**(5/2)/(15*e*(d + e*x)**15)

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Mathematica [A]  time = 0.232173, size = 295, normalized size = 0.81 \[ -\frac{\sqrt{(a+b x)^2} \left (3003 a^6 e^6+1287 a^5 b e^5 (d+15 e x)+495 a^4 b^2 e^4 \left (d^2+15 d e x+105 e^2 x^2\right )+165 a^3 b^3 e^3 \left (d^3+15 d^2 e x+105 d e^2 x^2+455 e^3 x^3\right )+45 a^2 b^4 e^2 \left (d^4+15 d^3 e x+105 d^2 e^2 x^2+455 d e^3 x^3+1365 e^4 x^4\right )+9 a b^5 e \left (d^5+15 d^4 e x+105 d^3 e^2 x^2+455 d^2 e^3 x^3+1365 d e^4 x^4+3003 e^5 x^5\right )+b^6 \left (d^6+15 d^5 e x+105 d^4 e^2 x^2+455 d^3 e^3 x^3+1365 d^2 e^4 x^4+3003 d e^5 x^5+5005 e^6 x^6\right )\right )}{45045 e^7 (a+b x) (d+e x)^{15}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^16,x]

[Out]

-(Sqrt[(a + b*x)^2]*(3003*a^6*e^6 + 1287*a^5*b*e^5*(d + 15*e*x) + 495*a^4*b^2*e^
4*(d^2 + 15*d*e*x + 105*e^2*x^2) + 165*a^3*b^3*e^3*(d^3 + 15*d^2*e*x + 105*d*e^2
*x^2 + 455*e^3*x^3) + 45*a^2*b^4*e^2*(d^4 + 15*d^3*e*x + 105*d^2*e^2*x^2 + 455*d
*e^3*x^3 + 1365*e^4*x^4) + 9*a*b^5*e*(d^5 + 15*d^4*e*x + 105*d^3*e^2*x^2 + 455*d
^2*e^3*x^3 + 1365*d*e^4*x^4 + 3003*e^5*x^5) + b^6*(d^6 + 15*d^5*e*x + 105*d^4*e^
2*x^2 + 455*d^3*e^3*x^3 + 1365*d^2*e^4*x^4 + 3003*d*e^5*x^5 + 5005*e^6*x^6)))/(4
5045*e^7*(a + b*x)*(d + e*x)^15)

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Maple [A]  time = 0.02, size = 392, normalized size = 1.1 \[ -{\frac{5005\,{x}^{6}{b}^{6}{e}^{6}+27027\,{x}^{5}a{b}^{5}{e}^{6}+3003\,{x}^{5}{b}^{6}d{e}^{5}+61425\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+12285\,{x}^{4}a{b}^{5}d{e}^{5}+1365\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+75075\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+20475\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+4095\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+455\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+51975\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+17325\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+4725\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+945\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+105\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+19305\,x{a}^{5}b{e}^{6}+7425\,x{a}^{4}{b}^{2}d{e}^{5}+2475\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+675\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+135\,xa{b}^{5}{d}^{4}{e}^{2}+15\,x{b}^{6}{d}^{5}e+3003\,{a}^{6}{e}^{6}+1287\,{a}^{5}bd{e}^{5}+495\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+165\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+45\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+9\,{d}^{5}a{b}^{5}e+{b}^{6}{d}^{6}}{45045\,{e}^{7} \left ( ex+d \right ) ^{15} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^16,x)

[Out]

-1/45045/e^7*(5005*b^6*e^6*x^6+27027*a*b^5*e^6*x^5+3003*b^6*d*e^5*x^5+61425*a^2*
b^4*e^6*x^4+12285*a*b^5*d*e^5*x^4+1365*b^6*d^2*e^4*x^4+75075*a^3*b^3*e^6*x^3+204
75*a^2*b^4*d*e^5*x^3+4095*a*b^5*d^2*e^4*x^3+455*b^6*d^3*e^3*x^3+51975*a^4*b^2*e^
6*x^2+17325*a^3*b^3*d*e^5*x^2+4725*a^2*b^4*d^2*e^4*x^2+945*a*b^5*d^3*e^3*x^2+105
*b^6*d^4*e^2*x^2+19305*a^5*b*e^6*x+7425*a^4*b^2*d*e^5*x+2475*a^3*b^3*d^2*e^4*x+6
75*a^2*b^4*d^3*e^3*x+135*a*b^5*d^4*e^2*x+15*b^6*d^5*e*x+3003*a^6*e^6+1287*a^5*b*
d*e^5+495*a^4*b^2*d^2*e^4+165*a^3*b^3*d^3*e^3+45*a^2*b^4*d^4*e^2+9*a*b^5*d^5*e+b
^6*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^15/(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)^16,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294904, size = 684, normalized size = 1.89 \[ -\frac{5005 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 9 \, a b^{5} d^{5} e + 45 \, a^{2} b^{4} d^{4} e^{2} + 165 \, a^{3} b^{3} d^{3} e^{3} + 495 \, a^{4} b^{2} d^{2} e^{4} + 1287 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} + 3003 \,{\left (b^{6} d e^{5} + 9 \, a b^{5} e^{6}\right )} x^{5} + 1365 \,{\left (b^{6} d^{2} e^{4} + 9 \, a b^{5} d e^{5} + 45 \, a^{2} b^{4} e^{6}\right )} x^{4} + 455 \,{\left (b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 45 \, a^{2} b^{4} d e^{5} + 165 \, a^{3} b^{3} e^{6}\right )} x^{3} + 105 \,{\left (b^{6} d^{4} e^{2} + 9 \, a b^{5} d^{3} e^{3} + 45 \, a^{2} b^{4} d^{2} e^{4} + 165 \, a^{3} b^{3} d e^{5} + 495 \, a^{4} b^{2} e^{6}\right )} x^{2} + 15 \,{\left (b^{6} d^{5} e + 9 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} + 165 \, a^{3} b^{3} d^{2} e^{4} + 495 \, a^{4} b^{2} d e^{5} + 1287 \, a^{5} b e^{6}\right )} x}{45045 \,{\left (e^{22} x^{15} + 15 \, d e^{21} x^{14} + 105 \, d^{2} e^{20} x^{13} + 455 \, d^{3} e^{19} x^{12} + 1365 \, d^{4} e^{18} x^{11} + 3003 \, d^{5} e^{17} x^{10} + 5005 \, d^{6} e^{16} x^{9} + 6435 \, d^{7} e^{15} x^{8} + 6435 \, d^{8} e^{14} x^{7} + 5005 \, d^{9} e^{13} x^{6} + 3003 \, d^{10} e^{12} x^{5} + 1365 \, d^{11} e^{11} x^{4} + 455 \, d^{12} e^{10} x^{3} + 105 \, d^{13} e^{9} x^{2} + 15 \, d^{14} e^{8} x + d^{15} e^{7}\right )}} \]

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)^16,x, algorithm="fricas")

[Out]

-1/45045*(5005*b^6*e^6*x^6 + b^6*d^6 + 9*a*b^5*d^5*e + 45*a^2*b^4*d^4*e^2 + 165*
a^3*b^3*d^3*e^3 + 495*a^4*b^2*d^2*e^4 + 1287*a^5*b*d*e^5 + 3003*a^6*e^6 + 3003*(
b^6*d*e^5 + 9*a*b^5*e^6)*x^5 + 1365*(b^6*d^2*e^4 + 9*a*b^5*d*e^5 + 45*a^2*b^4*e^
6)*x^4 + 455*(b^6*d^3*e^3 + 9*a*b^5*d^2*e^4 + 45*a^2*b^4*d*e^5 + 165*a^3*b^3*e^6
)*x^3 + 105*(b^6*d^4*e^2 + 9*a*b^5*d^3*e^3 + 45*a^2*b^4*d^2*e^4 + 165*a^3*b^3*d*
e^5 + 495*a^4*b^2*e^6)*x^2 + 15*(b^6*d^5*e + 9*a*b^5*d^4*e^2 + 45*a^2*b^4*d^3*e^
3 + 165*a^3*b^3*d^2*e^4 + 495*a^4*b^2*d*e^5 + 1287*a^5*b*e^6)*x)/(e^22*x^15 + 15
*d*e^21*x^14 + 105*d^2*e^20*x^13 + 455*d^3*e^19*x^12 + 1365*d^4*e^18*x^11 + 3003
*d^5*e^17*x^10 + 5005*d^6*e^16*x^9 + 6435*d^7*e^15*x^8 + 6435*d^8*e^14*x^7 + 500
5*d^9*e^13*x^6 + 3003*d^10*e^12*x^5 + 1365*d^11*e^11*x^4 + 455*d^12*e^10*x^3 + 1
05*d^13*e^9*x^2 + 15*d^14*e^8*x + d^15*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)**16,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.295959, size = 702, normalized size = 1.94 \[ -\frac{{\left (5005 \, b^{6} x^{6} e^{6}{\rm sign}\left (b x + a\right ) + 3003 \, b^{6} d x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 1365 \, b^{6} d^{2} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 455 \, b^{6} d^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 105 \, b^{6} d^{4} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 15 \, b^{6} d^{5} x e{\rm sign}\left (b x + a\right ) + b^{6} d^{6}{\rm sign}\left (b x + a\right ) + 27027 \, a b^{5} x^{5} e^{6}{\rm sign}\left (b x + a\right ) + 12285 \, a b^{5} d x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 4095 \, a b^{5} d^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 945 \, a b^{5} d^{3} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 135 \, a b^{5} d^{4} x e^{2}{\rm sign}\left (b x + a\right ) + 9 \, a b^{5} d^{5} e{\rm sign}\left (b x + a\right ) + 61425 \, a^{2} b^{4} x^{4} e^{6}{\rm sign}\left (b x + a\right ) + 20475 \, a^{2} b^{4} d x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 4725 \, a^{2} b^{4} d^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 675 \, a^{2} b^{4} d^{3} x e^{3}{\rm sign}\left (b x + a\right ) + 45 \, a^{2} b^{4} d^{4} e^{2}{\rm sign}\left (b x + a\right ) + 75075 \, a^{3} b^{3} x^{3} e^{6}{\rm sign}\left (b x + a\right ) + 17325 \, a^{3} b^{3} d x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 2475 \, a^{3} b^{3} d^{2} x e^{4}{\rm sign}\left (b x + a\right ) + 165 \, a^{3} b^{3} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 51975 \, a^{4} b^{2} x^{2} e^{6}{\rm sign}\left (b x + a\right ) + 7425 \, a^{4} b^{2} d x e^{5}{\rm sign}\left (b x + a\right ) + 495 \, a^{4} b^{2} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 19305 \, a^{5} b x e^{6}{\rm sign}\left (b x + a\right ) + 1287 \, a^{5} b d e^{5}{\rm sign}\left (b x + a\right ) + 3003 \, a^{6} e^{6}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}}{45045 \,{\left (x e + d\right )}^{15}} \]

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)^16,x, algorithm="giac")

[Out]

-1/45045*(5005*b^6*x^6*e^6*sign(b*x + a) + 3003*b^6*d*x^5*e^5*sign(b*x + a) + 13
65*b^6*d^2*x^4*e^4*sign(b*x + a) + 455*b^6*d^3*x^3*e^3*sign(b*x + a) + 105*b^6*d
^4*x^2*e^2*sign(b*x + a) + 15*b^6*d^5*x*e*sign(b*x + a) + b^6*d^6*sign(b*x + a)
+ 27027*a*b^5*x^5*e^6*sign(b*x + a) + 12285*a*b^5*d*x^4*e^5*sign(b*x + a) + 4095
*a*b^5*d^2*x^3*e^4*sign(b*x + a) + 945*a*b^5*d^3*x^2*e^3*sign(b*x + a) + 135*a*b
^5*d^4*x*e^2*sign(b*x + a) + 9*a*b^5*d^5*e*sign(b*x + a) + 61425*a^2*b^4*x^4*e^6
*sign(b*x + a) + 20475*a^2*b^4*d*x^3*e^5*sign(b*x + a) + 4725*a^2*b^4*d^2*x^2*e^
4*sign(b*x + a) + 675*a^2*b^4*d^3*x*e^3*sign(b*x + a) + 45*a^2*b^4*d^4*e^2*sign(
b*x + a) + 75075*a^3*b^3*x^3*e^6*sign(b*x + a) + 17325*a^3*b^3*d*x^2*e^5*sign(b*
x + a) + 2475*a^3*b^3*d^2*x*e^4*sign(b*x + a) + 165*a^3*b^3*d^3*e^3*sign(b*x + a
) + 51975*a^4*b^2*x^2*e^6*sign(b*x + a) + 7425*a^4*b^2*d*x*e^5*sign(b*x + a) + 4
95*a^4*b^2*d^2*e^4*sign(b*x + a) + 19305*a^5*b*x*e^6*sign(b*x + a) + 1287*a^5*b*
d*e^5*sign(b*x + a) + 3003*a^6*e^6*sign(b*x + a))*e^(-7)/(x*e + d)^15