3.1752 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=106 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B)}{6 (d+e x)^6 (b d-a e)^2}+\frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2} (B d-A e)}{7 (d+e x)^7 (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.207727, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B)}{6 (d+e x)^6 (b d-a e)^2}+\frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2} (B d-A e)}{7 (d+e x)^7 (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 36.6355, size = 95, normalized size = 0.9 \[ \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 \left (d + e x\right )^{6} \left (a e - b d\right )^{2}} - \frac{\left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{7 \left (d + e x\right )^{7} \left (a e - b d\right )^{2}} \]

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)**8,x)

[Out]

(2*a + 2*b*x)*(A*b - B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(12*(d + e*x)**6*(
a*e - b*d)**2) - (A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(7*(d + e*x)**7
*(a*e - b*d)**2)

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Mathematica [B]  time = 0.477091, size = 465, normalized size = 4.39 \[ -\frac{\sqrt{(a+b x)^2} \left (a^5 e^5 (6 A e+B (d+7 e x))+a^4 b e^4 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+a^3 b^2 e^3 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+a^2 b^3 e^2 \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+a b^4 e \left (2 A e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 B \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )+b^5 \left (A e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+6 B \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )\right )}{42 e^7 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^8,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a^5*e^5*(6*A*e + B*(d + 7*e*x)) + a^4*b*e^4*(5*A*e*(d + 7*e
*x) + 2*B*(d^2 + 7*d*e*x + 21*e^2*x^2)) + a^3*b^2*e^3*(4*A*e*(d^2 + 7*d*e*x + 21
*e^2*x^2) + 3*B*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)) + a^2*b^3*e^2*(3*
A*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*B*(d^4 + 7*d^3*e*x + 21*d^
2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + a*b^4*e*(2*A*e*(d^4 + 7*d^3*e*x + 21*d
^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*B*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2
+ 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)) + b^5*(A*e*(d^5 + 7*d^4*e*x + 21*
d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + 6*B*(d^6 + 7*d^5*e*x
+ 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6))
))/(42*e^7*(a + b*x)*(d + e*x)^7)

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Maple [B]  time = 0.016, size = 687, normalized size = 6.5 \[ -{\frac{42\,B{x}^{6}{b}^{5}{e}^{6}+21\,A{x}^{5}{b}^{5}{e}^{6}+105\,B{x}^{5}a{b}^{4}{e}^{6}+126\,B{x}^{5}{b}^{5}d{e}^{5}+70\,A{x}^{4}a{b}^{4}{e}^{6}+35\,A{x}^{4}{b}^{5}d{e}^{5}+140\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+175\,B{x}^{4}a{b}^{4}d{e}^{5}+210\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+105\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+70\,A{x}^{3}a{b}^{4}d{e}^{5}+35\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+105\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+140\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+175\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+210\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+84\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+63\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+42\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+21\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+42\,B{x}^{2}{a}^{4}b{e}^{6}+63\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+84\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+105\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+126\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+35\,Ax{a}^{4}b{e}^{6}+28\,Ax{a}^{3}{b}^{2}d{e}^{5}+21\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+14\,Axa{b}^{4}{d}^{3}{e}^{3}+7\,Ax{b}^{5}{d}^{4}{e}^{2}+7\,Bx{a}^{5}{e}^{6}+14\,Bx{a}^{4}bd{e}^{5}+21\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+28\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+35\,Bxa{b}^{4}{d}^{4}{e}^{2}+42\,Bx{b}^{5}{d}^{5}e+6\,A{a}^{5}{e}^{6}+5\,Ad{e}^{5}{a}^{4}b+4\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+3\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+2\,Aa{b}^{4}{d}^{4}{e}^{2}+A{b}^{5}{d}^{5}e+Bd{e}^{5}{a}^{5}+2\,B{a}^{4}b{d}^{2}{e}^{4}+3\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+4\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+5\,Ba{b}^{4}{d}^{5}e+6\,B{b}^{5}{d}^{6}}{42\, \left ( ex+d \right ) ^{7}{e}^{7} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^8,x)

[Out]

-1/42*(42*B*b^5*e^6*x^6+21*A*b^5*e^6*x^5+105*B*a*b^4*e^6*x^5+126*B*b^5*d*e^5*x^5
+70*A*a*b^4*e^6*x^4+35*A*b^5*d*e^5*x^4+140*B*a^2*b^3*e^6*x^4+175*B*a*b^4*d*e^5*x
^4+210*B*b^5*d^2*e^4*x^4+105*A*a^2*b^3*e^6*x^3+70*A*a*b^4*d*e^5*x^3+35*A*b^5*d^2
*e^4*x^3+105*B*a^3*b^2*e^6*x^3+140*B*a^2*b^3*d*e^5*x^3+175*B*a*b^4*d^2*e^4*x^3+2
10*B*b^5*d^3*e^3*x^3+84*A*a^3*b^2*e^6*x^2+63*A*a^2*b^3*d*e^5*x^2+42*A*a*b^4*d^2*
e^4*x^2+21*A*b^5*d^3*e^3*x^2+42*B*a^4*b*e^6*x^2+63*B*a^3*b^2*d*e^5*x^2+84*B*a^2*
b^3*d^2*e^4*x^2+105*B*a*b^4*d^3*e^3*x^2+126*B*b^5*d^4*e^2*x^2+35*A*a^4*b*e^6*x+2
8*A*a^3*b^2*d*e^5*x+21*A*a^2*b^3*d^2*e^4*x+14*A*a*b^4*d^3*e^3*x+7*A*b^5*d^4*e^2*
x+7*B*a^5*e^6*x+14*B*a^4*b*d*e^5*x+21*B*a^3*b^2*d^2*e^4*x+28*B*a^2*b^3*d^3*e^3*x
+35*B*a*b^4*d^4*e^2*x+42*B*b^5*d^5*e*x+6*A*a^5*e^6+5*A*a^4*b*d*e^5+4*A*a^3*b^2*d
^2*e^4+3*A*a^2*b^3*d^3*e^3+2*A*a*b^4*d^4*e^2+A*b^5*d^5*e+B*a^5*d*e^5+2*B*a^4*b*d
^2*e^4+3*B*a^3*b^2*d^3*e^3+4*B*a^2*b^3*d^4*e^2+5*B*a*b^4*d^5*e+6*B*b^5*d^6)*((b*
x+a)^2)^(5/2)/(e*x+d)^7/e^7/(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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277347, size = 838, normalized size = 7.91 \[ -\frac{42 \, B b^{5} e^{6} x^{6} + 6 \, B b^{5} d^{6} + 6 \, A a^{5} e^{6} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 2 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 3 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 2 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} +{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 21 \,{\left (6 \, B b^{5} d e^{5} +{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 35 \,{\left (6 \, B b^{5} d^{2} e^{4} +{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 2 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 35 \,{\left (6 \, B b^{5} d^{3} e^{3} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 2 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 3 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 21 \,{\left (6 \, B b^{5} d^{4} e^{2} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 2 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 3 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 2 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 7 \,{\left (6 \, B b^{5} d^{5} e +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 2 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 3 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 2 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} +{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{42 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]

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)^8,x, algorithm="fricas")

[Out]

-1/42*(42*B*b^5*e^6*x^6 + 6*B*b^5*d^6 + 6*A*a^5*e^6 + (5*B*a*b^4 + A*b^5)*d^5*e
+ 2*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 3*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 2*(B*a
^4*b + 2*A*a^3*b^2)*d^2*e^4 + (B*a^5 + 5*A*a^4*b)*d*e^5 + 21*(6*B*b^5*d*e^5 + (5
*B*a*b^4 + A*b^5)*e^6)*x^5 + 35*(6*B*b^5*d^2*e^4 + (5*B*a*b^4 + A*b^5)*d*e^5 + 2
*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 35*(6*B*b^5*d^3*e^3 + (5*B*a*b^4 + A*b^5)*d^
2*e^4 + 2*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + 3*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 2
1*(6*B*b^5*d^4*e^2 + (5*B*a*b^4 + A*b^5)*d^3*e^3 + 2*(2*B*a^2*b^3 + A*a*b^4)*d^2
*e^4 + 3*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 2*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 7*
(6*B*b^5*d^5*e + (5*B*a*b^4 + A*b^5)*d^4*e^2 + 2*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3
+ 3*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 2*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + (B*a^5
+ 5*A*a^4*b)*e^6)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^
4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*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)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293677, 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)^8,x, algorithm="giac")

[Out]

Done