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

Optimal. Leaf size=277 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1) (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2) (a+b x)}+\frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3) (a+b x)}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+5}}{e^5 (m+5) (a+b x)}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+4}}{e^5 (m+4) (a+b x)} \]

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Rubi [A]  time = 0.15, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1) (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2) (a+b x)}+\frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3) (a+b x)}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+4}}{e^5 (m+4) (a+b x)}+\frac {b^4 \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)^4*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(1 + m)*(a + b*x)) - (4*b*(b*d - a*e)^3*(
d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(2 + m)*(a + b*x)) + (6*b^2*(b*d - a*e)^2*(d + e*x)^(3 +
m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(3 + m)*(a + b*x)) - (4*b^3*(b*d - a*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^4*(d + e*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(5 + m)
*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 (a+b x) \left (a b+b^2 x\right )^3 (d+e x)^m \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^4 (d+e x)^m \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4 (d+e x)^m}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{1+m}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{2+m}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{3+m}}{e^4}+\frac {b^4 (d+e x)^{4+m}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^4 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (1+m) (a+b x)}-\frac {4 b (b d-a e)^3 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (2+m) (a+b x)}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (3+m) (a+b x)}-\frac {4 b^3 (b d-a 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^4 (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 = 0.16, size = 139, normalized size = 0.50 \begin {gather*} \frac {\sqrt {(a+b x)^2} (d+e x)^{m+1} \left (-\frac {4 b^3 (d+e x)^3 (b d-a e)}{m+4}+\frac {6 b^2 (d+e x)^2 (b d-a e)^2}{m+3}-\frac {4 b (d+e x) (b d-a e)^3}{m+2}+\frac {(b d-a e)^4}{m+1}+\frac {b^4 (d+e x)^4}{m+5}\right )}{e^5 (a+b x)} \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]

(Sqrt[(a + b*x)^2]*(d + e*x)^(1 + m)*((b*d - a*e)^4/(1 + m) - (4*b*(b*d - a*e)^3*(d + e*x))/(2 + m) + (6*b^2*(
b*d - a*e)^2*(d + e*x)^2)/(3 + m) - (4*b^3*(b*d - a*e)*(d + e*x)^3)/(4 + m) + (b^4*(d + e*x)^4)/(5 + m)))/(e^5
*(a + b*x))

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IntegrateAlgebraic [F]  time = 1.42, 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 = 0.45, size = 901, normalized size = 3.25 \begin {gather*} \frac {{\left (a^{4} d e^{4} m^{4} + 24 \, b^{4} d^{5} - 120 \, a b^{3} d^{4} e + 240 \, a^{2} b^{2} d^{3} e^{2} - 240 \, a^{3} b d^{2} e^{3} + 120 \, a^{4} d e^{4} + {\left (b^{4} e^{5} m^{4} + 10 \, b^{4} e^{5} m^{3} + 35 \, b^{4} e^{5} m^{2} + 50 \, b^{4} e^{5} m + 24 \, b^{4} e^{5}\right )} x^{5} + {\left (120 \, a b^{3} e^{5} + {\left (b^{4} d e^{4} + 4 \, a b^{3} e^{5}\right )} m^{4} + 2 \, {\left (3 \, b^{4} d e^{4} + 22 \, a b^{3} e^{5}\right )} m^{3} + {\left (11 \, b^{4} d e^{4} + 164 \, a b^{3} e^{5}\right )} m^{2} + 2 \, {\left (3 \, b^{4} d e^{4} + 122 \, a b^{3} e^{5}\right )} m\right )} x^{4} - 2 \, {\left (2 \, a^{3} b d^{2} e^{3} - 7 \, a^{4} d e^{4}\right )} m^{3} + 2 \, {\left (120 \, a^{2} b^{2} e^{5} + {\left (2 \, a b^{3} d e^{4} + 3 \, a^{2} b^{2} e^{5}\right )} m^{4} - 2 \, {\left (b^{4} d^{2} e^{3} - 8 \, a b^{3} d e^{4} - 18 \, a^{2} b^{2} e^{5}\right )} m^{3} - {\left (6 \, b^{4} d^{2} e^{3} - 34 \, a b^{3} d e^{4} - 147 \, a^{2} b^{2} e^{5}\right )} m^{2} - 2 \, {\left (2 \, b^{4} d^{2} e^{3} - 10 \, a b^{3} d e^{4} - 117 \, a^{2} b^{2} e^{5}\right )} m\right )} x^{3} + {\left (12 \, a^{2} b^{2} d^{3} e^{2} - 48 \, a^{3} b d^{2} e^{3} + 71 \, a^{4} d e^{4}\right )} m^{2} + 2 \, {\left (120 \, a^{3} b e^{5} + {\left (3 \, a^{2} b^{2} d e^{4} + 2 \, a^{3} b e^{5}\right )} m^{4} - 2 \, {\left (3 \, a b^{3} d^{2} e^{3} - 15 \, a^{2} b^{2} d e^{4} - 13 \, a^{3} b e^{5}\right )} m^{3} + {\left (6 \, b^{4} d^{3} e^{2} - 36 \, a b^{3} d^{2} e^{3} + 87 \, a^{2} b^{2} d e^{4} + 118 \, a^{3} b e^{5}\right )} m^{2} + 2 \, {\left (3 \, b^{4} d^{3} e^{2} - 15 \, a b^{3} d^{2} e^{3} + 30 \, a^{2} b^{2} d e^{4} + 107 \, a^{3} b e^{5}\right )} m\right )} x^{2} - 2 \, {\left (12 \, a b^{3} d^{4} e - 54 \, a^{2} b^{2} d^{3} e^{2} + 94 \, a^{3} b d^{2} e^{3} - 77 \, a^{4} d e^{4}\right )} m + {\left (120 \, a^{4} e^{5} + {\left (4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} m^{4} - 2 \, {\left (6 \, a^{2} b^{2} d^{2} e^{3} - 24 \, a^{3} b d e^{4} - 7 \, a^{4} e^{5}\right )} m^{3} + {\left (24 \, a b^{3} d^{3} e^{2} - 108 \, a^{2} b^{2} d^{2} e^{3} + 188 \, a^{3} b d e^{4} + 71 \, a^{4} e^{5}\right )} m^{2} - 2 \, {\left (12 \, b^{4} d^{4} e - 60 \, a b^{3} d^{3} e^{2} + 120 \, a^{2} b^{2} d^{2} e^{3} - 120 \, a^{3} b d e^{4} - 77 \, a^{4} e^{5}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{5} m^{5} + 15 \, e^{5} m^{4} + 85 \, e^{5} m^{3} + 225 \, e^{5} m^{2} + 274 \, e^{5} m + 120 \, 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="fricas")

[Out]

(a^4*d*e^4*m^4 + 24*b^4*d^5 - 120*a*b^3*d^4*e + 240*a^2*b^2*d^3*e^2 - 240*a^3*b*d^2*e^3 + 120*a^4*d*e^4 + (b^4
*e^5*m^4 + 10*b^4*e^5*m^3 + 35*b^4*e^5*m^2 + 50*b^4*e^5*m + 24*b^4*e^5)*x^5 + (120*a*b^3*e^5 + (b^4*d*e^4 + 4*
a*b^3*e^5)*m^4 + 2*(3*b^4*d*e^4 + 22*a*b^3*e^5)*m^3 + (11*b^4*d*e^4 + 164*a*b^3*e^5)*m^2 + 2*(3*b^4*d*e^4 + 12
2*a*b^3*e^5)*m)*x^4 - 2*(2*a^3*b*d^2*e^3 - 7*a^4*d*e^4)*m^3 + 2*(120*a^2*b^2*e^5 + (2*a*b^3*d*e^4 + 3*a^2*b^2*
e^5)*m^4 - 2*(b^4*d^2*e^3 - 8*a*b^3*d*e^4 - 18*a^2*b^2*e^5)*m^3 - (6*b^4*d^2*e^3 - 34*a*b^3*d*e^4 - 147*a^2*b^
2*e^5)*m^2 - 2*(2*b^4*d^2*e^3 - 10*a*b^3*d*e^4 - 117*a^2*b^2*e^5)*m)*x^3 + (12*a^2*b^2*d^3*e^2 - 48*a^3*b*d^2*
e^3 + 71*a^4*d*e^4)*m^2 + 2*(120*a^3*b*e^5 + (3*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*m^4 - 2*(3*a*b^3*d^2*e^3 - 15*a^2
*b^2*d*e^4 - 13*a^3*b*e^5)*m^3 + (6*b^4*d^3*e^2 - 36*a*b^3*d^2*e^3 + 87*a^2*b^2*d*e^4 + 118*a^3*b*e^5)*m^2 + 2
*(3*b^4*d^3*e^2 - 15*a*b^3*d^2*e^3 + 30*a^2*b^2*d*e^4 + 107*a^3*b*e^5)*m)*x^2 - 2*(12*a*b^3*d^4*e - 54*a^2*b^2
*d^3*e^2 + 94*a^3*b*d^2*e^3 - 77*a^4*d*e^4)*m + (120*a^4*e^5 + (4*a^3*b*d*e^4 + a^4*e^5)*m^4 - 2*(6*a^2*b^2*d^
2*e^3 - 24*a^3*b*d*e^4 - 7*a^4*e^5)*m^3 + (24*a*b^3*d^3*e^2 - 108*a^2*b^2*d^2*e^3 + 188*a^3*b*d*e^4 + 71*a^4*e
^5)*m^2 - 2*(12*b^4*d^4*e - 60*a*b^3*d^3*e^2 + 120*a^2*b^2*d^2*e^3 - 120*a^3*b*d*e^4 - 77*a^4*e^5)*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 + 274*e^5*m + 120*e^5)

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giac [B]  time = 0.33, size = 1949, normalized size = 7.04

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^4*m^4*x^5*e^5*sgn(b*x + a) + (x*e + d)^m*b^4*d*m^4*x^4*e^4*sgn(b*x + a) + 4*(x*e + d)^m*a*b^3*m
^4*x^4*e^5*sgn(b*x + a) + 10*(x*e + d)^m*b^4*m^3*x^5*e^5*sgn(b*x + a) + 4*(x*e + d)^m*a*b^3*d*m^4*x^3*e^4*sgn(
b*x + a) + 6*(x*e + d)^m*b^4*d*m^3*x^4*e^4*sgn(b*x + a) - 4*(x*e + d)^m*b^4*d^2*m^3*x^3*e^3*sgn(b*x + a) + 6*(
x*e + d)^m*a^2*b^2*m^4*x^3*e^5*sgn(b*x + a) + 44*(x*e + d)^m*a*b^3*m^3*x^4*e^5*sgn(b*x + a) + 35*(x*e + d)^m*b
^4*m^2*x^5*e^5*sgn(b*x + a) + 6*(x*e + d)^m*a^2*b^2*d*m^4*x^2*e^4*sgn(b*x + a) + 32*(x*e + d)^m*a*b^3*d*m^3*x^
3*e^4*sgn(b*x + a) + 11*(x*e + d)^m*b^4*d*m^2*x^4*e^4*sgn(b*x + a) - 12*(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^4*d^2*m^2*x^3*e^3*sgn(b*x + a) + 12*(x*e + d)^m*b^4*d^3*m^2*x^2*e^2*sgn(b*x + a) +
4*(x*e + d)^m*a^3*b*m^4*x^2*e^5*sgn(b*x + a) + 72*(x*e + d)^m*a^2*b^2*m^3*x^3*e^5*sgn(b*x + a) + 164*(x*e + d
)^m*a*b^3*m^2*x^4*e^5*sgn(b*x + a) + 50*(x*e + d)^m*b^4*m*x^5*e^5*sgn(b*x + a) + 4*(x*e + d)^m*a^3*b*d*m^4*x*e
^4*sgn(b*x + a) + 60*(x*e + d)^m*a^2*b^2*d*m^3*x^2*e^4*sgn(b*x + a) + 68*(x*e + d)^m*a*b^3*d*m^2*x^3*e^4*sgn(b
*x + a) + 6*(x*e + d)^m*b^4*d*m*x^4*e^4*sgn(b*x + a) - 12*(x*e + d)^m*a^2*b^2*d^2*m^3*x*e^3*sgn(b*x + a) - 72*
(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^4*d^2*m*x^3*e^3*sgn(b*x + a) + 24*(x*e + d)^m
*a*b^3*d^3*m^2*x*e^2*sgn(b*x + a) + 12*(x*e + d)^m*b^4*d^3*m*x^2*e^2*sgn(b*x + a) - 24*(x*e + d)^m*b^4*d^4*m*x
*e*sgn(b*x + a) + (x*e + d)^m*a^4*m^4*x*e^5*sgn(b*x + a) + 52*(x*e + d)^m*a^3*b*m^3*x^2*e^5*sgn(b*x + a) + 294
*(x*e + d)^m*a^2*b^2*m^2*x^3*e^5*sgn(b*x + a) + 244*(x*e + d)^m*a*b^3*m*x^4*e^5*sgn(b*x + a) + 24*(x*e + d)^m*
b^4*x^5*e^5*sgn(b*x + a) + (x*e + d)^m*a^4*d*m^4*e^4*sgn(b*x + a) + 48*(x*e + d)^m*a^3*b*d*m^3*x*e^4*sgn(b*x +
a) + 174*(x*e + d)^m*a^2*b^2*d*m^2*x^2*e^4*sgn(b*x + a) + 40*(x*e + d)^m*a*b^3*d*m*x^3*e^4*sgn(b*x + a) - 4*(
x*e + d)^m*a^3*b*d^2*m^3*e^3*sgn(b*x + a) - 108*(x*e + d)^m*a^2*b^2*d^2*m^2*x*e^3*sgn(b*x + a) - 60*(x*e + d)^
m*a*b^3*d^2*m*x^2*e^3*sgn(b*x + a) + 12*(x*e + d)^m*a^2*b^2*d^3*m^2*e^2*sgn(b*x + a) + 120*(x*e + d)^m*a*b^3*d
^3*m*x*e^2*sgn(b*x + a) - 24*(x*e + d)^m*a*b^3*d^4*m*e*sgn(b*x + a) + 24*(x*e + d)^m*b^4*d^5*sgn(b*x + a) + 14
*(x*e + d)^m*a^4*m^3*x*e^5*sgn(b*x + a) + 236*(x*e + d)^m*a^3*b*m^2*x^2*e^5*sgn(b*x + a) + 468*(x*e + d)^m*a^2
*b^2*m*x^3*e^5*sgn(b*x + a) + 120*(x*e + d)^m*a*b^3*x^4*e^5*sgn(b*x + a) + 14*(x*e + d)^m*a^4*d*m^3*e^4*sgn(b*
x + a) + 188*(x*e + d)^m*a^3*b*d*m^2*x*e^4*sgn(b*x + a) + 120*(x*e + d)^m*a^2*b^2*d*m*x^2*e^4*sgn(b*x + a) - 4
8*(x*e + d)^m*a^3*b*d^2*m^2*e^3*sgn(b*x + a) - 240*(x*e + d)^m*a^2*b^2*d^2*m*x*e^3*sgn(b*x + a) + 108*(x*e + d
)^m*a^2*b^2*d^3*m*e^2*sgn(b*x + a) - 120*(x*e + d)^m*a*b^3*d^4*e*sgn(b*x + a) + 71*(x*e + d)^m*a^4*m^2*x*e^5*s
gn(b*x + a) + 428*(x*e + d)^m*a^3*b*m*x^2*e^5*sgn(b*x + a) + 240*(x*e + d)^m*a^2*b^2*x^3*e^5*sgn(b*x + a) + 71
*(x*e + d)^m*a^4*d*m^2*e^4*sgn(b*x + a) + 240*(x*e + d)^m*a^3*b*d*m*x*e^4*sgn(b*x + a) - 188*(x*e + d)^m*a^3*b
*d^2*m*e^3*sgn(b*x + a) + 240*(x*e + d)^m*a^2*b^2*d^3*e^2*sgn(b*x + a) + 154*(x*e + d)^m*a^4*m*x*e^5*sgn(b*x +
a) + 240*(x*e + d)^m*a^3*b*x^2*e^5*sgn(b*x + a) + 154*(x*e + d)^m*a^4*d*m*e^4*sgn(b*x + a) - 240*(x*e + d)^m*
a^3*b*d^2*e^3*sgn(b*x + a) + 120*(x*e + d)^m*a^4*x*e^5*sgn(b*x + a) + 120*(x*e + d)^m*a^4*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 = 0.05, size = 784, normalized size = 2.83 \begin {gather*} \frac {\left (b^{4} e^{4} m^{4} x^{4}+4 a \,b^{3} e^{4} m^{4} x^{3}+10 b^{4} e^{4} m^{3} x^{4}+6 a^{2} b^{2} e^{4} m^{4} x^{2}+44 a \,b^{3} e^{4} m^{3} x^{3}-4 b^{4} d \,e^{3} m^{3} x^{3}+35 b^{4} e^{4} m^{2} x^{4}+4 a^{3} b \,e^{4} m^{4} x +72 a^{2} b^{2} e^{4} m^{3} x^{2}-12 a \,b^{3} d \,e^{3} m^{3} x^{2}+164 a \,b^{3} e^{4} m^{2} x^{3}-24 b^{4} d \,e^{3} m^{2} x^{3}+50 b^{4} e^{4} m \,x^{4}+a^{4} e^{4} m^{4}+52 a^{3} b \,e^{4} m^{3} x -12 a^{2} b^{2} d \,e^{3} m^{3} x +294 a^{2} b^{2} e^{4} m^{2} x^{2}-96 a \,b^{3} d \,e^{3} m^{2} x^{2}+244 a \,b^{3} e^{4} m \,x^{3}+12 b^{4} d^{2} e^{2} m^{2} x^{2}-44 b^{4} d \,e^{3} m \,x^{3}+24 b^{4} e^{4} x^{4}+14 a^{4} e^{4} m^{3}-4 a^{3} b d \,e^{3} m^{3}+236 a^{3} b \,e^{4} m^{2} x -120 a^{2} b^{2} d \,e^{3} m^{2} x +468 a^{2} b^{2} e^{4} m \,x^{2}+24 a \,b^{3} d^{2} e^{2} m^{2} x -204 a \,b^{3} d \,e^{3} m \,x^{2}+120 a \,b^{3} e^{4} x^{3}+36 b^{4} d^{2} e^{2} m \,x^{2}-24 b^{4} d \,e^{3} x^{3}+71 a^{4} e^{4} m^{2}-48 a^{3} b d \,e^{3} m^{2}+428 a^{3} b \,e^{4} m x +12 a^{2} b^{2} d^{2} e^{2} m^{2}-348 a^{2} b^{2} d \,e^{3} m x +240 a^{2} b^{2} e^{4} x^{2}+144 a \,b^{3} d^{2} e^{2} m x -120 a \,b^{3} d \,e^{3} x^{2}-24 b^{4} d^{3} e m x +24 b^{4} d^{2} e^{2} x^{2}+154 a^{4} e^{4} m -188 a^{3} b d \,e^{3} m +240 a^{3} b \,e^{4} x +108 a^{2} b^{2} d^{2} e^{2} m -240 a^{2} b^{2} d \,e^{3} x -24 a \,b^{3} d^{3} e m +120 a \,b^{3} d^{2} e^{2} x -24 b^{4} d^{3} e x +120 a^{4} e^{4}-240 a^{3} b d \,e^{3}+240 a^{2} b^{2} d^{2} e^{2}-120 a \,b^{3} d^{3} e +24 b^{4} 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^4*e^4*m^4*x^4+4*a*b^3*e^4*m^4*x^3+10*b^4*e^4*m^3*x^4+6*a^2*b^2*e^4*m^4*x^2+44*a*b^3*e^4*m^3*x
^3-4*b^4*d*e^3*m^3*x^3+35*b^4*e^4*m^2*x^4+4*a^3*b*e^4*m^4*x+72*a^2*b^2*e^4*m^3*x^2-12*a*b^3*d*e^3*m^3*x^2+164*
a*b^3*e^4*m^2*x^3-24*b^4*d*e^3*m^2*x^3+50*b^4*e^4*m*x^4+a^4*e^4*m^4+52*a^3*b*e^4*m^3*x-12*a^2*b^2*d*e^3*m^3*x+
294*a^2*b^2*e^4*m^2*x^2-96*a*b^3*d*e^3*m^2*x^2+244*a*b^3*e^4*m*x^3+12*b^4*d^2*e^2*m^2*x^2-44*b^4*d*e^3*m*x^3+2
4*b^4*e^4*x^4+14*a^4*e^4*m^3-4*a^3*b*d*e^3*m^3+236*a^3*b*e^4*m^2*x-120*a^2*b^2*d*e^3*m^2*x+468*a^2*b^2*e^4*m*x
^2+24*a*b^3*d^2*e^2*m^2*x-204*a*b^3*d*e^3*m*x^2+120*a*b^3*e^4*x^3+36*b^4*d^2*e^2*m*x^2-24*b^4*d*e^3*x^3+71*a^4
*e^4*m^2-48*a^3*b*d*e^3*m^2+428*a^3*b*e^4*m*x+12*a^2*b^2*d^2*e^2*m^2-348*a^2*b^2*d*e^3*m*x+240*a^2*b^2*e^4*x^2
+144*a*b^3*d^2*e^2*m*x-120*a*b^3*d*e^3*x^2-24*b^4*d^3*e*m*x+24*b^4*d^2*e^2*x^2+154*a^4*e^4*m-188*a^3*b*d*e^3*m
+240*a^3*b*e^4*x+108*a^2*b^2*d^2*e^2*m-240*a^2*b^2*d*e^3*x-24*a*b^3*d^3*e*m+120*a*b^3*d^2*e^2*x-24*b^4*d^3*e*x
+120*a^4*e^4-240*a^3*b*d*e^3+240*a^2*b^2*d^2*e^2-120*a*b^3*d^3*e+24*b^4*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 = 0.76, size = 756, normalized size = 2.73 \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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