3.1747 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=424 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{2 e^7 (a+b x) (d+e x)^2}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x) (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}-\frac{10 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^6 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (-5 a B e-A b e+6 b B d)}{3 e^7 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e) (-2 a B e-A b e+3 b B d)}{2 e^7 (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)} \]

[Out]

(-10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(e^6*(a + b*x)) - ((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2
*e^7*(a + b*x)*(d + e*x)^2) + ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) + (5*b^3*(b*d - a*e)*(3*b*B*d
- A*b*e - 2*a*B*e)*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x))
- (b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3
*e^7*(a + b*x)) + (b^5*B*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a +
b*x)) + (5*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]*Log[d + e*x])/(e^7*(a + b*x))

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Rubi [A]  time = 1.21364, antiderivative size = 424, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{2 e^7 (a+b x) (d+e x)^2}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x) (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}-\frac{10 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^6 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (-5 a B e-A b e+6 b B d)}{3 e^7 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e) (-2 a B e-A b e+3 b B d)}{2 e^7 (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(-10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(e^6*(a + b*x)) - ((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2
*e^7*(a + b*x)*(d + e*x)^2) + ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) + (5*b^3*(b*d - a*e)*(3*b*B*d
- A*b*e - 2*a*B*e)*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x))
- (b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3
*e^7*(a + b*x)) + (b^5*B*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a +
b*x)) + (5*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]*Log[d + e*x])/(e^7*(a + b*x))

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Rubi in Sympy [A]  time = 81.5803, size = 396, normalized size = 0.93 \[ \frac{5 b \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (2 A b e + B a e - 3 B b d\right )}{4 e^{3} \left (a e - b d\right )} + \frac{5 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (2 A b e + B a e - 3 B b d\right )}{3 e^{4}} + \frac{5 b \left (3 a + 3 b x\right ) \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (2 A b e + B a e - 3 B b d\right )}{6 e^{5}} + \frac{5 b \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (2 A b e + B a e - 3 B b d\right )}{e^{6}} + \frac{5 b \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (2 A b e + B a e - 3 B b d\right ) \log{\left (d + e x \right )}}{e^{7} \left (a + b x\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{4 e \left (d + e x\right )^{2} \left (a e - b d\right )} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (2 A b e + B a e - 3 B b d\right )}{e^{2} \left (d + e x\right ) \left (a e - b d\right )} \]

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)**3,x)

[Out]

5*b*(a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(2*A*b*e + B*a*e - 3*B*b*d)/(4
*e**3*(a*e - b*d)) + 5*b*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(2*A*b*e + B*a*e -
3*B*b*d)/(3*e**4) + 5*b*(3*a + 3*b*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**
2)*(2*A*b*e + B*a*e - 3*B*b*d)/(6*e**5) + 5*b*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x
+ b**2*x**2)*(2*A*b*e + B*a*e - 3*B*b*d)/e**6 + 5*b*(a*e - b*d)**3*sqrt(a**2 +
2*a*b*x + b**2*x**2)*(2*A*b*e + B*a*e - 3*B*b*d)*log(d + e*x)/(e**7*(a + b*x)) -
(2*a + 2*b*x)*(A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(4*e*(d + e*x)**2
*(a*e - b*d)) - (a**2 + 2*a*b*x + b**2*x**2)**(5/2)*(2*A*b*e + B*a*e - 3*B*b*d)/
(e**2*(d + e*x)*(a*e - b*d))

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Mathematica [A]  time = 0.576529, size = 501, normalized size = 1.18 \[ \frac{\sqrt{(a+b x)^2} \left (-6 a^5 e^5 (A e+B (d+2 e x))-30 a^4 b e^4 (A e (d+2 e x)-B d (3 d+4 e x))+60 a^3 b^2 e^3 \left (A d e (3 d+4 e x)+B \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )\right )+60 a^2 b^3 e^2 \left (A e \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+B \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )\right )+10 a b^4 e \left (3 A e \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+B \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )\right )+60 b (d+e x)^2 (b d-a e)^3 \log (d+e x) (-a B e-2 A b e+3 b B d)+b^5 \left (2 A e \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )+3 B \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )\right )\right )}{12 e^7 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(Sqrt[(a + b*x)^2]*(-6*a^5*e^5*(A*e + B*(d + 2*e*x)) - 30*a^4*b*e^4*(A*e*(d + 2*
e*x) - B*d*(3*d + 4*e*x)) + 60*a^3*b^2*e^3*(A*d*e*(3*d + 4*e*x) + B*(-5*d^3 - 4*
d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3)) + 60*a^2*b^3*e^2*(A*e*(-5*d^3 - 4*d^2*e*x +
4*d*e^2*x^2 + 2*e^3*x^3) + B*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 +
e^4*x^4)) + 10*a*b^4*e*(3*A*e*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3
+ e^4*x^4) + B*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4
*x^4 + 2*e^5*x^5)) + b^5*(2*A*e*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e
^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + 3*B*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 -
20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6)) + 60*b*(b*d - a*e)^3*(3
*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2*Log[d + e*x]))/(12*e^7*(a + b*x)*(d + e*x)
^2)

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Maple [B]  time = 0.034, size = 1205, normalized size = 2.8 \[ \text{result too large to display} \]

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)^3,x)

[Out]

1/12*((b*x+a)^2)^(5/2)*(-54*A*b^5*d^5*e+240*B*x^2*a^3*b^2*d*e^5+120*B*x^3*a^3*b^
2*e^6+180*A*a^3*b^2*d^2*e^4-300*A*a^2*b^3*d^3*e^3-6*B*d*e^5*a^5-6*A*a^5*e^6+66*B
*b^5*d^6-10*A*x^4*b^5*d*e^5+120*B*ln(e*x+d)*x*a^4*b*d*e^5+1440*B*ln(e*x+d)*x*a^2
*b^3*d^3*e^3+60*B*x^4*a^2*b^3*e^6+20*B*x^5*a*b^4*e^6-270*B*a*b^4*d^5*e-300*B*a^3
*b^2*d^3*e^3-204*B*x^2*b^5*d^4*e^2-60*A*x*a^4*b*e^6+12*A*x*b^5*d^4*e^2+720*B*ln(
e*x+d)*x^2*a^2*b^3*d^2*e^4-600*B*ln(e*x+d)*x^2*a*b^4*d^3*e^3-360*A*ln(e*x+d)*x^2
*a^2*b^3*d*e^5+360*A*ln(e*x+d)*x^2*a*b^4*d^2*e^4-360*B*ln(e*x+d)*x^2*a^3*b^2*d*e
^5+60*B*ln(e*x+d)*x^2*a^4*b*e^6+180*B*ln(e*x+d)*x^2*b^5*d^4*e^2+120*A*ln(e*x+d)*
x^2*a^3*b^2*e^6-120*A*ln(e*x+d)*x^2*b^5*d^3*e^3-660*B*x^2*a^2*b^3*d^2*e^4+240*A*
x^2*a^2*b^3*d*e^5-330*A*x^2*a*b^4*d^2*e^4-120*A*x^3*a*b^4*d*e^5-240*B*x^3*a^2*b^
3*d*e^5+200*B*x^3*a*b^4*d^2*e^4-50*B*x^4*a*b^4*d*e^5-240*A*ln(e*x+d)*x*b^5*d^4*e
^2+360*B*ln(e*x+d)*x*b^5*d^5*e-360*B*ln(e*x+d)*a^3*b^2*d^3*e^3+720*B*ln(e*x+d)*a
^2*b^3*d^4*e^2-600*B*ln(e*x+d)*a*b^4*d^5*e+120*A*ln(e*x+d)*a^3*b^2*d^2*e^4-60*B*
x^3*b^5*d^3*e^3+126*A*x^2*b^5*d^3*e^3-30*A*d*e^5*a^4*b+15*B*x^4*b^5*d^2*e^4+120*
A*x^3*a^2*b^3*e^6+40*A*x^3*b^5*d^2*e^4-360*A*ln(e*x+d)*a^2*b^3*d^3*e^3+360*A*ln(
e*x+d)*a*b^4*d^4*e^2+60*B*ln(e*x+d)*a^4*b*d^2*e^4-240*A*x*a^2*b^3*d^2*e^4+60*A*x
*a*b^4*d^3*e^3+120*B*x*a^4*b*d*e^5-240*B*x*a^3*b^2*d^2*e^4+120*B*x*a^2*b^3*d^3*e
^3+60*B*x*a*b^4*d^4*e^2+630*B*x^2*a*b^4*d^3*e^3+240*A*x*a^3*b^2*d*e^5+3*B*x^6*b^
5*e^6+4*A*x^5*b^5*e^6-12*B*x*a^5*e^6+180*B*ln(e*x+d)*b^5*d^6-720*A*ln(e*x+d)*x*a
^2*b^3*d^2*e^4+720*A*ln(e*x+d)*x*a*b^4*d^3*e^3+420*B*a^2*b^3*d^4*e^2+210*A*a*b^4
*d^4*e^2-720*B*ln(e*x+d)*x*a^3*b^2*d^2*e^4-1200*B*ln(e*x+d)*x*a*b^4*d^4*e^2+240*
A*ln(e*x+d)*x*a^3*b^2*d*e^5+90*B*a^4*b*d^2*e^4-48*B*x*b^5*d^5*e-120*A*ln(e*x+d)*
b^5*d^5*e-6*B*x^5*b^5*d*e^5+30*A*x^4*a*b^4*e^6)/(b*x+a)^5/e^7/(e*x+d)^2

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283284, size = 1176, normalized size = 2.77 \[ \text{result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

1/12*(3*B*b^5*e^6*x^6 + 66*B*b^5*d^6 - 6*A*a^5*e^6 - 54*(5*B*a*b^4 + A*b^5)*d^5*
e + 210*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 - 300*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 +
90*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - 6*(B*a^5 + 5*A*a^4*b)*d*e^5 - 2*(3*B*b^5*d*
e^5 - 2*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(3*B*b^5*d^2*e^4 - 2*(5*B*a*b^4 + A*b^5
)*d*e^5 + 6*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 20*(3*B*b^5*d^3*e^3 - 2*(5*B*a*b^
4 + A*b^5)*d^2*e^4 + 6*(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 - 6*(34*B*b^5*d^4*e^2 - 21*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 55*(2*B*a^2*b
^3 + A*a*b^4)*d^2*e^4 - 40*(B*a^3*b^2 + A*a^2*b^3)*d*e^5)*x^2 - 12*(4*B*b^5*d^5*
e - (5*B*a*b^4 + A*b^5)*d^4*e^2 - 5*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 20*(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 + (B*a^5 + 5*A*a^4*b
)*e^6)*x + 60*(3*B*b^5*d^6 - 2*(5*B*a*b^4 + A*b^5)*d^5*e + 6*(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 + (B*a^4*b + 2*A*a^3*b^2)*d^2*e
^4 + (3*B*b^5*d^4*e^2 - 2*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 6*(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 + (B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 +
2*(3*B*b^5*d^5*e - 2*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 6*(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 + (B*a^4*b + 2*A*a^3*b^2)*d*e^5)*x)*lo
g(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*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)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.302336, 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)^3,x, algorithm="giac")

[Out]

Done