3.1727 \(\int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=219 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1) (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2) (a+b x)}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+3}}{e^4 (m+3) (a+b x)}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+4}}{e^4 (m+4) (a+b x)} \]

[Out]

-(((b*d - a*e)^3*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^4*(1 + m)*(
a + b*x))) + (3*b*(b*d - a*e)^2*(d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(e^4*(2 + m)*(a + b*x)) - (3*b^2*(b*d - a*e)*(d + e*x)^(3 + m)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(e^4*(3 + m)*(a + b*x)) + (b^3*(d + e*x)^(4 + m)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(e^4*(4 + m)*(a + b*x))

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Rubi [A]  time = 0.252031, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1) (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2) (a+b x)}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+3}}{e^4 (m+3) (a+b x)}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+4}}{e^4 (m+4) (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(((b*d - a*e)^3*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^4*(1 + m)*(
a + b*x))) + (3*b*(b*d - a*e)^2*(d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(e^4*(2 + m)*(a + b*x)) - (3*b^2*(b*d - a*e)*(d + e*x)^(3 + m)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(e^4*(3 + m)*(a + b*x)) + (b^3*(d + e*x)^(4 + m)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(e^4*(4 + m)*(a + b*x))

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Rubi in Sympy [A]  time = 45.3883, size = 201, normalized size = 0.92 \[ \frac{\left (d + e x\right )^{m + 1} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e \left (m + 4\right )} + \frac{\left (3 a + 3 b x\right ) \left (d + e x\right )^{m + 1} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{2} \left (m + 3\right ) \left (m + 4\right )} + \frac{6 \left (d + e x\right )^{m + 1} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{3} \left (m + 2\right ) \left (m + 3\right ) \left (m + 4\right )} + \frac{6 \left (d + e x\right )^{m + 1} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{4} \left (a + b x\right ) \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right ) \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

(d + e*x)**(m + 1)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(e*(m + 4)) + (3*a + 3*b*
x)*(d + e*x)**(m + 1)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(e**2*(m + 3)
*(m + 4)) + 6*(d + e*x)**(m + 1)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)
/(e**3*(m + 2)*(m + 3)*(m + 4)) + 6*(d + e*x)**(m + 1)*(a*e - b*d)**3*sqrt(a**2
+ 2*a*b*x + b**2*x**2)/(e**4*(a + b*x)*(m + 1)*(m + 2)*(m + 3)*(m + 4))

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Mathematica [A]  time = 0.263546, size = 196, normalized size = 0.89 \[ \frac{\sqrt{(a+b x)^2} (d+e x)^{m+1} \left (a^3 e^3 \left (m^3+9 m^2+26 m+24\right )-3 a^2 b e^2 \left (m^2+7 m+12\right ) (d-e (m+1) x)+3 a b^2 e (m+4) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+b^3 \left (-\left (6 d^3-6 d^2 e (m+1) x+3 d e^2 \left (m^2+3 m+2\right ) x^2-e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right )\right )}{e^4 (m+1) (m+2) (m+3) (m+4) (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(d + e*x)^(1 + m)*(a^3*e^3*(24 + 26*m + 9*m^2 + m^3) - 3*a^2*
b*e^2*(12 + 7*m + m^2)*(d - e*(1 + m)*x) + 3*a*b^2*e*(4 + m)*(2*d^2 - 2*d*e*(1 +
m)*x + e^2*(2 + 3*m + m^2)*x^2) - b^3*(6*d^3 - 6*d^2*e*(1 + m)*x + 3*d*e^2*(2 +
3*m + m^2)*x^2 - e^3*(6 + 11*m + 6*m^2 + m^3)*x^3)))/(e^4*(1 + m)*(2 + m)*(3 +
m)*(4 + m)*(a + b*x))

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Maple [B]  time = 0.013, size = 402, normalized size = 1.8 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ({b}^{3}{e}^{3}{m}^{3}{x}^{3}+3\,a{b}^{2}{e}^{3}{m}^{3}{x}^{2}+6\,{b}^{3}{e}^{3}{m}^{2}{x}^{3}+3\,{a}^{2}b{e}^{3}{m}^{3}x+21\,a{b}^{2}{e}^{3}{m}^{2}{x}^{2}-3\,{b}^{3}d{e}^{2}{m}^{2}{x}^{2}+11\,{b}^{3}{e}^{3}m{x}^{3}+{a}^{3}{e}^{3}{m}^{3}+24\,{a}^{2}b{e}^{3}{m}^{2}x-6\,a{b}^{2}d{e}^{2}{m}^{2}x+42\,a{b}^{2}{e}^{3}m{x}^{2}-9\,{b}^{3}d{e}^{2}m{x}^{2}+6\,{x}^{3}{b}^{3}{e}^{3}+9\,{a}^{3}{e}^{3}{m}^{2}-3\,{a}^{2}bd{e}^{2}{m}^{2}+57\,{a}^{2}b{e}^{3}mx-30\,a{b}^{2}d{e}^{2}mx+24\,{x}^{2}a{b}^{2}{e}^{3}+6\,{b}^{3}{d}^{2}emx-6\,{x}^{2}{b}^{3}d{e}^{2}+26\,{a}^{3}{e}^{3}m-21\,{a}^{2}bd{e}^{2}m+36\,x{a}^{2}b{e}^{3}+6\,a{b}^{2}{d}^{2}em-24\,xa{b}^{2}d{e}^{2}+6\,x{b}^{3}{d}^{2}e+24\,{a}^{3}{e}^{3}-36\,{a}^{2}bd{e}^{2}+24\,a{b}^{2}{d}^{2}e-6\,{b}^{3}{d}^{3} \right ) }{ \left ( bx+a \right ) ^{3}{e}^{4} \left ({m}^{4}+10\,{m}^{3}+35\,{m}^{2}+50\,m+24 \right ) } \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

((b*x+a)^2)^(3/2)*(e*x+d)^(1+m)*(b^3*e^3*m^3*x^3+3*a*b^2*e^3*m^3*x^2+6*b^3*e^3*m
^2*x^3+3*a^2*b*e^3*m^3*x+21*a*b^2*e^3*m^2*x^2-3*b^3*d*e^2*m^2*x^2+11*b^3*e^3*m*x
^3+a^3*e^3*m^3+24*a^2*b*e^3*m^2*x-6*a*b^2*d*e^2*m^2*x+42*a*b^2*e^3*m*x^2-9*b^3*d
*e^2*m*x^2+6*b^3*e^3*x^3+9*a^3*e^3*m^2-3*a^2*b*d*e^2*m^2+57*a^2*b*e^3*m*x-30*a*b
^2*d*e^2*m*x+24*a*b^2*e^3*x^2+6*b^3*d^2*e*m*x-6*b^3*d*e^2*x^2+26*a^3*e^3*m-21*a^
2*b*d*e^2*m+36*a^2*b*e^3*x+6*a*b^2*d^2*e*m-24*a*b^2*d*e^2*x+6*b^3*d^2*e*x+24*a^3
*e^3-36*a^2*b*d*e^2+24*a*b^2*d^2*e-6*b^3*d^3)/(b*x+a)^3/e^4/(m^4+10*m^3+35*m^2+5
0*m+24)

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Maxima [A]  time = 0.767819, size = 412, normalized size = 1.88 \[ \frac{{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3} e^{4} x^{4} - 3 \,{\left (m^{2} + 7 \, m + 12\right )} a^{2} b d^{2} e^{2} +{\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} a^{3} d e^{3} + 6 \, a b^{2} d^{3} e{\left (m + 4\right )} - 6 \, b^{3} d^{4} +{\left ({\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{3} d e^{3} + 3 \,{\left (m^{3} + 7 \, m^{2} + 14 \, m + 8\right )} a b^{2} e^{4}\right )} x^{3} - 3 \,{\left ({\left (m^{2} + m\right )} b^{3} d^{2} e^{2} -{\left (m^{3} + 5 \, m^{2} + 4 \, m\right )} a b^{2} d e^{3} -{\left (m^{3} + 8 \, m^{2} + 19 \, m + 12\right )} a^{2} b e^{4}\right )} x^{2} -{\left (6 \,{\left (m^{2} + 4 \, m\right )} a b^{2} d^{2} e^{2} - 3 \,{\left (m^{3} + 7 \, m^{2} + 12 \, m\right )} a^{2} b d e^{3} -{\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} a^{3} e^{4} - 6 \, b^{3} d^{3} e m\right )} x\right )}{\left (e x + d\right )}^{m}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^m,x, algorithm="maxima")

[Out]

((m^3 + 6*m^2 + 11*m + 6)*b^3*e^4*x^4 - 3*(m^2 + 7*m + 12)*a^2*b*d^2*e^2 + (m^3
+ 9*m^2 + 26*m + 24)*a^3*d*e^3 + 6*a*b^2*d^3*e*(m + 4) - 6*b^3*d^4 + ((m^3 + 3*m
^2 + 2*m)*b^3*d*e^3 + 3*(m^3 + 7*m^2 + 14*m + 8)*a*b^2*e^4)*x^3 - 3*((m^2 + m)*b
^3*d^2*e^2 - (m^3 + 5*m^2 + 4*m)*a*b^2*d*e^3 - (m^3 + 8*m^2 + 19*m + 12)*a^2*b*e
^4)*x^2 - (6*(m^2 + 4*m)*a*b^2*d^2*e^2 - 3*(m^3 + 7*m^2 + 12*m)*a^2*b*d*e^3 - (m
^3 + 9*m^2 + 26*m + 24)*a^3*e^4 - 6*b^3*d^3*e*m)*x)*(e*x + d)^m/((m^4 + 10*m^3 +
35*m^2 + 50*m + 24)*e^4)

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Fricas [A]  time = 0.224722, size = 670, normalized size = 3.06 \[ \frac{{\left (a^{3} d e^{3} m^{3} - 6 \, b^{3} d^{4} + 24 \, a b^{2} d^{3} e - 36 \, a^{2} b d^{2} e^{2} + 24 \, a^{3} d e^{3} +{\left (b^{3} e^{4} m^{3} + 6 \, b^{3} e^{4} m^{2} + 11 \, b^{3} e^{4} m + 6 \, b^{3} e^{4}\right )} x^{4} +{\left (24 \, a b^{2} e^{4} +{\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} m^{3} + 3 \,{\left (b^{3} d e^{3} + 7 \, a b^{2} e^{4}\right )} m^{2} + 2 \,{\left (b^{3} d e^{3} + 21 \, a b^{2} e^{4}\right )} m\right )} x^{3} - 3 \,{\left (a^{2} b d^{2} e^{2} - 3 \, a^{3} d e^{3}\right )} m^{2} + 3 \,{\left (12 \, a^{2} b e^{4} +{\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} m^{3} -{\left (b^{3} d^{2} e^{2} - 5 \, a b^{2} d e^{3} - 8 \, a^{2} b e^{4}\right )} m^{2} -{\left (b^{3} d^{2} e^{2} - 4 \, a b^{2} d e^{3} - 19 \, a^{2} b e^{4}\right )} m\right )} x^{2} +{\left (6 \, a b^{2} d^{3} e - 21 \, a^{2} b d^{2} e^{2} + 26 \, a^{3} d e^{3}\right )} m +{\left (24 \, a^{3} e^{4} +{\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} m^{3} - 3 \,{\left (2 \, a b^{2} d^{2} e^{2} - 7 \, a^{2} b d e^{3} - 3 \, a^{3} e^{4}\right )} m^{2} + 2 \,{\left (3 \, b^{3} d^{3} e - 12 \, a b^{2} d^{2} e^{2} + 18 \, a^{2} b d e^{3} + 13 \, a^{3} e^{4}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^m,x, algorithm="fricas")

[Out]

(a^3*d*e^3*m^3 - 6*b^3*d^4 + 24*a*b^2*d^3*e - 36*a^2*b*d^2*e^2 + 24*a^3*d*e^3 +
(b^3*e^4*m^3 + 6*b^3*e^4*m^2 + 11*b^3*e^4*m + 6*b^3*e^4)*x^4 + (24*a*b^2*e^4 + (
b^3*d*e^3 + 3*a*b^2*e^4)*m^3 + 3*(b^3*d*e^3 + 7*a*b^2*e^4)*m^2 + 2*(b^3*d*e^3 +
21*a*b^2*e^4)*m)*x^3 - 3*(a^2*b*d^2*e^2 - 3*a^3*d*e^3)*m^2 + 3*(12*a^2*b*e^4 + (
a*b^2*d*e^3 + a^2*b*e^4)*m^3 - (b^3*d^2*e^2 - 5*a*b^2*d*e^3 - 8*a^2*b*e^4)*m^2 -
(b^3*d^2*e^2 - 4*a*b^2*d*e^3 - 19*a^2*b*e^4)*m)*x^2 + (6*a*b^2*d^3*e - 21*a^2*b
*d^2*e^2 + 26*a^3*d*e^3)*m + (24*a^3*e^4 + (3*a^2*b*d*e^3 + a^3*e^4)*m^3 - 3*(2*
a*b^2*d^2*e^2 - 7*a^2*b*d*e^3 - 3*a^3*e^4)*m^2 + 2*(3*b^3*d^3*e - 12*a*b^2*d^2*e
^2 + 18*a^2*b*d*e^3 + 13*a^3*e^4)*m)*x)*(e*x + d)^m/(e^4*m^4 + 10*e^4*m^3 + 35*e
^4*m^2 + 50*e^4*m + 24*e^4)

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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((e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227999, 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)^(3/2)*(e*x + d)^m,x, algorithm="giac")

[Out]

Done