\(\int (d+e x)^m (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [1737]

Optimal result

Integrand size = 28, antiderivative size = 337 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {(b d-a e)^5 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (1+m) (a+b x)}+\frac {5 b (b d-a e)^4 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (2+m) (a+b x)}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (3+m) (a+b x)}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (4+m) (a+b x)}-\frac {5 b^4 (b d-a e) (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (5+m) (a+b x)}+\frac {b^5 (d+e x)^{6+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (6+m) (a+b x)} \]

[Out]

-(-a*e+b*d)^5*(e*x+d)^(1+m)*((b*x+a)^2)^(1/2)/e^6/(1+m)/(b*x+a)+5*b*(-a*e+b*d)^4*(e*x+d)^(2+m)*((b*x+a)^2)^(1/
2)/e^6/(2+m)/(b*x+a)-10*b^2*(-a*e+b*d)^3*(e*x+d)^(3+m)*((b*x+a)^2)^(1/2)/e^6/(3+m)/(b*x+a)+10*b^3*(-a*e+b*d)^2
*(e*x+d)^(4+m)*((b*x+a)^2)^(1/2)/e^6/(4+m)/(b*x+a)-5*b^4*(-a*e+b*d)*(e*x+d)^(5+m)*((b*x+a)^2)^(1/2)/e^6/(5+m)/
(b*x+a)+b^5*(e*x+d)^(6+m)*((b*x+a)^2)^(1/2)/e^6/(6+m)/(b*x+a)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (d+e x)^{m+1}}{e^6 (m+1) (a+b x)}+\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+2}}{e^6 (m+2) (a+b x)}-\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+3}}{e^6 (m+3) (a+b x)}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+6}}{e^6 (m+6) (a+b x)}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+5}}{e^6 (m+5) (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+4}}{e^6 (m+4) (a+b x)} \]

[In]

Int[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

-(((b*d - a*e)^5*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(1 + m)*(a + b*x))) + (5*b*(b*d - a*e)^
4*(d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(2 + m)*(a + b*x)) - (10*b^2*(b*d - a*e)^3*(d + e*x)^(
3 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(3 + m)*(a + b*x)) + (10*b^3*(b*d - a*e)^2*(d + e*x)^(4 + m)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(e^6*(4 + m)*(a + b*x)) - (5*b^4*(b*d - a*e)*(d + e*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(e^6*(5 + m)*(a + b*x)) + (b^5*(d + e*x)^(6 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(6 + m)*(a + b*x
))

Rule 45

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 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (d+e x)^m \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (d+e x)^m}{e^5}+\frac {5 b^6 (b d-a e)^4 (d+e x)^{1+m}}{e^5}-\frac {10 b^7 (b d-a e)^3 (d+e x)^{2+m}}{e^5}+\frac {10 b^8 (b d-a e)^2 (d+e x)^{3+m}}{e^5}-\frac {5 b^9 (b d-a e) (d+e x)^{4+m}}{e^5}+\frac {b^{10} (d+e x)^{5+m}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {(b d-a e)^5 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (1+m) (a+b x)}+\frac {5 b (b d-a e)^4 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (2+m) (a+b x)}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (3+m) (a+b x)}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (4+m) (a+b x)}-\frac {5 b^4 (b d-a e) (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (5+m) (a+b x)}+\frac {b^5 (d+e x)^{6+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (6+m) (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.50 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {\left ((a+b x)^2\right )^{5/2} (d+e x)^{1+m} \left (-\frac {(b d-a e)^5}{1+m}+\frac {5 b (b d-a e)^4 (d+e x)}{2+m}-\frac {10 b^2 (b d-a e)^3 (d+e x)^2}{3+m}+\frac {10 b^3 (b d-a e)^2 (d+e x)^3}{4+m}-\frac {5 b^4 (b d-a e) (d+e x)^4}{5+m}+\frac {b^5 (d+e x)^5}{6+m}\right )}{e^6 (a+b x)^5} \]

[In]

Integrate[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(((a + b*x)^2)^(5/2)*(d + e*x)^(1 + m)*(-((b*d - a*e)^5/(1 + m)) + (5*b*(b*d - a*e)^4*(d + e*x))/(2 + m) - (10
*b^2*(b*d - a*e)^3*(d + e*x)^2)/(3 + m) + (10*b^3*(b*d - a*e)^2*(d + e*x)^3)/(4 + m) - (5*b^4*(b*d - a*e)*(d +
e*x)^4)/(5 + m) + (b^5*(d + e*x)^5)/(6 + m)))/(e^6*(a + b*x)^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1360\) vs. \(2(271)=542\).

Time = 2.32 (sec) , antiderivative size = 1361, normalized size of antiderivative = 4.04

method	result	size
gosper	\(\text {Expression too large to display}\)	\(1361\)
risch	\(\text {Expression too large to display}\)	\(1745\)

[In]

int((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/e^6*(e*x+d)^(1+m)/(b*x+a)^5*((b*x+a)^2)^(5/2)/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+720)*(b^5*e^5*m^5*
x^5+5*a*b^4*e^5*m^5*x^4+15*b^5*e^5*m^4*x^5+10*a^2*b^3*e^5*m^5*x^3+80*a*b^4*e^5*m^4*x^4-5*b^5*d*e^4*m^4*x^4+85*
b^5*e^5*m^3*x^5+10*a^3*b^2*e^5*m^5*x^2+170*a^2*b^3*e^5*m^4*x^3-20*a*b^4*d*e^4*m^4*x^3+475*a*b^4*e^5*m^3*x^4-50
*b^5*d*e^4*m^3*x^4+225*b^5*e^5*m^2*x^5+5*a^4*b*e^5*m^5*x+180*a^3*b^2*e^5*m^4*x^2-30*a^2*b^3*d*e^4*m^4*x^2+1070
*a^2*b^3*e^5*m^3*x^3-240*a*b^4*d*e^4*m^3*x^3+1300*a*b^4*e^5*m^2*x^4+20*b^5*d^2*e^3*m^3*x^3-175*b^5*d*e^4*m^2*x
^4+274*b^5*e^5*m*x^5+a^5*e^5*m^5+95*a^4*b*e^5*m^4*x-20*a^3*b^2*d*e^4*m^4*x+1210*a^3*b^2*e^5*m^3*x^2-420*a^2*b^
3*d*e^4*m^3*x^2+3070*a^2*b^3*e^5*m^2*x^3+60*a*b^4*d^2*e^3*m^3*x^2-940*a*b^4*d*e^4*m^2*x^3+1620*a*b^4*e^5*m*x^4
+120*b^5*d^2*e^3*m^2*x^3-250*b^5*d*e^4*m*x^4+120*b^5*e^5*x^5+20*a^5*e^5*m^4-5*a^4*b*d*e^4*m^4+685*a^4*b*e^5*m^
3*x-320*a^3*b^2*d*e^4*m^3*x+3720*a^3*b^2*e^5*m^2*x^2+60*a^2*b^3*d^2*e^3*m^3*x-1950*a^2*b^3*d*e^4*m^2*x^2+3960*
a^2*b^3*e^5*m*x^3+540*a*b^4*d^2*e^3*m^2*x^2-1440*a*b^4*d*e^4*m*x^3+720*a*b^4*e^5*x^4-60*b^5*d^3*e^2*m^2*x^2+22
0*b^5*d^2*e^3*m*x^3-120*b^5*d*e^4*x^4+155*a^5*e^5*m^3-90*a^4*b*d*e^4*m^3+2305*a^4*b*e^5*m^2*x+20*a^3*b^2*d^2*e
^3*m^3-1780*a^3*b^2*d*e^4*m^2*x+5080*a^3*b^2*e^5*m*x^2+720*a^2*b^3*d^2*e^3*m^2*x-3360*a^2*b^3*d*e^4*m*x^2+1800
*a^2*b^3*e^5*x^3-120*a*b^4*d^3*e^2*m^2*x+1200*a*b^4*d^2*e^3*m*x^2-720*a*b^4*d*e^4*x^3-180*b^5*d^3*e^2*m*x^2+12
0*b^5*d^2*e^3*x^3+580*a^5*e^5*m^2-595*a^4*b*d*e^4*m^2+3510*a^4*b*e^5*m*x+300*a^3*b^2*d^2*e^3*m^2-3880*a^3*b^2*
d*e^4*m*x+2400*a^3*b^2*e^5*x^2-60*a^2*b^3*d^3*e^2*m^2+2460*a^2*b^3*d^2*e^3*m*x-1800*a^2*b^3*d*e^4*x^2-840*a*b^
4*d^3*e^2*m*x+720*a*b^4*d^2*e^3*x^2+120*b^5*d^4*e*m*x-120*b^5*d^3*e^2*x^2+1044*a^5*e^5*m-1710*a^4*b*d*e^4*m+18
00*a^4*b*e^5*x+1480*a^3*b^2*d^2*e^3*m-2400*a^3*b^2*d*e^4*x-660*a^2*b^3*d^3*e^2*m+1800*a^2*b^3*d^2*e^3*x+120*a*
b^4*d^4*e*m-720*a*b^4*d^3*e^2*x+120*b^5*d^4*e*x+720*a^5*e^5-1800*a^4*b*d*e^4+2400*a^3*b^2*d^2*e^3-1800*a^2*b^3
*d^3*e^2+720*a*b^4*d^4*e-120*b^5*d^5)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1460 vs. \(2 (271) = 542\).

Time = 0.39 (sec) , antiderivative size = 1460, normalized size of antiderivative = 4.33 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

(a^5*d*e^5*m^5 - 120*b^5*d^6 + 720*a*b^4*d^5*e - 1800*a^2*b^3*d^4*e^2 + 2400*a^3*b^2*d^3*e^3 - 1800*a^4*b*d^2*
e^4 + 720*a^5*d*e^5 + (b^5*e^6*m^5 + 15*b^5*e^6*m^4 + 85*b^5*e^6*m^3 + 225*b^5*e^6*m^2 + 274*b^5*e^6*m + 120*b
^5*e^6)*x^6 + (720*a*b^4*e^6 + (b^5*d*e^5 + 5*a*b^4*e^6)*m^5 + 10*(b^5*d*e^5 + 8*a*b^4*e^6)*m^4 + 5*(7*b^5*d*e
^5 + 95*a*b^4*e^6)*m^3 + 50*(b^5*d*e^5 + 26*a*b^4*e^6)*m^2 + 12*(2*b^5*d*e^5 + 135*a*b^4*e^6)*m)*x^5 - 5*(a^4*
b*d^2*e^4 - 4*a^5*d*e^5)*m^4 + 5*(360*a^2*b^3*e^6 + (a*b^4*d*e^5 + 2*a^2*b^3*e^6)*m^5 - (b^5*d^2*e^4 - 12*a*b^
4*d*e^5 - 34*a^2*b^3*e^6)*m^4 - (6*b^5*d^2*e^4 - 47*a*b^4*d*e^5 - 214*a^2*b^3*e^6)*m^3 - (11*b^5*d^2*e^4 - 72*
a*b^4*d*e^5 - 614*a^2*b^3*e^6)*m^2 - 6*(b^5*d^2*e^4 - 6*a*b^4*d*e^5 - 132*a^2*b^3*e^6)*m)*x^4 + 5*(4*a^3*b^2*d
^3*e^3 - 18*a^4*b*d^2*e^4 + 31*a^5*d*e^5)*m^3 + 10*(240*a^3*b^2*e^6 + (a^2*b^3*d*e^5 + a^3*b^2*e^6)*m^5 - 2*(a
*b^4*d^2*e^4 - 7*a^2*b^3*d*e^5 - 9*a^3*b^2*e^6)*m^4 + (2*b^5*d^3*e^3 - 18*a*b^4*d^2*e^4 + 65*a^2*b^3*d*e^5 + 1
21*a^3*b^2*e^6)*m^3 + 2*(3*b^5*d^3*e^3 - 20*a*b^4*d^2*e^4 + 56*a^2*b^3*d*e^5 + 186*a^3*b^2*e^6)*m^2 + 4*(b^5*d
^3*e^3 - 6*a*b^4*d^2*e^4 + 15*a^2*b^3*d*e^5 + 127*a^3*b^2*e^6)*m)*x^3 - 5*(12*a^2*b^3*d^4*e^2 - 60*a^3*b^2*d^3
*e^3 + 119*a^4*b*d^2*e^4 - 116*a^5*d*e^5)*m^2 + 5*(360*a^4*b*e^6 + (2*a^3*b^2*d*e^5 + a^4*b*e^6)*m^5 - (6*a^2*
b^3*d^2*e^4 - 32*a^3*b^2*d*e^5 - 19*a^4*b*e^6)*m^4 + (12*a*b^4*d^3*e^3 - 72*a^2*b^3*d^2*e^4 + 178*a^3*b^2*d*e^
5 + 137*a^4*b*e^6)*m^3 - (12*b^5*d^4*e^2 - 84*a*b^4*d^3*e^3 + 246*a^2*b^3*d^2*e^4 - 388*a^3*b^2*d*e^5 - 461*a^
4*b*e^6)*m^2 - 6*(2*b^5*d^4*e^2 - 12*a*b^4*d^3*e^3 + 30*a^2*b^3*d^2*e^4 - 40*a^3*b^2*d*e^5 - 117*a^4*b*e^6)*m)
*x^2 + 2*(60*a*b^4*d^5*e - 330*a^2*b^3*d^4*e^2 + 740*a^3*b^2*d^3*e^3 - 855*a^4*b*d^2*e^4 + 522*a^5*d*e^5)*m +
(720*a^5*e^6 + (5*a^4*b*d*e^5 + a^5*e^6)*m^5 - 10*(2*a^3*b^2*d^2*e^4 - 9*a^4*b*d*e^5 - 2*a^5*e^6)*m^4 + 5*(12*
a^2*b^3*d^3*e^3 - 60*a^3*b^2*d^2*e^4 + 119*a^4*b*d*e^5 + 31*a^5*e^6)*m^3 - 10*(12*a*b^4*d^4*e^2 - 66*a^2*b^3*d
^3*e^3 + 148*a^3*b^2*d^2*e^4 - 171*a^4*b*d*e^5 - 58*a^5*e^6)*m^2 + 12*(10*b^5*d^5*e - 60*a*b^4*d^4*e^2 + 150*a
^2*b^3*d^3*e^3 - 200*a^3*b^2*d^2*e^4 + 150*a^4*b*d*e^5 + 87*a^5*e^6)*m)*x)*(e*x + d)^m/(e^6*m^6 + 21*e^6*m^5 +
175*e^6*m^4 + 735*e^6*m^3 + 1624*e^6*m^2 + 1764*e^6*m + 720*e^6)

Sympy [F(-2)]

Exception generated. \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate((e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 792 vs. \(2 (271) = 542\).

Time = 0.20 (sec) , antiderivative size = 792, normalized size of antiderivative = 2.35 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {{\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} b^{5} e^{6} x^{6} - 60 \, {\left (m^{2} + 11 \, m + 30\right )} a^{2} b^{3} d^{4} e^{2} + 20 \, {\left (m^{3} + 15 \, m^{2} + 74 \, m + 120\right )} a^{3} b^{2} d^{3} e^{3} - 5 \, {\left (m^{4} + 18 \, m^{3} + 119 \, m^{2} + 342 \, m + 360\right )} a^{4} b d^{2} e^{4} + {\left (m^{5} + 20 \, m^{4} + 155 \, m^{3} + 580 \, m^{2} + 1044 \, m + 720\right )} a^{5} d e^{5} + 120 \, a b^{4} d^{5} e {\left (m + 6\right )} - 120 \, b^{5} d^{6} + {\left ({\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} b^{5} d e^{5} + 5 \, {\left (m^{5} + 16 \, m^{4} + 95 \, m^{3} + 260 \, m^{2} + 324 \, m + 144\right )} a b^{4} e^{6}\right )} x^{5} - 5 \, {\left ({\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b^{5} d^{2} e^{4} - {\left (m^{5} + 12 \, m^{4} + 47 \, m^{3} + 72 \, m^{2} + 36 \, m\right )} a b^{4} d e^{5} - 2 \, {\left (m^{5} + 17 \, m^{4} + 107 \, m^{3} + 307 \, m^{2} + 396 \, m + 180\right )} a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (2 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{5} d^{3} e^{3} - 2 \, {\left (m^{4} + 9 \, m^{3} + 20 \, m^{2} + 12 \, m\right )} a b^{4} d^{2} e^{4} + {\left (m^{5} + 14 \, m^{4} + 65 \, m^{3} + 112 \, m^{2} + 60 \, m\right )} a^{2} b^{3} d e^{5} + {\left (m^{5} + 18 \, m^{4} + 121 \, m^{3} + 372 \, m^{2} + 508 \, m + 240\right )} a^{3} b^{2} e^{6}\right )} x^{3} - 5 \, {\left (12 \, {\left (m^{2} + m\right )} b^{5} d^{4} e^{2} - 12 \, {\left (m^{3} + 7 \, m^{2} + 6 \, m\right )} a b^{4} d^{3} e^{3} + 6 \, {\left (m^{4} + 12 \, m^{3} + 41 \, m^{2} + 30 \, m\right )} a^{2} b^{3} d^{2} e^{4} - 2 \, {\left (m^{5} + 16 \, m^{4} + 89 \, m^{3} + 194 \, m^{2} + 120 \, m\right )} a^{3} b^{2} d e^{5} - {\left (m^{5} + 19 \, m^{4} + 137 \, m^{3} + 461 \, m^{2} + 702 \, m + 360\right )} a^{4} b e^{6}\right )} x^{2} - {\left (120 \, {\left (m^{2} + 6 \, m\right )} a b^{4} d^{4} e^{2} - 60 \, {\left (m^{3} + 11 \, m^{2} + 30 \, m\right )} a^{2} b^{3} d^{3} e^{3} + 20 \, {\left (m^{4} + 15 \, m^{3} + 74 \, m^{2} + 120 \, m\right )} a^{3} b^{2} d^{2} e^{4} - 5 \, {\left (m^{5} + 18 \, m^{4} + 119 \, m^{3} + 342 \, m^{2} + 360 \, m\right )} a^{4} b d e^{5} - {\left (m^{5} + 20 \, m^{4} + 155 \, m^{3} + 580 \, m^{2} + 1044 \, m + 720\right )} a^{5} e^{6} - 120 \, b^{5} d^{5} e m\right )} x\right )} {\left (e x + d\right )}^{m}}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{6}} \]

[In]

integrate((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b^5*e^6*x^6 - 60*(m^2 + 11*m + 30)*a^2*b^3*d^4*e^2 + 20*(m^3
+ 15*m^2 + 74*m + 120)*a^3*b^2*d^3*e^3 - 5*(m^4 + 18*m^3 + 119*m^2 + 342*m + 360)*a^4*b*d^2*e^4 + (m^5 + 20*m^
4 + 155*m^3 + 580*m^2 + 1044*m + 720)*a^5*d*e^5 + 120*a*b^4*d^5*e*(m + 6) - 120*b^5*d^6 + ((m^5 + 10*m^4 + 35*
m^3 + 50*m^2 + 24*m)*b^5*d*e^5 + 5*(m^5 + 16*m^4 + 95*m^3 + 260*m^2 + 324*m + 144)*a*b^4*e^6)*x^5 - 5*((m^4 +
6*m^3 + 11*m^2 + 6*m)*b^5*d^2*e^4 - (m^5 + 12*m^4 + 47*m^3 + 72*m^2 + 36*m)*a*b^4*d*e^5 - 2*(m^5 + 17*m^4 + 10
7*m^3 + 307*m^2 + 396*m + 180)*a^2*b^3*e^6)*x^4 + 10*(2*(m^3 + 3*m^2 + 2*m)*b^5*d^3*e^3 - 2*(m^4 + 9*m^3 + 20*
m^2 + 12*m)*a*b^4*d^2*e^4 + (m^5 + 14*m^4 + 65*m^3 + 112*m^2 + 60*m)*a^2*b^3*d*e^5 + (m^5 + 18*m^4 + 121*m^3 +
372*m^2 + 508*m + 240)*a^3*b^2*e^6)*x^3 - 5*(12*(m^2 + m)*b^5*d^4*e^2 - 12*(m^3 + 7*m^2 + 6*m)*a*b^4*d^3*e^3
+ 6*(m^4 + 12*m^3 + 41*m^2 + 30*m)*a^2*b^3*d^2*e^4 - 2*(m^5 + 16*m^4 + 89*m^3 + 194*m^2 + 120*m)*a^3*b^2*d*e^5
- (m^5 + 19*m^4 + 137*m^3 + 461*m^2 + 702*m + 360)*a^4*b*e^6)*x^2 - (120*(m^2 + 6*m)*a*b^4*d^4*e^2 - 60*(m^3
+ 11*m^2 + 30*m)*a^2*b^3*d^3*e^3 + 20*(m^4 + 15*m^3 + 74*m^2 + 120*m)*a^3*b^2*d^2*e^4 - 5*(m^5 + 18*m^4 + 119*
m^3 + 342*m^2 + 360*m)*a^4*b*d*e^5 - (m^5 + 20*m^4 + 155*m^3 + 580*m^2 + 1044*m + 720)*a^5*e^6 - 120*b^5*d^5*e
*m)*x)*(e*x + d)^m/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3197 vs. \(2 (271) = 542\).

Time = 0.35 (sec) , antiderivative size = 3197, normalized size of antiderivative = 9.49 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

((e*x + d)^m*b^5*e^6*m^5*x^6*sgn(b*x + a) + (e*x + d)^m*b^5*d*e^5*m^5*x^5*sgn(b*x + a) + 5*(e*x + d)^m*a*b^4*e
^6*m^5*x^5*sgn(b*x + a) + 15*(e*x + d)^m*b^5*e^6*m^4*x^6*sgn(b*x + a) + 5*(e*x + d)^m*a*b^4*d*e^5*m^5*x^4*sgn(
b*x + a) + 10*(e*x + d)^m*a^2*b^3*e^6*m^5*x^4*sgn(b*x + a) + 10*(e*x + d)^m*b^5*d*e^5*m^4*x^5*sgn(b*x + a) + 8
0*(e*x + d)^m*a*b^4*e^6*m^4*x^5*sgn(b*x + a) + 85*(e*x + d)^m*b^5*e^6*m^3*x^6*sgn(b*x + a) + 10*(e*x + d)^m*a^
2*b^3*d*e^5*m^5*x^3*sgn(b*x + a) + 10*(e*x + d)^m*a^3*b^2*e^6*m^5*x^3*sgn(b*x + a) - 5*(e*x + d)^m*b^5*d^2*e^4
*m^4*x^4*sgn(b*x + a) + 60*(e*x + d)^m*a*b^4*d*e^5*m^4*x^4*sgn(b*x + a) + 170*(e*x + d)^m*a^2*b^3*e^6*m^4*x^4*
sgn(b*x + a) + 35*(e*x + d)^m*b^5*d*e^5*m^3*x^5*sgn(b*x + a) + 475*(e*x + d)^m*a*b^4*e^6*m^3*x^5*sgn(b*x + a)
+ 225*(e*x + d)^m*b^5*e^6*m^2*x^6*sgn(b*x + a) + 10*(e*x + d)^m*a^3*b^2*d*e^5*m^5*x^2*sgn(b*x + a) + 5*(e*x +
d)^m*a^4*b*e^6*m^5*x^2*sgn(b*x + a) - 20*(e*x + d)^m*a*b^4*d^2*e^4*m^4*x^3*sgn(b*x + a) + 140*(e*x + d)^m*a^2*
b^3*d*e^5*m^4*x^3*sgn(b*x + a) + 180*(e*x + d)^m*a^3*b^2*e^6*m^4*x^3*sgn(b*x + a) - 30*(e*x + d)^m*b^5*d^2*e^4
*m^3*x^4*sgn(b*x + a) + 235*(e*x + d)^m*a*b^4*d*e^5*m^3*x^4*sgn(b*x + a) + 1070*(e*x + d)^m*a^2*b^3*e^6*m^3*x^
4*sgn(b*x + a) + 50*(e*x + d)^m*b^5*d*e^5*m^2*x^5*sgn(b*x + a) + 1300*(e*x + d)^m*a*b^4*e^6*m^2*x^5*sgn(b*x +
a) + 274*(e*x + d)^m*b^5*e^6*m*x^6*sgn(b*x + a) + 5*(e*x + d)^m*a^4*b*d*e^5*m^5*x*sgn(b*x + a) + (e*x + d)^m*a
^5*e^6*m^5*x*sgn(b*x + a) - 30*(e*x + d)^m*a^2*b^3*d^2*e^4*m^4*x^2*sgn(b*x + a) + 160*(e*x + d)^m*a^3*b^2*d*e^
5*m^4*x^2*sgn(b*x + a) + 95*(e*x + d)^m*a^4*b*e^6*m^4*x^2*sgn(b*x + a) + 20*(e*x + d)^m*b^5*d^3*e^3*m^3*x^3*sg
n(b*x + a) - 180*(e*x + d)^m*a*b^4*d^2*e^4*m^3*x^3*sgn(b*x + a) + 650*(e*x + d)^m*a^2*b^3*d*e^5*m^3*x^3*sgn(b*
x + a) + 1210*(e*x + d)^m*a^3*b^2*e^6*m^3*x^3*sgn(b*x + a) - 55*(e*x + d)^m*b^5*d^2*e^4*m^2*x^4*sgn(b*x + a) +
360*(e*x + d)^m*a*b^4*d*e^5*m^2*x^4*sgn(b*x + a) + 3070*(e*x + d)^m*a^2*b^3*e^6*m^2*x^4*sgn(b*x + a) + 24*(e*
x + d)^m*b^5*d*e^5*m*x^5*sgn(b*x + a) + 1620*(e*x + d)^m*a*b^4*e^6*m*x^5*sgn(b*x + a) + 120*(e*x + d)^m*b^5*e^
6*x^6*sgn(b*x + a) + (e*x + d)^m*a^5*d*e^5*m^5*sgn(b*x + a) - 20*(e*x + d)^m*a^3*b^2*d^2*e^4*m^4*x*sgn(b*x + a
) + 90*(e*x + d)^m*a^4*b*d*e^5*m^4*x*sgn(b*x + a) + 20*(e*x + d)^m*a^5*e^6*m^4*x*sgn(b*x + a) + 60*(e*x + d)^m
*a*b^4*d^3*e^3*m^3*x^2*sgn(b*x + a) - 360*(e*x + d)^m*a^2*b^3*d^2*e^4*m^3*x^2*sgn(b*x + a) + 890*(e*x + d)^m*a
^3*b^2*d*e^5*m^3*x^2*sgn(b*x + a) + 685*(e*x + d)^m*a^4*b*e^6*m^3*x^2*sgn(b*x + a) + 60*(e*x + d)^m*b^5*d^3*e^
3*m^2*x^3*sgn(b*x + a) - 400*(e*x + d)^m*a*b^4*d^2*e^4*m^2*x^3*sgn(b*x + a) + 1120*(e*x + d)^m*a^2*b^3*d*e^5*m
^2*x^3*sgn(b*x + a) + 3720*(e*x + d)^m*a^3*b^2*e^6*m^2*x^3*sgn(b*x + a) - 30*(e*x + d)^m*b^5*d^2*e^4*m*x^4*sgn
(b*x + a) + 180*(e*x + d)^m*a*b^4*d*e^5*m*x^4*sgn(b*x + a) + 3960*(e*x + d)^m*a^2*b^3*e^6*m*x^4*sgn(b*x + a) +
720*(e*x + d)^m*a*b^4*e^6*x^5*sgn(b*x + a) - 5*(e*x + d)^m*a^4*b*d^2*e^4*m^4*sgn(b*x + a) + 20*(e*x + d)^m*a^
5*d*e^5*m^4*sgn(b*x + a) + 60*(e*x + d)^m*a^2*b^3*d^3*e^3*m^3*x*sgn(b*x + a) - 300*(e*x + d)^m*a^3*b^2*d^2*e^4
*m^3*x*sgn(b*x + a) + 595*(e*x + d)^m*a^4*b*d*e^5*m^3*x*sgn(b*x + a) + 155*(e*x + d)^m*a^5*e^6*m^3*x*sgn(b*x +
a) - 60*(e*x + d)^m*b^5*d^4*e^2*m^2*x^2*sgn(b*x + a) + 420*(e*x + d)^m*a*b^4*d^3*e^3*m^2*x^2*sgn(b*x + a) - 1
230*(e*x + d)^m*a^2*b^3*d^2*e^4*m^2*x^2*sgn(b*x + a) + 1940*(e*x + d)^m*a^3*b^2*d*e^5*m^2*x^2*sgn(b*x + a) + 2
305*(e*x + d)^m*a^4*b*e^6*m^2*x^2*sgn(b*x + a) + 40*(e*x + d)^m*b^5*d^3*e^3*m*x^3*sgn(b*x + a) - 240*(e*x + d)
^m*a*b^4*d^2*e^4*m*x^3*sgn(b*x + a) + 600*(e*x + d)^m*a^2*b^3*d*e^5*m*x^3*sgn(b*x + a) + 5080*(e*x + d)^m*a^3*
b^2*e^6*m*x^3*sgn(b*x + a) + 1800*(e*x + d)^m*a^2*b^3*e^6*x^4*sgn(b*x + a) + 20*(e*x + d)^m*a^3*b^2*d^3*e^3*m^
3*sgn(b*x + a) - 90*(e*x + d)^m*a^4*b*d^2*e^4*m^3*sgn(b*x + a) + 155*(e*x + d)^m*a^5*d*e^5*m^3*sgn(b*x + a) -
120*(e*x + d)^m*a*b^4*d^4*e^2*m^2*x*sgn(b*x + a) + 660*(e*x + d)^m*a^2*b^3*d^3*e^3*m^2*x*sgn(b*x + a) - 1480*(
e*x + d)^m*a^3*b^2*d^2*e^4*m^2*x*sgn(b*x + a) + 1710*(e*x + d)^m*a^4*b*d*e^5*m^2*x*sgn(b*x + a) + 580*(e*x + d
)^m*a^5*e^6*m^2*x*sgn(b*x + a) - 60*(e*x + d)^m*b^5*d^4*e^2*m*x^2*sgn(b*x + a) + 360*(e*x + d)^m*a*b^4*d^3*e^3
*m*x^2*sgn(b*x + a) - 900*(e*x + d)^m*a^2*b^3*d^2*e^4*m*x^2*sgn(b*x + a) + 1200*(e*x + d)^m*a^3*b^2*d*e^5*m*x^
2*sgn(b*x + a) + 3510*(e*x + d)^m*a^4*b*e^6*m*x^2*sgn(b*x + a) + 2400*(e*x + d)^m*a^3*b^2*e^6*x^3*sgn(b*x + a)
- 60*(e*x + d)^m*a^2*b^3*d^4*e^2*m^2*sgn(b*x + a) + 300*(e*x + d)^m*a^3*b^2*d^3*e^3*m^2*sgn(b*x + a) - 595*(e
*x + d)^m*a^4*b*d^2*e^4*m^2*sgn(b*x + a) + 580*(e*x + d)^m*a^5*d*e^5*m^2*sgn(b*x + a) + 120*(e*x + d)^m*b^5*d^
5*e*m*x*sgn(b*x + a) - 720*(e*x + d)^m*a*b^4*d^4*e^2*m*x*sgn(b*x + a) + 1800*(e*x + d)^m*a^2*b^3*d^3*e^3*m*x*s
gn(b*x + a) - 2400*(e*x + d)^m*a^3*b^2*d^2*e^4*m*x*sgn(b*x + a) + 1800*(e*x + d)^m*a^4*b*d*e^5*m*x*sgn(b*x + a
) + 1044*(e*x + d)^m*a^5*e^6*m*x*sgn(b*x + a) + 1800*(e*x + d)^m*a^4*b*e^6*x^2*sgn(b*x + a) + 120*(e*x + d)^m*
a*b^4*d^5*e*m*sgn(b*x + a) - 660*(e*x + d)^m*a^2*b^3*d^4*e^2*m*sgn(b*x + a) + 1480*(e*x + d)^m*a^3*b^2*d^3*e^3
*m*sgn(b*x + a) - 1710*(e*x + d)^m*a^4*b*d^2*e^4*m*sgn(b*x + a) + 1044*(e*x + d)^m*a^5*d*e^5*m*sgn(b*x + a) +
720*(e*x + d)^m*a^5*e^6*x*sgn(b*x + a) - 120*(e*x + d)^m*b^5*d^6*sgn(b*x + a) + 720*(e*x + d)^m*a*b^4*d^5*e*sg
n(b*x + a) - 1800*(e*x + d)^m*a^2*b^3*d^4*e^2*sgn(b*x + a) + 2400*(e*x + d)^m*a^3*b^2*d^3*e^3*sgn(b*x + a) - 1
800*(e*x + d)^m*a^4*b*d^2*e^4*sgn(b*x + a) + 720*(e*x + d)^m*a^5*d*e^5*sgn(b*x + a))/(e^6*m^6 + 21*e^6*m^5 + 1
75*e^6*m^4 + 735*e^6*m^3 + 1624*e^6*m^2 + 1764*e^6*m + 720*e^6)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]

[In]

int((d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)