3.17.67 \(\int (A+B x) (d+e x)^m (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)
Optimal. Leaf size=321 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e) (d+e x)^{m+1}}{e^5 (m+1) (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+2} (-a B e-3 A b e+4 b B d)}{e^5 (m+2) (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+3} (-a B e-A b e+2 b B d)}{e^5 (m+3) (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+4} (-3 a B e-A b e+4 b B d)}{e^5 (m+4) (a+b x)}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+5}}{e^5 (m+5) (a+b x)} \]
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Rubi [A] time = 0.21, antiderivative size = 321, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used =
{770, 77} \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e) (d+e x)^{m+1}}{e^5 (m+1) (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+2} (-a B e-3 A b e+4 b B d)}{e^5 (m+2) (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+3} (-a B e-A b e+2 b B d)}{e^5 (m+3) (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+4} (-3 a B e-A b e+4 b B d)}{e^5 (m+4) (a+b x)}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+5}}{e^5 (m+5) (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
((b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(1 + m)*(a + b*x)) - ((b*d -
a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(2 + m)*(a + b*x)) +
(3*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(3 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(3 + m)*(a +
b*x)) - (b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(4 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(4 + m)*(a + b*
x)) + (b^3*B*(d + e*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(5 + m)*(a + b*x))
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rule 770
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^m \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (-B d+A e) (d+e x)^m}{e^4}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{1+m}}{e^4}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{2+m}}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{3+m}}{e^4}+\frac {b^6 B (d+e x)^{4+m}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^3 (B d-A e) (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (1+m) (a+b x)}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (2+m) (a+b x)}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (3+m) (a+b x)}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (4+m) (a+b x)}+\frac {b^3 B (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (5+m) (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 183, normalized size = 0.57 \begin {gather*} \frac {\left ((a+b x)^2\right )^{3/2} (d+e x)^{m+1} \left (-\frac {b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)}{m+4}+\frac {3 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)}{m+3}-\frac {(d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{m+2}+\frac {(b d-a e)^3 (B d-A e)}{m+1}+\frac {b^3 B (d+e x)^4}{m+5}\right )}{e^5 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
(((a + b*x)^2)^(3/2)*(d + e*x)^(1 + m)*(((b*d - a*e)^3*(B*d - A*e))/(1 + m) - ((b*d - a*e)^2*(4*b*B*d - 3*A*b*
e - a*B*e)*(d + e*x))/(2 + m) + (3*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2)/(3 + m) - (b^2*(4*b*B*
d - A*b*e - 3*a*B*e)*(d + e*x)^3)/(4 + m) + (b^3*B*(d + e*x)^4)/(5 + m)))/(e^5*(a + b*x)^3)
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IntegrateAlgebraic [F] time = 3.76, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(A + B*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
Defer[IntegrateAlgebraic][(A + B*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(3/2), x]
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fricas [B] time = 0.47, size = 1282, normalized size = 3.99
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")
[Out]
(A*a^3*d*e^4*m^4 + 24*B*b^3*d^5 + 120*A*a^3*d*e^4 - 30*(3*B*a*b^2 + A*b^3)*d^4*e + 120*(B*a^2*b + A*a*b^2)*d^3
*e^2 - 60*(B*a^3 + 3*A*a^2*b)*d^2*e^3 + (B*b^3*e^5*m^4 + 10*B*b^3*e^5*m^3 + 35*B*b^3*e^5*m^2 + 50*B*b^3*e^5*m
+ 24*B*b^3*e^5)*x^5 + (30*(3*B*a*b^2 + A*b^3)*e^5 + (B*b^3*d*e^4 + (3*B*a*b^2 + A*b^3)*e^5)*m^4 + (6*B*b^3*d*e
^4 + 11*(3*B*a*b^2 + A*b^3)*e^5)*m^3 + (11*B*b^3*d*e^4 + 41*(3*B*a*b^2 + A*b^3)*e^5)*m^2 + (6*B*b^3*d*e^4 + 61
*(3*B*a*b^2 + A*b^3)*e^5)*m)*x^4 + (14*A*a^3*d*e^4 - (B*a^3 + 3*A*a^2*b)*d^2*e^3)*m^3 + (120*(B*a^2*b + A*a*b^
2)*e^5 + ((3*B*a*b^2 + A*b^3)*d*e^4 + 3*(B*a^2*b + A*a*b^2)*e^5)*m^4 - 4*(B*b^3*d^2*e^3 - 2*(3*B*a*b^2 + A*b^3
)*d*e^4 - 9*(B*a^2*b + A*a*b^2)*e^5)*m^3 - (12*B*b^3*d^2*e^3 - 17*(3*B*a*b^2 + A*b^3)*d*e^4 - 147*(B*a^2*b + A
*a*b^2)*e^5)*m^2 - 2*(4*B*b^3*d^2*e^3 - 5*(3*B*a*b^2 + A*b^3)*d*e^4 - 117*(B*a^2*b + A*a*b^2)*e^5)*m)*x^3 + (7
1*A*a^3*d*e^4 + 6*(B*a^2*b + A*a*b^2)*d^3*e^2 - 12*(B*a^3 + 3*A*a^2*b)*d^2*e^3)*m^2 + (60*(B*a^3 + 3*A*a^2*b)*
e^5 + (3*(B*a^2*b + A*a*b^2)*d*e^4 + (B*a^3 + 3*A*a^2*b)*e^5)*m^4 - (3*(3*B*a*b^2 + A*b^3)*d^2*e^3 - 30*(B*a^2
*b + A*a*b^2)*d*e^4 - 13*(B*a^3 + 3*A*a^2*b)*e^5)*m^3 + (12*B*b^3*d^3*e^2 - 18*(3*B*a*b^2 + A*b^3)*d^2*e^3 + 8
7*(B*a^2*b + A*a*b^2)*d*e^4 + 59*(B*a^3 + 3*A*a^2*b)*e^5)*m^2 + (12*B*b^3*d^3*e^2 - 15*(3*B*a*b^2 + A*b^3)*d^2
*e^3 + 60*(B*a^2*b + A*a*b^2)*d*e^4 + 107*(B*a^3 + 3*A*a^2*b)*e^5)*m)*x^2 + (154*A*a^3*d*e^4 - 6*(3*B*a*b^2 +
A*b^3)*d^4*e + 54*(B*a^2*b + A*a*b^2)*d^3*e^2 - 47*(B*a^3 + 3*A*a^2*b)*d^2*e^3)*m + (120*A*a^3*e^5 + (A*a^3*e^
5 + (B*a^3 + 3*A*a^2*b)*d*e^4)*m^4 + 2*(7*A*a^3*e^5 - 3*(B*a^2*b + A*a*b^2)*d^2*e^3 + 6*(B*a^3 + 3*A*a^2*b)*d*
e^4)*m^3 + (71*A*a^3*e^5 + 6*(3*B*a*b^2 + A*b^3)*d^3*e^2 - 54*(B*a^2*b + A*a*b^2)*d^2*e^3 + 47*(B*a^3 + 3*A*a^
2*b)*d*e^4)*m^2 - 2*(12*B*b^3*d^4*e - 77*A*a^3*e^5 - 15*(3*B*a*b^2 + A*b^3)*d^3*e^2 + 60*(B*a^2*b + A*a*b^2)*d
^2*e^3 - 30*(B*a^3 + 3*A*a^2*b)*d*e^4)*m)*x)*(e*x + d)^m/(e^5*m^5 + 15*e^5*m^4 + 85*e^5*m^3 + 225*e^5*m^2 + 27
4*e^5*m + 120*e^5)
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giac [B] time = 0.42, size = 3208, normalized size = 9.99
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")
[Out]
((x*e + d)^m*B*b^3*m^4*x^5*e^5*sgn(b*x + a) + (x*e + d)^m*B*b^3*d*m^4*x^4*e^4*sgn(b*x + a) + 3*(x*e + d)^m*B*a
*b^2*m^4*x^4*e^5*sgn(b*x + a) + (x*e + d)^m*A*b^3*m^4*x^4*e^5*sgn(b*x + a) + 10*(x*e + d)^m*B*b^3*m^3*x^5*e^5*
sgn(b*x + a) + 3*(x*e + d)^m*B*a*b^2*d*m^4*x^3*e^4*sgn(b*x + a) + (x*e + d)^m*A*b^3*d*m^4*x^3*e^4*sgn(b*x + a)
+ 6*(x*e + d)^m*B*b^3*d*m^3*x^4*e^4*sgn(b*x + a) - 4*(x*e + d)^m*B*b^3*d^2*m^3*x^3*e^3*sgn(b*x + a) + 3*(x*e
+ d)^m*B*a^2*b*m^4*x^3*e^5*sgn(b*x + a) + 3*(x*e + d)^m*A*a*b^2*m^4*x^3*e^5*sgn(b*x + a) + 33*(x*e + d)^m*B*a*
b^2*m^3*x^4*e^5*sgn(b*x + a) + 11*(x*e + d)^m*A*b^3*m^3*x^4*e^5*sgn(b*x + a) + 35*(x*e + d)^m*B*b^3*m^2*x^5*e^
5*sgn(b*x + a) + 3*(x*e + d)^m*B*a^2*b*d*m^4*x^2*e^4*sgn(b*x + a) + 3*(x*e + d)^m*A*a*b^2*d*m^4*x^2*e^4*sgn(b*
x + a) + 24*(x*e + d)^m*B*a*b^2*d*m^3*x^3*e^4*sgn(b*x + a) + 8*(x*e + d)^m*A*b^3*d*m^3*x^3*e^4*sgn(b*x + a) +
11*(x*e + d)^m*B*b^3*d*m^2*x^4*e^4*sgn(b*x + a) - 9*(x*e + d)^m*B*a*b^2*d^2*m^3*x^2*e^3*sgn(b*x + a) - 3*(x*e
+ d)^m*A*b^3*d^2*m^3*x^2*e^3*sgn(b*x + a) - 12*(x*e + d)^m*B*b^3*d^2*m^2*x^3*e^3*sgn(b*x + a) + 12*(x*e + d)^m
*B*b^3*d^3*m^2*x^2*e^2*sgn(b*x + a) + (x*e + d)^m*B*a^3*m^4*x^2*e^5*sgn(b*x + a) + 3*(x*e + d)^m*A*a^2*b*m^4*x
^2*e^5*sgn(b*x + a) + 36*(x*e + d)^m*B*a^2*b*m^3*x^3*e^5*sgn(b*x + a) + 36*(x*e + d)^m*A*a*b^2*m^3*x^3*e^5*sgn
(b*x + a) + 123*(x*e + d)^m*B*a*b^2*m^2*x^4*e^5*sgn(b*x + a) + 41*(x*e + d)^m*A*b^3*m^2*x^4*e^5*sgn(b*x + a) +
50*(x*e + d)^m*B*b^3*m*x^5*e^5*sgn(b*x + a) + (x*e + d)^m*B*a^3*d*m^4*x*e^4*sgn(b*x + a) + 3*(x*e + d)^m*A*a^
2*b*d*m^4*x*e^4*sgn(b*x + a) + 30*(x*e + d)^m*B*a^2*b*d*m^3*x^2*e^4*sgn(b*x + a) + 30*(x*e + d)^m*A*a*b^2*d*m^
3*x^2*e^4*sgn(b*x + a) + 51*(x*e + d)^m*B*a*b^2*d*m^2*x^3*e^4*sgn(b*x + a) + 17*(x*e + d)^m*A*b^3*d*m^2*x^3*e^
4*sgn(b*x + a) + 6*(x*e + d)^m*B*b^3*d*m*x^4*e^4*sgn(b*x + a) - 6*(x*e + d)^m*B*a^2*b*d^2*m^3*x*e^3*sgn(b*x +
a) - 6*(x*e + d)^m*A*a*b^2*d^2*m^3*x*e^3*sgn(b*x + a) - 54*(x*e + d)^m*B*a*b^2*d^2*m^2*x^2*e^3*sgn(b*x + a) -
18*(x*e + d)^m*A*b^3*d^2*m^2*x^2*e^3*sgn(b*x + a) - 8*(x*e + d)^m*B*b^3*d^2*m*x^3*e^3*sgn(b*x + a) + 18*(x*e +
d)^m*B*a*b^2*d^3*m^2*x*e^2*sgn(b*x + a) + 6*(x*e + d)^m*A*b^3*d^3*m^2*x*e^2*sgn(b*x + a) + 12*(x*e + d)^m*B*b
^3*d^3*m*x^2*e^2*sgn(b*x + a) - 24*(x*e + d)^m*B*b^3*d^4*m*x*e*sgn(b*x + a) + (x*e + d)^m*A*a^3*m^4*x*e^5*sgn(
b*x + a) + 13*(x*e + d)^m*B*a^3*m^3*x^2*e^5*sgn(b*x + a) + 39*(x*e + d)^m*A*a^2*b*m^3*x^2*e^5*sgn(b*x + a) + 1
47*(x*e + d)^m*B*a^2*b*m^2*x^3*e^5*sgn(b*x + a) + 147*(x*e + d)^m*A*a*b^2*m^2*x^3*e^5*sgn(b*x + a) + 183*(x*e
+ d)^m*B*a*b^2*m*x^4*e^5*sgn(b*x + a) + 61*(x*e + d)^m*A*b^3*m*x^4*e^5*sgn(b*x + a) + 24*(x*e + d)^m*B*b^3*x^5
*e^5*sgn(b*x + a) + (x*e + d)^m*A*a^3*d*m^4*e^4*sgn(b*x + a) + 12*(x*e + d)^m*B*a^3*d*m^3*x*e^4*sgn(b*x + a) +
36*(x*e + d)^m*A*a^2*b*d*m^3*x*e^4*sgn(b*x + a) + 87*(x*e + d)^m*B*a^2*b*d*m^2*x^2*e^4*sgn(b*x + a) + 87*(x*e
+ d)^m*A*a*b^2*d*m^2*x^2*e^4*sgn(b*x + a) + 30*(x*e + d)^m*B*a*b^2*d*m*x^3*e^4*sgn(b*x + a) + 10*(x*e + d)^m*
A*b^3*d*m*x^3*e^4*sgn(b*x + a) - (x*e + d)^m*B*a^3*d^2*m^3*e^3*sgn(b*x + a) - 3*(x*e + d)^m*A*a^2*b*d^2*m^3*e^
3*sgn(b*x + a) - 54*(x*e + d)^m*B*a^2*b*d^2*m^2*x*e^3*sgn(b*x + a) - 54*(x*e + d)^m*A*a*b^2*d^2*m^2*x*e^3*sgn(
b*x + a) - 45*(x*e + d)^m*B*a*b^2*d^2*m*x^2*e^3*sgn(b*x + a) - 15*(x*e + d)^m*A*b^3*d^2*m*x^2*e^3*sgn(b*x + a)
+ 6*(x*e + d)^m*B*a^2*b*d^3*m^2*e^2*sgn(b*x + a) + 6*(x*e + d)^m*A*a*b^2*d^3*m^2*e^2*sgn(b*x + a) + 90*(x*e +
d)^m*B*a*b^2*d^3*m*x*e^2*sgn(b*x + a) + 30*(x*e + d)^m*A*b^3*d^3*m*x*e^2*sgn(b*x + a) - 18*(x*e + d)^m*B*a*b^
2*d^4*m*e*sgn(b*x + a) - 6*(x*e + d)^m*A*b^3*d^4*m*e*sgn(b*x + a) + 24*(x*e + d)^m*B*b^3*d^5*sgn(b*x + a) + 14
*(x*e + d)^m*A*a^3*m^3*x*e^5*sgn(b*x + a) + 59*(x*e + d)^m*B*a^3*m^2*x^2*e^5*sgn(b*x + a) + 177*(x*e + d)^m*A*
a^2*b*m^2*x^2*e^5*sgn(b*x + a) + 234*(x*e + d)^m*B*a^2*b*m*x^3*e^5*sgn(b*x + a) + 234*(x*e + d)^m*A*a*b^2*m*x^
3*e^5*sgn(b*x + a) + 90*(x*e + d)^m*B*a*b^2*x^4*e^5*sgn(b*x + a) + 30*(x*e + d)^m*A*b^3*x^4*e^5*sgn(b*x + a) +
14*(x*e + d)^m*A*a^3*d*m^3*e^4*sgn(b*x + a) + 47*(x*e + d)^m*B*a^3*d*m^2*x*e^4*sgn(b*x + a) + 141*(x*e + d)^m
*A*a^2*b*d*m^2*x*e^4*sgn(b*x + a) + 60*(x*e + d)^m*B*a^2*b*d*m*x^2*e^4*sgn(b*x + a) + 60*(x*e + d)^m*A*a*b^2*d
*m*x^2*e^4*sgn(b*x + a) - 12*(x*e + d)^m*B*a^3*d^2*m^2*e^3*sgn(b*x + a) - 36*(x*e + d)^m*A*a^2*b*d^2*m^2*e^3*s
gn(b*x + a) - 120*(x*e + d)^m*B*a^2*b*d^2*m*x*e^3*sgn(b*x + a) - 120*(x*e + d)^m*A*a*b^2*d^2*m*x*e^3*sgn(b*x +
a) + 54*(x*e + d)^m*B*a^2*b*d^3*m*e^2*sgn(b*x + a) + 54*(x*e + d)^m*A*a*b^2*d^3*m*e^2*sgn(b*x + a) - 90*(x*e
+ d)^m*B*a*b^2*d^4*e*sgn(b*x + a) - 30*(x*e + d)^m*A*b^3*d^4*e*sgn(b*x + a) + 71*(x*e + d)^m*A*a^3*m^2*x*e^5*s
gn(b*x + a) + 107*(x*e + d)^m*B*a^3*m*x^2*e^5*sgn(b*x + a) + 321*(x*e + d)^m*A*a^2*b*m*x^2*e^5*sgn(b*x + a) +
120*(x*e + d)^m*B*a^2*b*x^3*e^5*sgn(b*x + a) + 120*(x*e + d)^m*A*a*b^2*x^3*e^5*sgn(b*x + a) + 71*(x*e + d)^m*A
*a^3*d*m^2*e^4*sgn(b*x + a) + 60*(x*e + d)^m*B*a^3*d*m*x*e^4*sgn(b*x + a) + 180*(x*e + d)^m*A*a^2*b*d*m*x*e^4*
sgn(b*x + a) - 47*(x*e + d)^m*B*a^3*d^2*m*e^3*sgn(b*x + a) - 141*(x*e + d)^m*A*a^2*b*d^2*m*e^3*sgn(b*x + a) +
120*(x*e + d)^m*B*a^2*b*d^3*e^2*sgn(b*x + a) + 120*(x*e + d)^m*A*a*b^2*d^3*e^2*sgn(b*x + a) + 154*(x*e + d)^m*
A*a^3*m*x*e^5*sgn(b*x + a) + 60*(x*e + d)^m*B*a^3*x^2*e^5*sgn(b*x + a) + 180*(x*e + d)^m*A*a^2*b*x^2*e^5*sgn(b
*x + a) + 154*(x*e + d)^m*A*a^3*d*m*e^4*sgn(b*x + a) - 60*(x*e + d)^m*B*a^3*d^2*e^3*sgn(b*x + a) - 180*(x*e +
d)^m*A*a^2*b*d^2*e^3*sgn(b*x + a) + 120*(x*e + d)^m*A*a^3*x*e^5*sgn(b*x + a) + 120*(x*e + d)^m*A*a^3*d*e^4*sgn
(b*x + a))/(m^5*e^5 + 15*m^4*e^5 + 85*m^3*e^5 + 225*m^2*e^5 + 274*m*e^5 + 120*e^5)
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maple [B] time = 0.06, size = 1286, normalized size = 4.01 \begin {gather*} \frac {\left (B \,b^{3} e^{4} m^{4} x^{4}+A \,b^{3} e^{4} m^{4} x^{3}+3 B a \,b^{2} e^{4} m^{4} x^{3}+10 B \,b^{3} e^{4} m^{3} x^{4}+3 A a \,b^{2} e^{4} m^{4} x^{2}+11 A \,b^{3} e^{4} m^{3} x^{3}+3 B \,a^{2} b \,e^{4} m^{4} x^{2}+33 B a \,b^{2} e^{4} m^{3} x^{3}-4 B \,b^{3} d \,e^{3} m^{3} x^{3}+35 B \,b^{3} e^{4} m^{2} x^{4}+3 A \,a^{2} b \,e^{4} m^{4} x +36 A a \,b^{2} e^{4} m^{3} x^{2}-3 A \,b^{3} d \,e^{3} m^{3} x^{2}+41 A \,b^{3} e^{4} m^{2} x^{3}+B \,a^{3} e^{4} m^{4} x +36 B \,a^{2} b \,e^{4} m^{3} x^{2}-9 B a \,b^{2} d \,e^{3} m^{3} x^{2}+123 B a \,b^{2} e^{4} m^{2} x^{3}-24 B \,b^{3} d \,e^{3} m^{2} x^{3}+50 B \,b^{3} e^{4} m \,x^{4}+A \,a^{3} e^{4} m^{4}+39 A \,a^{2} b \,e^{4} m^{3} x -6 A a \,b^{2} d \,e^{3} m^{3} x +147 A a \,b^{2} e^{4} m^{2} x^{2}-24 A \,b^{3} d \,e^{3} m^{2} x^{2}+61 A \,b^{3} e^{4} m \,x^{3}+13 B \,a^{3} e^{4} m^{3} x -6 B \,a^{2} b d \,e^{3} m^{3} x +147 B \,a^{2} b \,e^{4} m^{2} x^{2}-72 B a \,b^{2} d \,e^{3} m^{2} x^{2}+183 B a \,b^{2} e^{4} m \,x^{3}+12 B \,b^{3} d^{2} e^{2} m^{2} x^{2}-44 B \,b^{3} d \,e^{3} m \,x^{3}+24 b^{3} B \,x^{4} e^{4}+14 A \,a^{3} e^{4} m^{3}-3 A \,a^{2} b d \,e^{3} m^{3}+177 A \,a^{2} b \,e^{4} m^{2} x -60 A a \,b^{2} d \,e^{3} m^{2} x +234 A a \,b^{2} e^{4} m \,x^{2}+6 A \,b^{3} d^{2} e^{2} m^{2} x -51 A \,b^{3} d \,e^{3} m \,x^{2}+30 A \,b^{3} e^{4} x^{3}-B \,a^{3} d \,e^{3} m^{3}+59 B \,a^{3} e^{4} m^{2} x -60 B \,a^{2} b d \,e^{3} m^{2} x +234 B \,a^{2} b \,e^{4} m \,x^{2}+18 B a \,b^{2} d^{2} e^{2} m^{2} x -153 B a \,b^{2} d \,e^{3} m \,x^{2}+90 B a \,b^{2} e^{4} x^{3}+36 B \,b^{3} d^{2} e^{2} m \,x^{2}-24 B \,b^{3} d \,e^{3} x^{3}+71 A \,a^{3} e^{4} m^{2}-36 A \,a^{2} b d \,e^{3} m^{2}+321 A \,a^{2} b \,e^{4} m x +6 A a \,b^{2} d^{2} e^{2} m^{2}-174 A a \,b^{2} d \,e^{3} m x +120 A a \,b^{2} e^{4} x^{2}+36 A \,b^{3} d^{2} e^{2} m x -30 A \,b^{3} d \,e^{3} x^{2}-12 B \,a^{3} d \,e^{3} m^{2}+107 B \,a^{3} e^{4} m x +6 B \,a^{2} b \,d^{2} e^{2} m^{2}-174 B \,a^{2} b d \,e^{3} m x +120 B \,a^{2} b \,e^{4} x^{2}+108 B a \,b^{2} d^{2} e^{2} m x -90 B a \,b^{2} d \,e^{3} x^{2}-24 B \,b^{3} d^{3} e m x +24 B \,b^{3} d^{2} e^{2} x^{2}+154 A \,a^{3} e^{4} m -141 A \,a^{2} b d \,e^{3} m +180 A \,a^{2} b \,e^{4} x +54 A a \,b^{2} d^{2} e^{2} m -120 A a \,b^{2} d \,e^{3} x -6 A \,b^{3} d^{3} e m +30 A \,b^{3} d^{2} e^{2} x -47 B \,a^{3} d \,e^{3} m +60 B \,a^{3} e^{4} x +54 B \,a^{2} b \,d^{2} e^{2} m -120 B \,a^{2} b d \,e^{3} x -18 B a \,b^{2} d^{3} e m +90 B a \,b^{2} d^{2} e^{2} x -24 B \,b^{3} d^{3} e x +120 A \,a^{3} e^{4}-180 A \,a^{2} b d \,e^{3}+120 A a \,b^{2} d^{2} e^{2}-30 A \,b^{3} d^{3} e -60 B \,a^{3} d \,e^{3}+120 B \,a^{2} b \,d^{2} e^{2}-90 B a \,b^{2} d^{3} e +24 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (e x +d \right )^{m +1}}{\left (b x +a \right )^{3} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right ) e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
[Out]
(e*x+d)^(m+1)*(B*b^3*e^4*m^4*x^4+A*b^3*e^4*m^4*x^3+3*B*a*b^2*e^4*m^4*x^3+10*B*b^3*e^4*m^3*x^4+3*A*a*b^2*e^4*m^
4*x^2+11*A*b^3*e^4*m^3*x^3+3*B*a^2*b*e^4*m^4*x^2+33*B*a*b^2*e^4*m^3*x^3-4*B*b^3*d*e^3*m^3*x^3+35*B*b^3*e^4*m^2
*x^4+3*A*a^2*b*e^4*m^4*x+36*A*a*b^2*e^4*m^3*x^2-3*A*b^3*d*e^3*m^3*x^2+41*A*b^3*e^4*m^2*x^3+B*a^3*e^4*m^4*x+36*
B*a^2*b*e^4*m^3*x^2-9*B*a*b^2*d*e^3*m^3*x^2+123*B*a*b^2*e^4*m^2*x^3-24*B*b^3*d*e^3*m^2*x^3+50*B*b^3*e^4*m*x^4+
A*a^3*e^4*m^4+39*A*a^2*b*e^4*m^3*x-6*A*a*b^2*d*e^3*m^3*x+147*A*a*b^2*e^4*m^2*x^2-24*A*b^3*d*e^3*m^2*x^2+61*A*b
^3*e^4*m*x^3+13*B*a^3*e^4*m^3*x-6*B*a^2*b*d*e^3*m^3*x+147*B*a^2*b*e^4*m^2*x^2-72*B*a*b^2*d*e^3*m^2*x^2+183*B*a
*b^2*e^4*m*x^3+12*B*b^3*d^2*e^2*m^2*x^2-44*B*b^3*d*e^3*m*x^3+24*B*b^3*e^4*x^4+14*A*a^3*e^4*m^3-3*A*a^2*b*d*e^3
*m^3+177*A*a^2*b*e^4*m^2*x-60*A*a*b^2*d*e^3*m^2*x+234*A*a*b^2*e^4*m*x^2+6*A*b^3*d^2*e^2*m^2*x-51*A*b^3*d*e^3*m
*x^2+30*A*b^3*e^4*x^3-B*a^3*d*e^3*m^3+59*B*a^3*e^4*m^2*x-60*B*a^2*b*d*e^3*m^2*x+234*B*a^2*b*e^4*m*x^2+18*B*a*b
^2*d^2*e^2*m^2*x-153*B*a*b^2*d*e^3*m*x^2+90*B*a*b^2*e^4*x^3+36*B*b^3*d^2*e^2*m*x^2-24*B*b^3*d*e^3*x^3+71*A*a^3
*e^4*m^2-36*A*a^2*b*d*e^3*m^2+321*A*a^2*b*e^4*m*x+6*A*a*b^2*d^2*e^2*m^2-174*A*a*b^2*d*e^3*m*x+120*A*a*b^2*e^4*
x^2+36*A*b^3*d^2*e^2*m*x-30*A*b^3*d*e^3*x^2-12*B*a^3*d*e^3*m^2+107*B*a^3*e^4*m*x+6*B*a^2*b*d^2*e^2*m^2-174*B*a
^2*b*d*e^3*m*x+120*B*a^2*b*e^4*x^2+108*B*a*b^2*d^2*e^2*m*x-90*B*a*b^2*d*e^3*x^2-24*B*b^3*d^3*e*m*x+24*B*b^3*d^
2*e^2*x^2+154*A*a^3*e^4*m-141*A*a^2*b*d*e^3*m+180*A*a^2*b*e^4*x+54*A*a*b^2*d^2*e^2*m-120*A*a*b^2*d*e^3*x-6*A*b
^3*d^3*e*m+30*A*b^3*d^2*e^2*x-47*B*a^3*d*e^3*m+60*B*a^3*e^4*x+54*B*a^2*b*d^2*e^2*m-120*B*a^2*b*d*e^3*x-18*B*a*
b^2*d^3*e*m+90*B*a*b^2*d^2*e^2*x-24*B*b^3*d^3*e*x+120*A*a^3*e^4-180*A*a^2*b*d*e^3+120*A*a*b^2*d^2*e^2-30*A*b^3
*d^3*e-60*B*a^3*d*e^3+120*B*a^2*b*d^2*e^2-90*B*a*b^2*d^3*e+24*B*b^3*d^4)*((b*x+a)^2)^(3/2)/(b*x+a)^3/e^5/(m^5+
15*m^4+85*m^3+225*m^2+274*m+120)
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maxima [B] time = 0.65, size = 756, normalized size = 2.36 \begin {gather*} \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} A}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{3} e^{5} x^{5} + 6 \, {\left (m^{2} + 9 \, m + 20\right )} a^{2} b d^{3} e^{2} - {\left (m^{3} + 12 \, m^{2} + 47 \, m + 60\right )} a^{3} d^{2} e^{3} - 18 \, a b^{2} d^{4} e {\left (m + 5\right )} + 24 \, b^{3} d^{5} + {\left ({\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b^{3} d e^{4} + 3 \, {\left (m^{4} + 11 \, m^{3} + 41 \, m^{2} + 61 \, m + 30\right )} a b^{2} e^{5}\right )} x^{4} - {\left (4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{3} d^{2} e^{3} - 3 \, {\left (m^{4} + 8 \, m^{3} + 17 \, m^{2} + 10 \, m\right )} a b^{2} d e^{4} - 3 \, {\left (m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40\right )} a^{2} b e^{5}\right )} x^{3} + {\left (12 \, {\left (m^{2} + m\right )} b^{3} d^{3} e^{2} - 9 \, {\left (m^{3} + 6 \, m^{2} + 5 \, m\right )} a b^{2} d^{2} e^{3} + 3 \, {\left (m^{4} + 10 \, m^{3} + 29 \, m^{2} + 20 \, m\right )} a^{2} b d e^{4} + {\left (m^{4} + 13 \, m^{3} + 59 \, m^{2} + 107 \, m + 60\right )} a^{3} e^{5}\right )} x^{2} + {\left (18 \, {\left (m^{2} + 5 \, m\right )} a b^{2} d^{3} e^{2} - 6 \, {\left (m^{3} + 9 \, m^{2} + 20 \, m\right )} a^{2} b d^{2} e^{3} + {\left (m^{4} + 12 \, m^{3} + 47 \, m^{2} + 60 \, m\right )} a^{3} d e^{4} - 24 \, b^{3} d^{4} e m\right )} x\right )} {\left (e x + d\right )}^{m} B}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),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*A/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*b^3
*e^5*x^5 + 6*(m^2 + 9*m + 20)*a^2*b*d^3*e^2 - (m^3 + 12*m^2 + 47*m + 60)*a^3*d^2*e^3 - 18*a*b^2*d^4*e*(m + 5)
+ 24*b^3*d^5 + ((m^4 + 6*m^3 + 11*m^2 + 6*m)*b^3*d*e^4 + 3*(m^4 + 11*m^3 + 41*m^2 + 61*m + 30)*a*b^2*e^5)*x^4
- (4*(m^3 + 3*m^2 + 2*m)*b^3*d^2*e^3 - 3*(m^4 + 8*m^3 + 17*m^2 + 10*m)*a*b^2*d*e^4 - 3*(m^4 + 12*m^3 + 49*m^2
+ 78*m + 40)*a^2*b*e^5)*x^3 + (12*(m^2 + m)*b^3*d^3*e^2 - 9*(m^3 + 6*m^2 + 5*m)*a*b^2*d^2*e^3 + 3*(m^4 + 10*m^
3 + 29*m^2 + 20*m)*a^2*b*d*e^4 + (m^4 + 13*m^3 + 59*m^2 + 107*m + 60)*a^3*e^5)*x^2 + (18*(m^2 + 5*m)*a*b^2*d^3
*e^2 - 6*(m^3 + 9*m^2 + 20*m)*a^2*b*d^2*e^3 + (m^4 + 12*m^3 + 47*m^2 + 60*m)*a^3*d*e^4 - 24*b^3*d^4*e*m)*x)*(e
*x + d)^m*B/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)*(d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)
[Out]
int((A + B*x)*(d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + B x\right ) \left (d + e x\right )^{m} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
Integral((A + B*x)*(d + e*x)**m*((a + b*x)**2)**(3/2), x)
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