3.1754 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=262 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{504 e (d+e x)^6 (b d-a e)^4}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{84 e (d+e x)^7 (b d-a e)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{24 e (d+e x)^8 (b d-a e)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (B d-A e)}{9 e (d+e x)^9 (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e*(b*d - a*e)*(d + e
*x)^9) + ((2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(24*e*(b*d - a*e)^2*(d + e*x)^8) + (b*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^5*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(84*e*(b*d - a*e)^3*(d + e*x)^7) + (b^2*(2*b*B*d +
A*b*e - 3*a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(504*e*(b*d - a*e)^
4*(d + e*x)^6)

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Rubi [A]  time = 0.500849, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{504 e (d+e x)^6 (b d-a e)^4}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{84 e (d+e x)^7 (b d-a e)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{24 e (d+e x)^8 (b d-a e)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (B d-A e)}{9 e (d+e x)^9 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^10,x]

[Out]

-((B*d - A*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e*(b*d - a*e)*(d + e
*x)^9) + ((2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(24*e*(b*d - a*e)^2*(d + e*x)^8) + (b*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^5*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(84*e*(b*d - a*e)^3*(d + e*x)^7) + (b^2*(2*b*B*d +
A*b*e - 3*a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(504*e*(b*d - a*e)^
4*(d + e*x)^6)

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Rubi in Sympy [A]  time = 47.8315, size = 243, normalized size = 0.93 \[ \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}} \left (A b e - 3 B a e + 2 B b d\right )}{504 \left (d + e x\right )^{7} \left (a e - b d\right )^{4}} - \frac{b \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (A b e - 3 B a e + 2 B b d\right )}{144 e \left (d + e x\right )^{7} \left (a e - b d\right )^{3}} + \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (A b e - 3 B a e + 2 B b d\right )}{48 e \left (d + e x\right )^{8} \left (a e - b d\right )^{2}} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{18 e \left (d + e x\right )^{9} \left (a e - b d\right )} \]

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)**(5/2)/(e*x+d)**10,x)

[Out]

b*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)*(A*b*e - 3*B*a*e + 2*B*b*d)/(504*(d + e*x)
**7*(a*e - b*d)**4) - b*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)*(A*b*e
- 3*B*a*e + 2*B*b*d)/(144*e*(d + e*x)**7*(a*e - b*d)**3) + (2*a + 2*b*x)*(a**2
+ 2*a*b*x + b**2*x**2)**(5/2)*(A*b*e - 3*B*a*e + 2*B*b*d)/(48*e*(d + e*x)**8*(a*
e - b*d)**2) - (2*a + 2*b*x)*(A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(18
*e*(d + e*x)**9*(a*e - b*d))

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Mathematica [A]  time = 0.445951, size = 468, normalized size = 1.79 \[ -\frac{\sqrt{(a+b x)^2} \left (7 a^5 e^5 (8 A e+B (d+9 e x))+5 a^4 b e^4 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+2 a^2 b^3 e^2 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+a b^4 e \left (4 A e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+b^5 \left (A e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+2 B \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )\right )}{504 e^7 (a+b x) (d+e x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^10,x]

[Out]

-(Sqrt[(a + b*x)^2]*(7*a^5*e^5*(8*A*e + B*(d + 9*e*x)) + 5*a^4*b*e^4*(7*A*e*(d +
9*e*x) + 2*B*(d^2 + 9*d*e*x + 36*e^2*x^2)) + 10*a^3*b^2*e^3*(2*A*e*(d^2 + 9*d*e
*x + 36*e^2*x^2) + B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 2*a^2*b^3*
e^2*(5*A*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(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)) + a*b^4*e*(4*A*e*(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) + 5*B*(d^5 + 9*d^4*e*x + 36*d^3
*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) + b^5*(A*e*(d^5 + 9*d^
4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 2*B*(d^
6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^
5 + 84*e^6*x^6))))/(504*e^7*(a + b*x)*(d + e*x)^9)

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Maple [B]  time = 0.016, size = 688, normalized size = 2.6 \[ -{\frac{168\,B{x}^{6}{b}^{5}{e}^{6}+126\,A{x}^{5}{b}^{5}{e}^{6}+630\,B{x}^{5}a{b}^{4}{e}^{6}+252\,B{x}^{5}{b}^{5}d{e}^{5}+504\,A{x}^{4}a{b}^{4}{e}^{6}+126\,A{x}^{4}{b}^{5}d{e}^{5}+1008\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+630\,B{x}^{4}a{b}^{4}d{e}^{5}+252\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+840\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+336\,A{x}^{3}a{b}^{4}d{e}^{5}+84\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+840\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+672\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+420\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+168\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+720\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+360\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+144\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+36\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+360\,B{x}^{2}{a}^{4}b{e}^{6}+360\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+288\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+180\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+72\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+315\,Ax{a}^{4}b{e}^{6}+180\,Ax{a}^{3}{b}^{2}d{e}^{5}+90\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+36\,Axa{b}^{4}{d}^{3}{e}^{3}+9\,Ax{b}^{5}{d}^{4}{e}^{2}+63\,Bx{a}^{5}{e}^{6}+90\,Bx{a}^{4}bd{e}^{5}+90\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+72\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+45\,Bxa{b}^{4}{d}^{4}{e}^{2}+18\,Bx{b}^{5}{d}^{5}e+56\,A{a}^{5}{e}^{6}+35\,Ad{e}^{5}{a}^{4}b+20\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+10\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+4\,Aa{b}^{4}{d}^{4}{e}^{2}+A{b}^{5}{d}^{5}e+7\,Bd{e}^{5}{a}^{5}+10\,B{a}^{4}b{d}^{2}{e}^{4}+10\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+8\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+5\,Ba{b}^{4}{d}^{5}e+2\,B{b}^{5}{d}^{6}}{504\,{e}^{7} \left ( ex+d \right ) ^{9} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^10,x)

[Out]

-1/504/e^7*(168*B*b^5*e^6*x^6+126*A*b^5*e^6*x^5+630*B*a*b^4*e^6*x^5+252*B*b^5*d*
e^5*x^5+504*A*a*b^4*e^6*x^4+126*A*b^5*d*e^5*x^4+1008*B*a^2*b^3*e^6*x^4+630*B*a*b
^4*d*e^5*x^4+252*B*b^5*d^2*e^4*x^4+840*A*a^2*b^3*e^6*x^3+336*A*a*b^4*d*e^5*x^3+8
4*A*b^5*d^2*e^4*x^3+840*B*a^3*b^2*e^6*x^3+672*B*a^2*b^3*d*e^5*x^3+420*B*a*b^4*d^
2*e^4*x^3+168*B*b^5*d^3*e^3*x^3+720*A*a^3*b^2*e^6*x^2+360*A*a^2*b^3*d*e^5*x^2+14
4*A*a*b^4*d^2*e^4*x^2+36*A*b^5*d^3*e^3*x^2+360*B*a^4*b*e^6*x^2+360*B*a^3*b^2*d*e
^5*x^2+288*B*a^2*b^3*d^2*e^4*x^2+180*B*a*b^4*d^3*e^3*x^2+72*B*b^5*d^4*e^2*x^2+31
5*A*a^4*b*e^6*x+180*A*a^3*b^2*d*e^5*x+90*A*a^2*b^3*d^2*e^4*x+36*A*a*b^4*d^3*e^3*
x+9*A*b^5*d^4*e^2*x+63*B*a^5*e^6*x+90*B*a^4*b*d*e^5*x+90*B*a^3*b^2*d^2*e^4*x+72*
B*a^2*b^3*d^3*e^3*x+45*B*a*b^4*d^4*e^2*x+18*B*b^5*d^5*e*x+56*A*a^5*e^6+35*A*a^4*
b*d*e^5+20*A*a^3*b^2*d^2*e^4+10*A*a^2*b^3*d^3*e^3+4*A*a*b^4*d^4*e^2+A*b^5*d^5*e+
7*B*a^5*d*e^5+10*B*a^4*b*d^2*e^4+10*B*a^3*b^2*d^3*e^3+8*B*a^2*b^3*d^4*e^2+5*B*a*
b^4*d^5*e+2*B*b^5*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^9/(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)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277764, size = 871, normalized size = 3.32 \[ -\frac{168 \, B b^{5} e^{6} x^{6} + 2 \, B b^{5} d^{6} + 56 \, A a^{5} e^{6} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 10 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 126 \,{\left (2 \, B b^{5} d e^{5} +{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 126 \,{\left (2 \, B b^{5} d^{2} e^{4} +{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 84 \,{\left (2 \, B b^{5} d^{3} e^{3} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 36 \,{\left (2 \, B b^{5} d^{4} e^{2} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 10 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 9 \,{\left (2 \, B b^{5} d^{5} e +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 10 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{504 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

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)^10,x, algorithm="fricas")

[Out]

-1/504*(168*B*b^5*e^6*x^6 + 2*B*b^5*d^6 + 56*A*a^5*e^6 + (5*B*a*b^4 + A*b^5)*d^5
*e + 4*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 10*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 10
*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 7*(B*a^5 + 5*A*a^4*b)*d*e^5 + 126*(2*B*b^5*d*
e^5 + (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 126*(2*B*b^5*d^2*e^4 + (5*B*a*b^4 + A*b^5)*
d*e^5 + 4*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 84*(2*B*b^5*d^3*e^3 + (5*B*a*b^4 +
A*b^5)*d^2*e^4 + 4*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + 10*(B*a^3*b^2 + A*a^2*b^3)*e^
6)*x^3 + 36*(2*B*b^5*d^4*e^2 + (5*B*a*b^4 + A*b^5)*d^3*e^3 + 4*(2*B*a^2*b^3 + A*
a*b^4)*d^2*e^4 + 10*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 10*(B*a^4*b + 2*A*a^3*b^2)*e
^6)*x^2 + 9*(2*B*b^5*d^5*e + (5*B*a*b^4 + A*b^5)*d^4*e^2 + 4*(2*B*a^2*b^3 + A*a*
b^4)*d^3*e^3 + 10*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 10*(B*a^4*b + 2*A*a^3*b^2)*d
*e^5 + 7*(B*a^5 + 5*A*a^4*b)*e^6)*x)/(e^16*x^9 + 9*d*e^15*x^8 + 36*d^2*e^14*x^7
+ 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d
^7*e^9*x^2 + 9*d^8*e^8*x + d^9*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)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.292149, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^10,x, algorithm="giac")

[Out]

Done