3.22.51 \(\int (a+b x) (d+e x)^m (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [2151]
Optimal. Leaf size=395 \[ \frac {(b d-a e)^6 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (1+m) (a+b x)}-\frac {6 b (b d-a e)^5 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (2+m) (a+b x)}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (3+m) (a+b x)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (4+m) (a+b x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (5+m) (a+b x)}-\frac {6 b^5 (b d-a e) (d+e x)^{6+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (6+m) (a+b x)}+\frac {b^6 (d+e x)^{7+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (7+m) (a+b x)} \]
[Out]
(-a*e+b*d)^6*(e*x+d)^(1+m)*((b*x+a)^2)^(1/2)/e^7/(1+m)/(b*x+a)-6*b*(-a*e+b*d)^5*(e*x+d)^(2+m)*((b*x+a)^2)^(1/2
)/e^7/(2+m)/(b*x+a)+15*b^2*(-a*e+b*d)^4*(e*x+d)^(3+m)*((b*x+a)^2)^(1/2)/e^7/(3+m)/(b*x+a)-20*b^3*(-a*e+b*d)^3*
(e*x+d)^(4+m)*((b*x+a)^2)^(1/2)/e^7/(4+m)/(b*x+a)+15*b^4*(-a*e+b*d)^2*(e*x+d)^(5+m)*((b*x+a)^2)^(1/2)/e^7/(5+m
)/(b*x+a)-6*b^5*(-a*e+b*d)*(e*x+d)^(6+m)*((b*x+a)^2)^(1/2)/e^7/(6+m)/(b*x+a)+b^6*(e*x+d)^(7+m)*((b*x+a)^2)^(1/
2)/e^7/(7+m)/(b*x+a)
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Rubi [A]
time = 0.14, antiderivative size = 395, 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 = {784, 21, 45}
\begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6 (d+e x)^{m+1}}{e^7 (m+1) (a+b x)}-\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (d+e x)^{m+2}}{e^7 (m+2) (a+b x)}+\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+3}}{e^7 (m+3) (a+b x)}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+7}}{e^7 (m+7) (a+b x)}-\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+6}}{e^7 (m+6) (a+b x)}+\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+5}}{e^7 (m+5) (a+b x)}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+4}}{e^7 (m+4) (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)^(5/2),x]
[Out]
((b*d - a*e)^6*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(1 + m)*(a + b*x)) - (6*b*(b*d - a*e)^5*(
d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(2 + m)*(a + b*x)) + (15*b^2*(b*d - a*e)^4*(d + e*x)^(3 +
m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(3 + m)*(a + b*x)) - (20*b^3*(b*d - a*e)^3*(d + e*x)^(4 + m)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(e^7*(4 + m)*(a + b*x)) + (15*b^4*(b*d - a*e)^2*(d + e*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(e^7*(5 + m)*(a + b*x)) - (6*b^5*(b*d - a*e)*(d + e*x)^(6 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(6
+ m)*(a + b*x)) + (b^6*(d + e*x)^(7 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(7 + 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 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 784
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 )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^m \, dx}{b^4 \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)^6 (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)^6 (d+e x)^m}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{1+m}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{2+m}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{3+m}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{4+m}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{5+m}}{e^6}+\frac {b^6 (d+e x)^{6+m}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^6 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (1+m) (a+b x)}-\frac {6 b (b d-a e)^5 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (2+m) (a+b x)}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (3+m) (a+b x)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (4+m) (a+b x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (5+m) (a+b x)}-\frac {6 b^5 (b d-a e) (d+e x)^{6+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (6+m) (a+b x)}+\frac {b^6 (d+e x)^{7+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (7+m) (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 193, normalized size = 0.49 \begin {gather*} \frac {\sqrt {(a+b x)^2} (d+e x)^{1+m} \left (\frac {(b d-a e)^6}{1+m}-\frac {6 b (b d-a e)^5 (d+e x)}{2+m}+\frac {15 b^2 (b d-a e)^4 (d+e x)^2}{3+m}-\frac {20 b^3 (b d-a e)^3 (d+e x)^3}{4+m}+\frac {15 b^4 (b d-a e)^2 (d+e x)^4}{5+m}-\frac {6 b^5 (b d-a e) (d+e x)^5}{6+m}+\frac {b^6 (d+e x)^6}{7+m}\right )}{e^7 (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)^(5/2),x]
[Out]
(Sqrt[(a + b*x)^2]*(d + e*x)^(1 + m)*((b*d - a*e)^6/(1 + m) - (6*b*(b*d - a*e)^5*(d + e*x))/(2 + m) + (15*b^2*
(b*d - a*e)^4*(d + e*x)^2)/(3 + m) - (20*b^3*(b*d - a*e)^3*(d + e*x)^3)/(4 + m) + (15*b^4*(b*d - a*e)^2*(d + e
*x)^4)/(5 + m) - (6*b^5*(b*d - a*e)*(d + e*x)^5)/(6 + m) + (b^6*(d + e*x)^6)/(7 + m)))/(e^7*(a + b*x))
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2172\) vs.
\(2(318)=636\).
time = 0.06, size = 2173, normalized size = 5.50
| | |
| method |
result |
size |
| | |
| gosper |
\(\text {Expression too large to display}\) |
\(2173\) |
| risch |
\(\text {Expression too large to display}\) |
\(2703\) |
| | |
|
|
|
|
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)^(5/2),x,method=_RETURNVERBOSE)
[Out]
(e*x+d)^(1+m)*(b^6*e^6*m^6*x^6+6*a*b^5*e^6*m^6*x^5+21*b^6*e^6*m^5*x^6+15*a^2*b^4*e^6*m^6*x^4+132*a*b^5*e^6*m^5
*x^5-6*b^6*d*e^5*m^5*x^5+175*b^6*e^6*m^4*x^6+20*a^3*b^3*e^6*m^6*x^3+345*a^2*b^4*e^6*m^5*x^4-30*a*b^5*d*e^5*m^5
*x^4+1140*a*b^5*e^6*m^4*x^5-90*b^6*d*e^5*m^4*x^5+735*b^6*e^6*m^3*x^6+15*a^4*b^2*e^6*m^6*x^2+480*a^3*b^3*e^6*m^
5*x^3-60*a^2*b^4*d*e^5*m^5*x^3+3105*a^2*b^4*e^6*m^4*x^4-510*a*b^5*d*e^5*m^4*x^4+4920*a*b^5*e^6*m^3*x^5+30*b^6*
d^2*e^4*m^4*x^4-510*b^6*d*e^5*m^3*x^5+1624*b^6*e^6*m^2*x^6+6*a^5*b*e^6*m^6*x+375*a^4*b^2*e^6*m^5*x^2-60*a^3*b^
3*d*e^5*m^5*x^2+4520*a^3*b^3*e^6*m^4*x^3-1140*a^2*b^4*d*e^5*m^4*x^3+13875*a^2*b^4*e^6*m^3*x^4+120*a*b^5*d^2*e^
4*m^4*x^3-3150*a*b^5*d*e^5*m^3*x^4+11094*a*b^5*e^6*m^2*x^5+300*b^6*d^2*e^4*m^3*x^4-1350*b^6*d*e^5*m^2*x^5+1764
*b^6*e^6*m*x^6+a^6*e^6*m^6+156*a^5*b*e^6*m^5*x-30*a^4*b^2*d*e^5*m^5*x+3705*a^4*b^2*e^6*m^4*x^2-1260*a^3*b^3*d*
e^5*m^4*x^2+21120*a^3*b^3*e^6*m^3*x^3+180*a^2*b^4*d^2*e^4*m^4*x^2-7860*a^2*b^4*d*e^5*m^3*x^3+32160*a^2*b^4*e^6
*m^2*x^4+1560*a*b^5*d^2*e^4*m^3*x^3-8850*a*b^5*d*e^5*m^2*x^4+12228*a*b^5*e^6*m*x^5-120*b^6*d^3*e^3*m^3*x^3+105
0*b^6*d^2*e^4*m^2*x^4-1644*b^6*d*e^5*m*x^5+720*b^6*e^6*x^6+27*a^6*e^6*m^5-6*a^5*b*d*e^5*m^5+1620*a^5*b*e^6*m^4
*x-690*a^4*b^2*d*e^5*m^4*x+18285*a^4*b^2*e^6*m^3*x^2+120*a^3*b^3*d^2*e^4*m^4*x-9780*a^3*b^3*d*e^5*m^3*x^2+5090
0*a^3*b^3*e^6*m^2*x^3+2880*a^2*b^4*d^2*e^4*m^3*x^2-24060*a^2*b^4*d*e^5*m^2*x^3+36180*a^2*b^4*e^6*m*x^4-360*a*b
^5*d^3*e^3*m^3*x^2+6360*a*b^5*d^2*e^4*m^2*x^3-11220*a*b^5*d*e^5*m*x^4+5040*a*b^5*e^6*x^5-720*b^6*d^3*e^3*m^2*x
^3+1500*b^6*d^2*e^4*m*x^4-720*b^6*d*e^5*x^5+295*a^6*e^6*m^4-150*a^5*b*d*e^5*m^4+8520*a^5*b*e^6*m^3*x+30*a^4*b^
2*d^2*e^4*m^4-6030*a^4*b^2*d*e^5*m^3*x+46680*a^4*b^2*e^6*m^2*x^2+2280*a^3*b^3*d^2*e^4*m^3*x-34020*a^3*b^3*d*e^
5*m^2*x^2+59040*a^3*b^3*e^6*m*x^3-360*a^2*b^4*d^3*e^3*m^3*x+14940*a^2*b^4*d^2*e^4*m^2*x^2-32400*a^2*b^4*d*e^5*
m*x^3+15120*a^2*b^4*e^6*x^4-3600*a*b^5*d^3*e^3*m^2*x^2+9960*a*b^5*d^2*e^4*m*x^3-5040*a*b^5*d*e^5*x^4+360*b^6*d
^4*e^2*m^2*x^2-1320*b^6*d^3*e^3*m*x^3+720*b^6*d^2*e^4*x^4+1665*a^6*e^6*m^3-1470*a^5*b*d*e^5*m^3+23574*a^5*b*e^
6*m^2*x+660*a^4*b^2*d^2*e^4*m^3-24510*a^4*b^2*d*e^5*m^2*x+56940*a^4*b^2*e^6*m*x^2-120*a^3*b^3*d^3*e^3*m^3+1500
0*a^3*b^3*d^2*e^4*m^2*x-50640*a^3*b^3*d*e^5*m*x^2+25200*a^3*b^3*e^6*x^3-5040*a^2*b^4*d^3*e^3*m^2*x+27360*a^2*b
^4*d^2*e^4*m*x^2-15120*a^2*b^4*d*e^5*x^3+720*a*b^5*d^4*e^2*m^2*x-8280*a*b^5*d^3*e^3*m*x^2+5040*a*b^5*d^2*e^4*x
^3+1080*b^6*d^4*e^2*m*x^2-720*b^6*d^3*e^3*x^3+5104*a^6*e^6*m^2-7050*a^5*b*d*e^5*m^2+31644*a^5*b*e^6*m*x+5370*a
^4*b^2*d^2*e^4*m^2-44340*a^4*b^2*d*e^5*m*x+25200*a^4*b^2*e^6*x^2-2160*a^3*b^3*d^3*e^3*m^2+38040*a^3*b^3*d^2*e^
4*m*x-25200*a^3*b^3*d*e^5*x^2+360*a^2*b^4*d^4*e^2*m^2-19800*a^2*b^4*d^3*e^3*m*x+15120*a^2*b^4*d^2*e^4*x^2+5760
*a*b^5*d^4*e^2*m*x-5040*a*b^5*d^3*e^3*x^2-720*b^6*d^5*e*m*x+720*b^6*d^4*e^2*x^2+8028*a^6*e^6*m-16524*a^5*b*d*e
^5*m+15120*a^5*b*e^6*x+19140*a^4*b^2*d^2*e^4*m-25200*a^4*b^2*d*e^5*x-12840*a^3*b^3*d^3*e^3*m+25200*a^3*b^3*d^2
*e^4*x+4680*a^2*b^4*d^4*e^2*m-15120*a^2*b^4*d^3*e^3*x-720*a*b^5*d^5*e*m+5040*a*b^5*d^4*e^2*x-720*b^6*d^5*e*x+5
040*a^6*e^6-15120*a^5*b*d*e^5+25200*a^4*b^2*d^2*e^4-25200*a^3*b^3*d^3*e^3+15120*a^2*b^4*d^4*e^2-5040*a*b^5*d^5
*e+720*b^6*d^6)*((b*x+a)^2)^(5/2)/(b*x+a)^5/e^7/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1818 vs.
\(2 (324) = 648\).
time = 0.31, size = 1818, normalized size = 4.60 \begin {gather*} \text {Too large to display} \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)^(5/2),x, algorithm="maxima")
[Out]
((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b^5*x^6*e^6 + 120*a*b^4*d^5*(m + 6)*e - 120*b^5*d^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 + 3
42*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 + ((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*b^5*d^5*m*e - 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)*x)*a*e^(m*log(x*e + d) - 6)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720) + ((m^6 + 21*m^5
+ 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*b^5*x^7*e^7 - 600*a*b^4*d^6*(m + 7)*e + 720*b^5*d^7 + 240*(m^2
+ 13*m + 42)*a^2*b^3*d^5*e^2 - 60*(m^3 + 18*m^2 + 107*m + 210)*a^3*b^2*d^4*e^3 + 10*(m^4 + 22*m^3 + 179*m^2 +
638*m + 840)*a^4*b*d^3*e^4 - (m^5 + 25*m^4 + 245*m^3 + 1175*m^2 + 2754*m + 2520)*a^5*d^2*e^5 + ((m^6 + 15*m^5
+ 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*b^5*d*e^6 + 5*(m^6 + 22*m^5 + 190*m^4 + 820*m^3 + 1849*m^2 + 2038*m + 84
0)*a*b^4*e^7)*x^6 - (6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*b^5*d^2*e^5 - 5*(m^6 + 17*m^5 + 105*m^4 + 295*m
^3 + 374*m^2 + 168*m)*a*b^4*d*e^6 - 10*(m^6 + 23*m^5 + 207*m^4 + 925*m^3 + 2144*m^2 + 2412*m + 1008)*a^2*b^3*e
^7)*x^5 + 5*(6*(m^4 + 6*m^3 + 11*m^2 + 6*m)*b^5*d^3*e^4 - 5*(m^5 + 13*m^4 + 53*m^3 + 83*m^2 + 42*m)*a*b^4*d^2*
e^5 + 2*(m^6 + 19*m^5 + 131*m^4 + 401*m^3 + 540*m^2 + 252*m)*a^2*b^3*d*e^6 + 2*(m^6 + 24*m^5 + 226*m^4 + 1056*
m^3 + 2545*m^2 + 2952*m + 1260)*a^3*b^2*e^7)*x^4 - 5*(24*(m^3 + 3*m^2 + 2*m)*b^5*d^4*e^3 - 20*(m^4 + 10*m^3 +
23*m^2 + 14*m)*a*b^4*d^3*e^4 + 8*(m^5 + 16*m^4 + 83*m^3 + 152*m^2 + 84*m)*a^2*b^3*d^2*e^5 - 2*(m^6 + 21*m^5 +
163*m^4 + 567*m^3 + 844*m^2 + 420*m)*a^3*b^2*d*e^6 - (m^6 + 25*m^5 + 247*m^4 + 1219*m^3 + 3112*m^2 + 3796*m +
1680)*a^4*b*e^7)*x^3 + (360*(m^2 + m)*b^5*d^5*e^2 - 300*(m^3 + 8*m^2 + 7*m)*a*b^4*d^4*e^3 + 120*(m^4 + 14*m^3
+ 55*m^2 + 42*m)*a^2*b^3*d^3*e^4 - 30*(m^5 + 19*m^4 + 125*m^3 + 317*m^2 + 210*m)*a^3*b^2*d^2*e^5 + 5*(m^6 + 23
*m^5 + 201*m^4 + 817*m^3 + 1478*m^2 + 840*m)*a^4*b*d*e^6 + (m^6 + 26*m^5 + 270*m^4 + 1420*m^3 + 3929*m^2 + 527
4*m + 2520)*a^5*e^7)*x^2 - (720*b^5*d^6*m*e - 600*(m^2 + 7*m)*a*b^4*d^5*e^2 + 240*(m^3 + 13*m^2 + 42*m)*a^2*b^
3*d^4*e^3 - 60*(m^4 + 18*m^3 + 107*m^2 + 210*m)*a^3*b^2*d^3*e^4 + 10*(m^5 + 22*m^4 + 179*m^3 + 638*m^2 + 840*m
)*a^4*b*d^2*e^5 - (m^6 + 25*m^5 + 245*m^4 + 1175*m^3 + 2754*m^2 + 2520*m)*a^5*d*e^6)*x)*b*e^(m*log(x*e + d) -
7)/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1973 vs.
\(2 (324) = 648\).
time = 3.17, size = 1973, normalized size = 4.99 \begin {gather*} \text {Too large to display} \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)^(5/2),x, algorithm="fricas")
[Out]
(720*b^6*d^7 + ((b^6*m^6 + 21*b^6*m^5 + 175*b^6*m^4 + 735*b^6*m^3 + 1624*b^6*m^2 + 1764*b^6*m + 720*b^6)*x^7 +
6*(a*b^5*m^6 + 22*a*b^5*m^5 + 190*a*b^5*m^4 + 820*a*b^5*m^3 + 1849*a*b^5*m^2 + 2038*a*b^5*m + 840*a*b^5)*x^6
+ 15*(a^2*b^4*m^6 + 23*a^2*b^4*m^5 + 207*a^2*b^4*m^4 + 925*a^2*b^4*m^3 + 2144*a^2*b^4*m^2 + 2412*a^2*b^4*m + 1
008*a^2*b^4)*x^5 + 20*(a^3*b^3*m^6 + 24*a^3*b^3*m^5 + 226*a^3*b^3*m^4 + 1056*a^3*b^3*m^3 + 2545*a^3*b^3*m^2 +
2952*a^3*b^3*m + 1260*a^3*b^3)*x^4 + 15*(a^4*b^2*m^6 + 25*a^4*b^2*m^5 + 247*a^4*b^2*m^4 + 1219*a^4*b^2*m^3 + 3
112*a^4*b^2*m^2 + 3796*a^4*b^2*m + 1680*a^4*b^2)*x^3 + 6*(a^5*b*m^6 + 26*a^5*b*m^5 + 270*a^5*b*m^4 + 1420*a^5*
b*m^3 + 3929*a^5*b*m^2 + 5274*a^5*b*m + 2520*a^5*b)*x^2 + (a^6*m^6 + 27*a^6*m^5 + 295*a^6*m^4 + 1665*a^6*m^3 +
5104*a^6*m^2 + 8028*a^6*m + 5040*a^6)*x)*e^7 + (a^6*d*m^6 + 27*a^6*d*m^5 + 295*a^6*d*m^4 + 1665*a^6*d*m^3 + 5
104*a^6*d*m^2 + 8028*a^6*d*m + 5040*a^6*d + (b^6*d*m^6 + 15*b^6*d*m^5 + 85*b^6*d*m^4 + 225*b^6*d*m^3 + 274*b^6
*d*m^2 + 120*b^6*d*m)*x^6 + 6*(a*b^5*d*m^6 + 17*a*b^5*d*m^5 + 105*a*b^5*d*m^4 + 295*a*b^5*d*m^3 + 374*a*b^5*d*
m^2 + 168*a*b^5*d*m)*x^5 + 15*(a^2*b^4*d*m^6 + 19*a^2*b^4*d*m^5 + 131*a^2*b^4*d*m^4 + 401*a^2*b^4*d*m^3 + 540*
a^2*b^4*d*m^2 + 252*a^2*b^4*d*m)*x^4 + 20*(a^3*b^3*d*m^6 + 21*a^3*b^3*d*m^5 + 163*a^3*b^3*d*m^4 + 567*a^3*b^3*
d*m^3 + 844*a^3*b^3*d*m^2 + 420*a^3*b^3*d*m)*x^3 + 15*(a^4*b^2*d*m^6 + 23*a^4*b^2*d*m^5 + 201*a^4*b^2*d*m^4 +
817*a^4*b^2*d*m^3 + 1478*a^4*b^2*d*m^2 + 840*a^4*b^2*d*m)*x^2 + 6*(a^5*b*d*m^6 + 25*a^5*b*d*m^5 + 245*a^5*b*d*
m^4 + 1175*a^5*b*d*m^3 + 2754*a^5*b*d*m^2 + 2520*a^5*b*d*m)*x)*e^6 - 6*(a^5*b*d^2*m^5 + 25*a^5*b*d^2*m^4 + 245
*a^5*b*d^2*m^3 + 1175*a^5*b*d^2*m^2 + 2754*a^5*b*d^2*m + 2520*a^5*b*d^2 + (b^6*d^2*m^5 + 10*b^6*d^2*m^4 + 35*b
^6*d^2*m^3 + 50*b^6*d^2*m^2 + 24*b^6*d^2*m)*x^5 + 5*(a*b^5*d^2*m^5 + 13*a*b^5*d^2*m^4 + 53*a*b^5*d^2*m^3 + 83*
a*b^5*d^2*m^2 + 42*a*b^5*d^2*m)*x^4 + 10*(a^2*b^4*d^2*m^5 + 16*a^2*b^4*d^2*m^4 + 83*a^2*b^4*d^2*m^3 + 152*a^2*
b^4*d^2*m^2 + 84*a^2*b^4*d^2*m)*x^3 + 10*(a^3*b^3*d^2*m^5 + 19*a^3*b^3*d^2*m^4 + 125*a^3*b^3*d^2*m^3 + 317*a^3
*b^3*d^2*m^2 + 210*a^3*b^3*d^2*m)*x^2 + 5*(a^4*b^2*d^2*m^5 + 22*a^4*b^2*d^2*m^4 + 179*a^4*b^2*d^2*m^3 + 638*a^
4*b^2*d^2*m^2 + 840*a^4*b^2*d^2*m)*x)*e^5 + 30*(a^4*b^2*d^3*m^4 + 22*a^4*b^2*d^3*m^3 + 179*a^4*b^2*d^3*m^2 + 6
38*a^4*b^2*d^3*m + 840*a^4*b^2*d^3 + (b^6*d^3*m^4 + 6*b^6*d^3*m^3 + 11*b^6*d^3*m^2 + 6*b^6*d^3*m)*x^4 + 4*(a*b
^5*d^3*m^4 + 10*a*b^5*d^3*m^3 + 23*a*b^5*d^3*m^2 + 14*a*b^5*d^3*m)*x^3 + 6*(a^2*b^4*d^3*m^4 + 14*a^2*b^4*d^3*m
^3 + 55*a^2*b^4*d^3*m^2 + 42*a^2*b^4*d^3*m)*x^2 + 4*(a^3*b^3*d^3*m^4 + 18*a^3*b^3*d^3*m^3 + 107*a^3*b^3*d^3*m^
2 + 210*a^3*b^3*d^3*m)*x)*e^4 - 120*(a^3*b^3*d^4*m^3 + 18*a^3*b^3*d^4*m^2 + 107*a^3*b^3*d^4*m + 210*a^3*b^3*d^
4 + (b^6*d^4*m^3 + 3*b^6*d^4*m^2 + 2*b^6*d^4*m)*x^3 + 3*(a*b^5*d^4*m^3 + 8*a*b^5*d^4*m^2 + 7*a*b^5*d^4*m)*x^2
+ 3*(a^2*b^4*d^4*m^3 + 13*a^2*b^4*d^4*m^2 + 42*a^2*b^4*d^4*m)*x)*e^3 + 360*(a^2*b^4*d^5*m^2 + 13*a^2*b^4*d^5*m
+ 42*a^2*b^4*d^5 + (b^6*d^5*m^2 + b^6*d^5*m)*x^2 + 2*(a*b^5*d^5*m^2 + 7*a*b^5*d^5*m)*x)*e^2 - 720*(b^6*d^6*m*
x + a*b^5*d^6*m + 7*a*b^5*d^6)*e)*(x*e + d)^m*e^(-7)/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2
+ 13068*m + 5040)
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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)**(5/2),x)
[Out]
Exception raised: HeuristicGCDFailed >> no luck
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4885 vs.
\(2 (324) = 648\).
time = 1.22, size = 4885, normalized size = 12.37 \begin {gather*} \text {Too large to display} \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)^(5/2),x, algorithm="giac")
[Out]
((x*e + d)^m*b^6*m^6*x^7*e^7*sgn(b*x + a) + (x*e + d)^m*b^6*d*m^6*x^6*e^6*sgn(b*x + a) + 6*(x*e + d)^m*a*b^5*m
^6*x^6*e^7*sgn(b*x + a) + 21*(x*e + d)^m*b^6*m^5*x^7*e^7*sgn(b*x + a) + 6*(x*e + d)^m*a*b^5*d*m^6*x^5*e^6*sgn(
b*x + a) + 15*(x*e + d)^m*b^6*d*m^5*x^6*e^6*sgn(b*x + a) - 6*(x*e + d)^m*b^6*d^2*m^5*x^5*e^5*sgn(b*x + a) + 15
*(x*e + d)^m*a^2*b^4*m^6*x^5*e^7*sgn(b*x + a) + 132*(x*e + d)^m*a*b^5*m^5*x^6*e^7*sgn(b*x + a) + 175*(x*e + d)
^m*b^6*m^4*x^7*e^7*sgn(b*x + a) + 15*(x*e + d)^m*a^2*b^4*d*m^6*x^4*e^6*sgn(b*x + a) + 102*(x*e + d)^m*a*b^5*d*
m^5*x^5*e^6*sgn(b*x + a) + 85*(x*e + d)^m*b^6*d*m^4*x^6*e^6*sgn(b*x + a) - 30*(x*e + d)^m*a*b^5*d^2*m^5*x^4*e^
5*sgn(b*x + a) - 60*(x*e + d)^m*b^6*d^2*m^4*x^5*e^5*sgn(b*x + a) + 30*(x*e + d)^m*b^6*d^3*m^4*x^4*e^4*sgn(b*x
+ a) + 20*(x*e + d)^m*a^3*b^3*m^6*x^4*e^7*sgn(b*x + a) + 345*(x*e + d)^m*a^2*b^4*m^5*x^5*e^7*sgn(b*x + a) + 11
40*(x*e + d)^m*a*b^5*m^4*x^6*e^7*sgn(b*x + a) + 735*(x*e + d)^m*b^6*m^3*x^7*e^7*sgn(b*x + a) + 20*(x*e + d)^m*
a^3*b^3*d*m^6*x^3*e^6*sgn(b*x + a) + 285*(x*e + d)^m*a^2*b^4*d*m^5*x^4*e^6*sgn(b*x + a) + 630*(x*e + d)^m*a*b^
5*d*m^4*x^5*e^6*sgn(b*x + a) + 225*(x*e + d)^m*b^6*d*m^3*x^6*e^6*sgn(b*x + a) - 60*(x*e + d)^m*a^2*b^4*d^2*m^5
*x^3*e^5*sgn(b*x + a) - 390*(x*e + d)^m*a*b^5*d^2*m^4*x^4*e^5*sgn(b*x + a) - 210*(x*e + d)^m*b^6*d^2*m^3*x^5*e
^5*sgn(b*x + a) + 120*(x*e + d)^m*a*b^5*d^3*m^4*x^3*e^4*sgn(b*x + a) + 180*(x*e + d)^m*b^6*d^3*m^3*x^4*e^4*sgn
(b*x + a) - 120*(x*e + d)^m*b^6*d^4*m^3*x^3*e^3*sgn(b*x + a) + 15*(x*e + d)^m*a^4*b^2*m^6*x^3*e^7*sgn(b*x + a)
+ 480*(x*e + d)^m*a^3*b^3*m^5*x^4*e^7*sgn(b*x + a) + 3105*(x*e + d)^m*a^2*b^4*m^4*x^5*e^7*sgn(b*x + a) + 4920
*(x*e + d)^m*a*b^5*m^3*x^6*e^7*sgn(b*x + a) + 1624*(x*e + d)^m*b^6*m^2*x^7*e^7*sgn(b*x + a) + 15*(x*e + d)^m*a
^4*b^2*d*m^6*x^2*e^6*sgn(b*x + a) + 420*(x*e + d)^m*a^3*b^3*d*m^5*x^3*e^6*sgn(b*x + a) + 1965*(x*e + d)^m*a^2*
b^4*d*m^4*x^4*e^6*sgn(b*x + a) + 1770*(x*e + d)^m*a*b^5*d*m^3*x^5*e^6*sgn(b*x + a) + 274*(x*e + d)^m*b^6*d*m^2
*x^6*e^6*sgn(b*x + a) - 60*(x*e + d)^m*a^3*b^3*d^2*m^5*x^2*e^5*sgn(b*x + a) - 960*(x*e + d)^m*a^2*b^4*d^2*m^4*
x^3*e^5*sgn(b*x + a) - 1590*(x*e + d)^m*a*b^5*d^2*m^3*x^4*e^5*sgn(b*x + a) - 300*(x*e + d)^m*b^6*d^2*m^2*x^5*e
^5*sgn(b*x + a) + 180*(x*e + d)^m*a^2*b^4*d^3*m^4*x^2*e^4*sgn(b*x + a) + 1200*(x*e + d)^m*a*b^5*d^3*m^3*x^3*e^
4*sgn(b*x + a) + 330*(x*e + d)^m*b^6*d^3*m^2*x^4*e^4*sgn(b*x + a) - 360*(x*e + d)^m*a*b^5*d^4*m^3*x^2*e^3*sgn(
b*x + a) - 360*(x*e + d)^m*b^6*d^4*m^2*x^3*e^3*sgn(b*x + a) + 360*(x*e + d)^m*b^6*d^5*m^2*x^2*e^2*sgn(b*x + a)
+ 6*(x*e + d)^m*a^5*b*m^6*x^2*e^7*sgn(b*x + a) + 375*(x*e + d)^m*a^4*b^2*m^5*x^3*e^7*sgn(b*x + a) + 4520*(x*e
+ d)^m*a^3*b^3*m^4*x^4*e^7*sgn(b*x + a) + 13875*(x*e + d)^m*a^2*b^4*m^3*x^5*e^7*sgn(b*x + a) + 11094*(x*e + d
)^m*a*b^5*m^2*x^6*e^7*sgn(b*x + a) + 1764*(x*e + d)^m*b^6*m*x^7*e^7*sgn(b*x + a) + 6*(x*e + d)^m*a^5*b*d*m^6*x
*e^6*sgn(b*x + a) + 345*(x*e + d)^m*a^4*b^2*d*m^5*x^2*e^6*sgn(b*x + a) + 3260*(x*e + d)^m*a^3*b^3*d*m^4*x^3*e^
6*sgn(b*x + a) + 6015*(x*e + d)^m*a^2*b^4*d*m^3*x^4*e^6*sgn(b*x + a) + 2244*(x*e + d)^m*a*b^5*d*m^2*x^5*e^6*sg
n(b*x + a) + 120*(x*e + d)^m*b^6*d*m*x^6*e^6*sgn(b*x + a) - 30*(x*e + d)^m*a^4*b^2*d^2*m^5*x*e^5*sgn(b*x + a)
- 1140*(x*e + d)^m*a^3*b^3*d^2*m^4*x^2*e^5*sgn(b*x + a) - 4980*(x*e + d)^m*a^2*b^4*d^2*m^3*x^3*e^5*sgn(b*x + a
) - 2490*(x*e + d)^m*a*b^5*d^2*m^2*x^4*e^5*sgn(b*x + a) - 144*(x*e + d)^m*b^6*d^2*m*x^5*e^5*sgn(b*x + a) + 120
*(x*e + d)^m*a^3*b^3*d^3*m^4*x*e^4*sgn(b*x + a) + 2520*(x*e + d)^m*a^2*b^4*d^3*m^3*x^2*e^4*sgn(b*x + a) + 2760
*(x*e + d)^m*a*b^5*d^3*m^2*x^3*e^4*sgn(b*x + a) + 180*(x*e + d)^m*b^6*d^3*m*x^4*e^4*sgn(b*x + a) - 360*(x*e +
d)^m*a^2*b^4*d^4*m^3*x*e^3*sgn(b*x + a) - 2880*(x*e + d)^m*a*b^5*d^4*m^2*x^2*e^3*sgn(b*x + a) - 240*(x*e + d)^
m*b^6*d^4*m*x^3*e^3*sgn(b*x + a) + 720*(x*e + d)^m*a*b^5*d^5*m^2*x*e^2*sgn(b*x + a) + 360*(x*e + d)^m*b^6*d^5*
m*x^2*e^2*sgn(b*x + a) - 720*(x*e + d)^m*b^6*d^6*m*x*e*sgn(b*x + a) + (x*e + d)^m*a^6*m^6*x*e^7*sgn(b*x + a) +
156*(x*e + d)^m*a^5*b*m^5*x^2*e^7*sgn(b*x + a) + 3705*(x*e + d)^m*a^4*b^2*m^4*x^3*e^7*sgn(b*x + a) + 21120*(x
*e + d)^m*a^3*b^3*m^3*x^4*e^7*sgn(b*x + a) + 32160*(x*e + d)^m*a^2*b^4*m^2*x^5*e^7*sgn(b*x + a) + 12228*(x*e +
d)^m*a*b^5*m*x^6*e^7*sgn(b*x + a) + 720*(x*e + d)^m*b^6*x^7*e^7*sgn(b*x + a) + (x*e + d)^m*a^6*d*m^6*e^6*sgn(
b*x + a) + 150*(x*e + d)^m*a^5*b*d*m^5*x*e^6*sgn(b*x + a) + 3015*(x*e + d)^m*a^4*b^2*d*m^4*x^2*e^6*sgn(b*x + a
) + 11340*(x*e + d)^m*a^3*b^3*d*m^3*x^3*e^6*sgn(b*x + a) + 8100*(x*e + d)^m*a^2*b^4*d*m^2*x^4*e^6*sgn(b*x + a)
+ 1008*(x*e + d)^m*a*b^5*d*m*x^5*e^6*sgn(b*x + a) - 6*(x*e + d)^m*a^5*b*d^2*m^5*e^5*sgn(b*x + a) - 660*(x*e +
d)^m*a^4*b^2*d^2*m^4*x*e^5*sgn(b*x + a) - 7500*(x*e + d)^m*a^3*b^3*d^2*m^3*x^2*e^5*sgn(b*x + a) - 9120*(x*e +
d)^m*a^2*b^4*d^2*m^2*x^3*e^5*sgn(b*x + a) - 1260*(x*e + d)^m*a*b^5*d^2*m*x^4*e^5*sgn(b*x + a) + 30*(x*e + d)^
m*a^4*b^2*d^3*m^4*e^4*sgn(b*x + a) + 2160*(x*e + d)^m*a^3*b^3*d^3*m^3*x*e^4*sgn(b*x + a) + 9900*(x*e + d)^m*a^
2*b^4*d^3*m^2*x^2*e^4*sgn(b*x + a) + 1680*(x*e ...
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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 )}^{5/2} \,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)^(5/2),x)
[Out]
int((a + b*x)*(d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
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