3.1734 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=298 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{5 e^5 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{2 e^5 (a+b x) (d+e x)^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x) (d+e x)^7}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{8 e^5 (a+b x) (d+e x)^8}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.608338, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{5 e^5 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{2 e^5 (a+b x) (d+e x)^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x) (d+e x)^7}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{8 e^5 (a+b x) (d+e x)^8}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^9,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 53.7134, size = 291, normalized size = 0.98 \[ \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e - 2 B a e + B b d\right )}{56 e^{4} \left (d + e x\right )^{5} \left (a e - b d\right )} - \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e - 2 B a e + B b d\right )}{280 e^{5} \left (a + b x\right ) \left (d + e x\right )^{5}} + \frac{b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e - 2 B a e + B b d\right )}{84 e^{3} \left (d + e x\right )^{6} \left (a e - b d\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{16 e \left (d + e x\right )^{8} \left (a e - b d\right )} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (A b e - 2 B a e + B b d\right )}{14 e^{2} \left (d + e x\right )^{7} \left (a e - b d\right )} \]

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)**9,x)

[Out]

b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(A*b*e - 2*B*a*e + B*b*d)/(56*e**4*(d + e*
x)**5*(a*e - b*d)) - b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(A*b*e - 2*B*a*e + B*
b*d)/(280*e**5*(a + b*x)*(d + e*x)**5) + b*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b
**2*x**2)*(A*b*e - 2*B*a*e + B*b*d)/(84*e**3*(d + e*x)**6*(a*e - b*d)) - (2*a +
2*b*x)*(A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(16*e*(d + e*x)**8*(a*e -
b*d)) + (a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(A*b*e - 2*B*a*e + B*b*d)/(14*e**2*
(d + e*x)**7*(a*e - b*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.188507, size = 229, normalized size = 0.77 \[ -\frac{\sqrt{(a+b x)^2} \left (5 a^3 e^3 (7 A e+B (d+8 e x))+5 a^2 b e^2 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+a b^2 e \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )\right )}{280 e^5 (a+b x) (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^9,x]

[Out]

-(Sqrt[(a + b*x)^2]*(5*a^3*e^3*(7*A*e + B*(d + 8*e*x)) + 5*a^2*b*e^2*(3*A*e*(d +
8*e*x) + B*(d^2 + 8*d*e*x + 28*e^2*x^2)) + a*b^2*e*(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)) + b^3*(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))))/(280*e^5*(a + b*x)*(d + e*x)^8)

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 315, normalized size = 1.1 \[ -{\frac{70\,B{x}^{4}{b}^{3}{e}^{4}+56\,A{x}^{3}{b}^{3}{e}^{4}+168\,B{x}^{3}a{b}^{2}{e}^{4}+56\,B{x}^{3}{b}^{3}d{e}^{3}+140\,A{x}^{2}a{b}^{2}{e}^{4}+28\,A{x}^{2}{b}^{3}d{e}^{3}+140\,B{x}^{2}{a}^{2}b{e}^{4}+84\,B{x}^{2}a{b}^{2}d{e}^{3}+28\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+120\,Ax{a}^{2}b{e}^{4}+40\,Axa{b}^{2}d{e}^{3}+8\,Ax{b}^{3}{d}^{2}{e}^{2}+40\,Bx{a}^{3}{e}^{4}+40\,Bx{a}^{2}bd{e}^{3}+24\,Bxa{b}^{2}{d}^{2}{e}^{2}+8\,Bx{b}^{3}{d}^{3}e+35\,A{a}^{3}{e}^{4}+15\,Ad{e}^{3}{a}^{2}b+5\,Aa{b}^{2}{d}^{2}{e}^{2}+A{b}^{3}{d}^{3}e+5\,Bd{e}^{3}{a}^{3}+5\,B{a}^{2}b{d}^{2}{e}^{2}+3\,Ba{b}^{2}{d}^{3}e+B{b}^{3}{d}^{4}}{280\,{e}^{5} \left ( ex+d \right ) ^{8} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^9,x)

[Out]

-1/280/e^5*(70*B*b^3*e^4*x^4+56*A*b^3*e^4*x^3+168*B*a*b^2*e^4*x^3+56*B*b^3*d*e^3
*x^3+140*A*a*b^2*e^4*x^2+28*A*b^3*d*e^3*x^2+140*B*a^2*b*e^4*x^2+84*B*a*b^2*d*e^3
*x^2+28*B*b^3*d^2*e^2*x^2+120*A*a^2*b*e^4*x+40*A*a*b^2*d*e^3*x+8*A*b^3*d^2*e^2*x
+40*B*a^3*e^4*x+40*B*a^2*b*d*e^3*x+24*B*a*b^2*d^2*e^2*x+8*B*b^3*d^3*e*x+35*A*a^3
*e^4+15*A*a^2*b*d*e^3+5*A*a*b^2*d^2*e^2+A*b^3*d^3*e+5*B*a^3*d*e^3+5*B*a^2*b*d^2*
e^2+3*B*a*b^2*d^3*e+B*b^3*d^4)*((b*x+a)^2)^(3/2)/(e*x+d)^8/(b*x+a)^3

_______________________________________________________________________________________

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)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.312075, size = 452, normalized size = 1.52 \[ -\frac{70 \, B b^{3} e^{4} x^{4} + B b^{3} d^{4} + 35 \, A a^{3} e^{4} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 5 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 56 \,{\left (B b^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 28 \,{\left (B b^{3} d^{2} e^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 5 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 8 \,{\left (B b^{3} d^{3} e +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 5 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{280 \,{\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \]

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)^9,x, algorithm="fricas")

[Out]

-1/280*(70*B*b^3*e^4*x^4 + B*b^3*d^4 + 35*A*a^3*e^4 + (3*B*a*b^2 + A*b^3)*d^3*e
+ 5*(B*a^2*b + A*a*b^2)*d^2*e^2 + 5*(B*a^3 + 3*A*a^2*b)*d*e^3 + 56*(B*b^3*d*e^3
+ (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 28*(B*b^3*d^2*e^2 + (3*B*a*b^2 + A*b^3)*d*e^3 +
5*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 8*(B*b^3*d^3*e + (3*B*a*b^2 + A*b^3)*d^2*e^2 +
5*(B*a^2*b + A*a*b^2)*d*e^3 + 5*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^13*x^8 + 8*d*e^1
2*x^7 + 28*d^2*e^11*x^6 + 56*d^3*e^10*x^5 + 70*d^4*e^9*x^4 + 56*d^5*e^8*x^3 + 28
*d^6*e^7*x^2 + 8*d^7*e^6*x + d^8*e^5)

_______________________________________________________________________________________

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)**9,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.294142, size = 574, normalized size = 1.93 \[ -\frac{{\left (70 \, B b^{3} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 56 \, B b^{3} d x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 28 \, B b^{3} d^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 8 \, B b^{3} d^{3} x e{\rm sign}\left (b x + a\right ) + B b^{3} d^{4}{\rm sign}\left (b x + a\right ) + 168 \, B a b^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 56 \, A b^{3} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 84 \, B a b^{2} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 28 \, A b^{3} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 24 \, B a b^{2} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 8 \, A b^{3} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 3 \, B a b^{2} d^{3} e{\rm sign}\left (b x + a\right ) + A b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 140 \, B a^{2} b x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 140 \, A a b^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 40 \, B a^{2} b d x e^{3}{\rm sign}\left (b x + a\right ) + 40 \, A a b^{2} d x e^{3}{\rm sign}\left (b x + a\right ) + 5 \, B a^{2} b d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 5 \, A a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 40 \, B a^{3} x e^{4}{\rm sign}\left (b x + a\right ) + 120 \, A a^{2} b x e^{4}{\rm sign}\left (b x + a\right ) + 5 \, B a^{3} d e^{3}{\rm sign}\left (b x + a\right ) + 15 \, A a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + 35 \, A a^{3} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{280 \,{\left (x e + d\right )}^{8}} \]

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)^9,x, algorithm="giac")

[Out]

-1/280*(70*B*b^3*x^4*e^4*sign(b*x + a) + 56*B*b^3*d*x^3*e^3*sign(b*x + a) + 28*B
*b^3*d^2*x^2*e^2*sign(b*x + a) + 8*B*b^3*d^3*x*e*sign(b*x + a) + B*b^3*d^4*sign(
b*x + a) + 168*B*a*b^2*x^3*e^4*sign(b*x + a) + 56*A*b^3*x^3*e^4*sign(b*x + a) +
84*B*a*b^2*d*x^2*e^3*sign(b*x + a) + 28*A*b^3*d*x^2*e^3*sign(b*x + a) + 24*B*a*b
^2*d^2*x*e^2*sign(b*x + a) + 8*A*b^3*d^2*x*e^2*sign(b*x + a) + 3*B*a*b^2*d^3*e*s
ign(b*x + a) + A*b^3*d^3*e*sign(b*x + a) + 140*B*a^2*b*x^2*e^4*sign(b*x + a) + 1
40*A*a*b^2*x^2*e^4*sign(b*x + a) + 40*B*a^2*b*d*x*e^3*sign(b*x + a) + 40*A*a*b^2
*d*x*e^3*sign(b*x + a) + 5*B*a^2*b*d^2*e^2*sign(b*x + a) + 5*A*a*b^2*d^2*e^2*sig
n(b*x + a) + 40*B*a^3*x*e^4*sign(b*x + a) + 120*A*a^2*b*x*e^4*sign(b*x + a) + 5*
B*a^3*d*e^3*sign(b*x + a) + 15*A*a^2*b*d*e^3*sign(b*x + a) + 35*A*a^3*e^4*sign(b
*x + a))*e^(-5)/(x*e + d)^8