3.1753 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=193 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{28 e (d+e x)^7 (b d-a e)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e*(b*d - a*e)*(d + e
*x)^8) + ((3*b*B*d + A*b*e - 4*a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(28*e*(b*d - a*e)^2*(d + e*x)^7) + (b*(3*b*B*d + A*b*e - 4*a*B*e)*(a + b*x)^5*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(168*e*(b*d - a*e)^3*(d + e*x)^6)

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Rubi [A]  time = 0.397519, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{28 e (d+e x)^7 (b d-a e)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^9,x]

[Out]

-((B*d - A*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e*(b*d - a*e)*(d + e
*x)^8) + ((3*b*B*d + A*b*e - 4*a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(28*e*(b*d - a*e)^2*(d + e*x)^7) + (b*(3*b*B*d + A*b*e - 4*a*B*e)*(a + b*x)^5*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(168*e*(b*d - a*e)^3*(d + e*x)^6)

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Rubi in Sympy [A]  time = 32.9036, size = 173, normalized size = 0.9 \[ - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}} \left (A b e - 4 B a e + 3 B b d\right )}{168 \left (d + e x\right )^{7} \left (a e - b d\right )^{3}} + \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (A b e - 4 B a e + 3 B b d\right )}{48 e \left (d + e x\right )^{7} \left (a e - b d\right )^{2}} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{16 e \left (d + e x\right )^{8} \left (a e - b d\right )} \]

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)**9,x)

[Out]

-(a**2 + 2*a*b*x + b**2*x**2)**(7/2)*(A*b*e - 4*B*a*e + 3*B*b*d)/(168*(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 -
4*B*a*e + 3*B*b*d)/(48*e*(d + e*x)**7*(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)/(16*e*(d + e*x)**8*(a*e - b*d))

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Mathematica [B]  time = 0.531876, size = 466, normalized size = 2.41 \[ -\frac{\sqrt{(a+b x)^2} \left (3 a^5 e^5 (7 A e+B (d+8 e x))+5 a^4 b e^4 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+2 a^3 b^2 e^3 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+6 a^2 b^3 e^2 \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+a b^4 e \left (3 A e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )+b^5 \left (A e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+3 B \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )\right )}{168 e^7 (a+b x) (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^9,x]

[Out]

-(Sqrt[(a + b*x)^2]*(3*a^5*e^5*(7*A*e + B*(d + 8*e*x)) + 5*a^4*b*e^4*(3*A*e*(d +
8*e*x) + B*(d^2 + 8*d*e*x + 28*e^2*x^2)) + 2*a^3*b^2*e^3*(5*A*e*(d^2 + 8*d*e*x
+ 28*e^2*x^2) + 3*B*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + 6*a^2*b^3*e
^2*(A*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + B*(d^4 + 8*d^3*e*x + 28*
d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + a*b^4*e*(3*A*e*(d^4 + 8*d^3*e*x + 28
*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*B*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^
2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)) + b^5*(A*e*(d^5 + 8*d^4*e*x + 2
8*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + 3*B*(d^6 + 8*d^5*e
*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^
6))))/(168*e^7*(a + b*x)*(d + e*x)^8)

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Maple [B]  time = 0.015, size = 688, normalized size = 3.6 \[ -{\frac{84\,B{x}^{6}{b}^{5}{e}^{6}+56\,A{x}^{5}{b}^{5}{e}^{6}+280\,B{x}^{5}a{b}^{4}{e}^{6}+168\,B{x}^{5}{b}^{5}d{e}^{5}+210\,A{x}^{4}a{b}^{4}{e}^{6}+70\,A{x}^{4}{b}^{5}d{e}^{5}+420\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+350\,B{x}^{4}a{b}^{4}d{e}^{5}+210\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+336\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+168\,A{x}^{3}a{b}^{4}d{e}^{5}+56\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+336\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+336\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+280\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+168\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+280\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+168\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+84\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+28\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+140\,B{x}^{2}{a}^{4}b{e}^{6}+168\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+168\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+140\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+84\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+120\,Ax{a}^{4}b{e}^{6}+80\,Ax{a}^{3}{b}^{2}d{e}^{5}+48\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+24\,Axa{b}^{4}{d}^{3}{e}^{3}+8\,Ax{b}^{5}{d}^{4}{e}^{2}+24\,Bx{a}^{5}{e}^{6}+40\,Bx{a}^{4}bd{e}^{5}+48\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+48\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+40\,Bxa{b}^{4}{d}^{4}{e}^{2}+24\,Bx{b}^{5}{d}^{5}e+21\,A{a}^{5}{e}^{6}+15\,Ad{e}^{5}{a}^{4}b+10\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+6\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+3\,Aa{b}^{4}{d}^{4}{e}^{2}+A{b}^{5}{d}^{5}e+3\,Bd{e}^{5}{a}^{5}+5\,B{a}^{4}b{d}^{2}{e}^{4}+6\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+6\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+5\,Ba{b}^{4}{d}^{5}e+3\,B{b}^{5}{d}^{6}}{168\,{e}^{7} \left ( ex+d \right ) ^{8} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^9,x)

[Out]

-1/168/e^7*(84*B*b^5*e^6*x^6+56*A*b^5*e^6*x^5+280*B*a*b^4*e^6*x^5+168*B*b^5*d*e^
5*x^5+210*A*a*b^4*e^6*x^4+70*A*b^5*d*e^5*x^4+420*B*a^2*b^3*e^6*x^4+350*B*a*b^4*d
*e^5*x^4+210*B*b^5*d^2*e^4*x^4+336*A*a^2*b^3*e^6*x^3+168*A*a*b^4*d*e^5*x^3+56*A*
b^5*d^2*e^4*x^3+336*B*a^3*b^2*e^6*x^3+336*B*a^2*b^3*d*e^5*x^3+280*B*a*b^4*d^2*e^
4*x^3+168*B*b^5*d^3*e^3*x^3+280*A*a^3*b^2*e^6*x^2+168*A*a^2*b^3*d*e^5*x^2+84*A*a
*b^4*d^2*e^4*x^2+28*A*b^5*d^3*e^3*x^2+140*B*a^4*b*e^6*x^2+168*B*a^3*b^2*d*e^5*x^
2+168*B*a^2*b^3*d^2*e^4*x^2+140*B*a*b^4*d^3*e^3*x^2+84*B*b^5*d^4*e^2*x^2+120*A*a
^4*b*e^6*x+80*A*a^3*b^2*d*e^5*x+48*A*a^2*b^3*d^2*e^4*x+24*A*a*b^4*d^3*e^3*x+8*A*
b^5*d^4*e^2*x+24*B*a^5*e^6*x+40*B*a^4*b*d*e^5*x+48*B*a^3*b^2*d^2*e^4*x+48*B*a^2*
b^3*d^3*e^3*x+40*B*a*b^4*d^4*e^2*x+24*B*b^5*d^5*e*x+21*A*a^5*e^6+15*A*a^4*b*d*e^
5+10*A*a^3*b^2*d^2*e^4+6*A*a^2*b^3*d^3*e^3+3*A*a*b^4*d^4*e^2+A*b^5*d^5*e+3*B*a^5
*d*e^5+5*B*a^4*b*d^2*e^4+6*B*a^3*b^2*d^3*e^3+6*B*a^2*b^3*d^4*e^2+5*B*a*b^4*d^5*e
+3*B*b^5*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^8/(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)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276162, size = 856, normalized size = 4.44 \[ -\frac{84 \, B b^{5} e^{6} x^{6} + 3 \, B b^{5} d^{6} + 21 \, A a^{5} e^{6} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 3 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 6 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 56 \,{\left (3 \, B b^{5} d e^{5} +{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 70 \,{\left (3 \, B b^{5} d^{2} e^{4} +{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 3 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 56 \,{\left (3 \, B b^{5} d^{3} e^{3} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 3 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 6 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 28 \,{\left (3 \, B b^{5} d^{4} e^{2} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 3 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 6 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 8 \,{\left (3 \, B b^{5} d^{5} e +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 3 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 6 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{168 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]

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)^9,x, algorithm="fricas")

[Out]

-1/168*(84*B*b^5*e^6*x^6 + 3*B*b^5*d^6 + 21*A*a^5*e^6 + (5*B*a*b^4 + A*b^5)*d^5*
e + 3*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 6*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 5*(B
*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 3*(B*a^5 + 5*A*a^4*b)*d*e^5 + 56*(3*B*b^5*d*e^5
+ (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 70*(3*B*b^5*d^2*e^4 + (5*B*a*b^4 + A*b^5)*d*e^5
+ 3*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 56*(3*B*b^5*d^3*e^3 + (5*B*a*b^4 + A*b^5
)*d^2*e^4 + 3*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + 6*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3
+ 28*(3*B*b^5*d^4*e^2 + (5*B*a*b^4 + A*b^5)*d^3*e^3 + 3*(2*B*a^2*b^3 + A*a*b^4)
*d^2*e^4 + 6*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 5*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2
+ 8*(3*B*b^5*d^5*e + (5*B*a*b^4 + A*b^5)*d^4*e^2 + 3*(2*B*a^2*b^3 + A*a*b^4)*d^3
*e^3 + 6*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 5*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + 3*(
B*a^5 + 5*A*a^4*b)*e^6)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e
^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8
*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)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.295294, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

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)^9,x, algorithm="giac")

[Out]

Done