3.1985 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=254 \[ -\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^5 (a+b x) (d+e x)^7}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^5 (a+b x) (d+e x)^8}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^5 (a+b x) (d+e x)^9}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^5 (a+b x) (d+e x)^6} \]

[Out]

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

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Rubi [A]  time = 0.371039, antiderivative size = 254, 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 \[ -\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^5 (a+b x) (d+e x)^7}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^5 (a+b x) (d+e x)^8}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^5 (a+b x) (d+e x)^9}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^5 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^10,x]

[Out]

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

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Rubi in Sympy [A]  time = 70.8868, size = 196, normalized size = 0.77 \[ - \frac{b^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{630 \left (d + e x\right )^{5} \left (a e - b d\right )^{5}} + \frac{b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{126 \left (d + e x\right )^{6} \left (a e - b d\right )^{4}} - \frac{b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{42 \left (d + e x\right )^{7} \left (a e - b d\right )^{3}} + \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{18 \left (d + e x\right )^{8} \left (a e - b d\right )^{2}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{9 \left (d + e x\right )^{9} \left (a e - b d\right )} \]

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)**(3/2)/(e*x+d)**10,x)

[Out]

-b**4*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(630*(d + e*x)**5*(a*e - b*d)**5) + b*
*3*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(126*(d + e*x)**6*(a*e - b*d)**4) - b**2*
(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(42*(d + e*x)**7*(a*e - b*d)**3) + b*(a**2 +
2*a*b*x + b**2*x**2)**(5/2)/(18*(d + e*x)**8*(a*e - b*d)**2) - (a**2 + 2*a*b*x
+ b**2*x**2)**(5/2)/(9*(d + e*x)**9*(a*e - b*d))

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Mathematica [A]  time = 0.132012, size = 162, normalized size = 0.64 \[ -\frac{\sqrt{(a+b x)^2} \left (70 a^4 e^4+35 a^3 b e^3 (d+9 e x)+15 a^2 b^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+5 a b^3 e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+b^4 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )}{630 e^5 (a+b x) (d+e x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^10,x]

[Out]

-(Sqrt[(a + b*x)^2]*(70*a^4*e^4 + 35*a^3*b*e^3*(d + 9*e*x) + 15*a^2*b^2*e^2*(d^2
+ 9*d*e*x + 36*e^2*x^2) + 5*a*b^3*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^
3) + b^4*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)))/(630*
e^5*(a + b*x)*(d + e*x)^9)

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Maple [A]  time = 0.014, size = 201, normalized size = 0.8 \[ -{\frac{126\,{x}^{4}{b}^{4}{e}^{4}+420\,{x}^{3}a{b}^{3}{e}^{4}+84\,{x}^{3}{b}^{4}d{e}^{3}+540\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+180\,{x}^{2}a{b}^{3}d{e}^{3}+36\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+315\,x{a}^{3}b{e}^{4}+135\,x{a}^{2}{b}^{2}d{e}^{3}+45\,xa{b}^{3}{d}^{2}{e}^{2}+9\,x{b}^{4}{d}^{3}e+70\,{a}^{4}{e}^{4}+35\,{a}^{3}bd{e}^{3}+15\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+5\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4}}{630\,{e}^{5} \left ( ex+d \right ) ^{9} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^(3/2)/(e*x+d)^10,x)

[Out]

-1/630/e^5*(126*b^4*e^4*x^4+420*a*b^3*e^4*x^3+84*b^4*d*e^3*x^3+540*a^2*b^2*e^4*x
^2+180*a*b^3*d*e^3*x^2+36*b^4*d^2*e^2*x^2+315*a^3*b*e^4*x+135*a^2*b^2*d*e^3*x+45
*a*b^3*d^2*e^2*x+9*b^4*d^3*e*x+70*a^4*e^4+35*a^3*b*d*e^3+15*a^2*b^2*d^2*e^2+5*a*
b^3*d^3*e+b^4*d^4)*((b*x+a)^2)^(3/2)/(e*x+d)^9/(b*x+a)^3

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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)^(3/2)*(b*x + a)/(e*x + d)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300914, size = 363, normalized size = 1.43 \[ -\frac{126 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 5 \, a b^{3} d^{3} e + 15 \, a^{2} b^{2} d^{2} e^{2} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4} + 84 \,{\left (b^{4} d e^{3} + 5 \, a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (b^{4} d^{2} e^{2} + 5 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 9 \,{\left (b^{4} d^{3} e + 5 \, a b^{3} d^{2} e^{2} + 15 \, a^{2} b^{2} d e^{3} + 35 \, a^{3} b e^{4}\right )} x}{630 \,{\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^10,x, algorithm="fricas")

[Out]

-1/630*(126*b^4*e^4*x^4 + b^4*d^4 + 5*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 35*a^3*
b*d*e^3 + 70*a^4*e^4 + 84*(b^4*d*e^3 + 5*a*b^3*e^4)*x^3 + 36*(b^4*d^2*e^2 + 5*a*
b^3*d*e^3 + 15*a^2*b^2*e^4)*x^2 + 9*(b^4*d^3*e + 5*a*b^3*d^2*e^2 + 15*a^2*b^2*d*
e^3 + 35*a^3*b*e^4)*x)/(e^14*x^9 + 9*d*e^13*x^8 + 36*d^2*e^12*x^7 + 84*d^3*e^11*
x^6 + 126*d^4*e^10*x^5 + 126*d^5*e^9*x^4 + 84*d^6*e^8*x^3 + 36*d^7*e^7*x^2 + 9*d
^8*e^6*x + d^9*e^5)

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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)**(3/2)/(e*x+d)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280354, size = 356, normalized size = 1.4 \[ -\frac{{\left (126 \, b^{4} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 84 \, b^{4} d x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 36 \, b^{4} d^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 9 \, b^{4} d^{3} x e{\rm sign}\left (b x + a\right ) + b^{4} d^{4}{\rm sign}\left (b x + a\right ) + 420 \, a b^{3} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 180 \, a b^{3} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 45 \, a b^{3} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 5 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 540 \, a^{2} b^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 135 \, a^{2} b^{2} d x e^{3}{\rm sign}\left (b x + a\right ) + 15 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 315 \, a^{3} b x e^{4}{\rm sign}\left (b x + a\right ) + 35 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + 70 \, a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{630 \,{\left (x e + d\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^10,x, algorithm="giac")

[Out]

-1/630*(126*b^4*x^4*e^4*sign(b*x + a) + 84*b^4*d*x^3*e^3*sign(b*x + a) + 36*b^4*
d^2*x^2*e^2*sign(b*x + a) + 9*b^4*d^3*x*e*sign(b*x + a) + b^4*d^4*sign(b*x + a)
+ 420*a*b^3*x^3*e^4*sign(b*x + a) + 180*a*b^3*d*x^2*e^3*sign(b*x + a) + 45*a*b^3
*d^2*x*e^2*sign(b*x + a) + 5*a*b^3*d^3*e*sign(b*x + a) + 540*a^2*b^2*x^2*e^4*sig
n(b*x + a) + 135*a^2*b^2*d*x*e^3*sign(b*x + a) + 15*a^2*b^2*d^2*e^2*sign(b*x + a
) + 315*a^3*b*x*e^4*sign(b*x + a) + 35*a^3*b*d*e^3*sign(b*x + a) + 70*a^4*e^4*si
gn(b*x + a))*e^(-5)/(x*e + d)^9