3.22.97 \(\int (d+e x) (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\) [2197]
Optimal. Leaf size=371 \[ \frac {5 (2 c d-b e)^5 (16 c e f+2 c d g-9 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^5 e}+\frac {5 (2 c d-b e)^3 (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6144 c^4 e}+\frac {(2 c d-b e) (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{384 c^3 e}+\frac {(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac {5 (2 c d-b e)^7 (16 c e f+2 c d g-9 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{32768 c^{11/2} e^2} \]
[Out]
5/6144*(-b*e+2*c*d)^3*(-9*b*e*g+2*c*d*g+16*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^4/e+1/384
*(-b*e+2*c*d)*(-9*b*e*g+2*c*d*g+16*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^3/e+1/112*(9*b*e*
g-16*c*(d*g+e*f)-14*c*e*g*x)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^2/e^2+5/32768*(-b*e+2*c*d)^7*(-9*b*e*g+2
*c*d*g+16*c*e*f)*arctan(1/2*e*(2*c*x+b)/c^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/c^(11/2)/e^2+5/16384*(
-b*e+2*c*d)^5*(-9*b*e*g+2*c*d*g+16*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c^5/e
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Rubi [A]
time = 0.38, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {793, 626, 635,
210} \begin {gather*} \frac {5 (2 c d-b e)^7 \text {ArcTan}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-9 b e g+2 c d g+16 c e f)}{32768 c^{11/2} e^2}+\frac {5 (b+2 c x) (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+2 c d g+16 c e f)}{16384 c^5 e}+\frac {5 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-9 b e g+2 c d g+16 c e f)}{6144 c^4 e}+\frac {(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+2 c d g+16 c e f)}{384 c^3 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-16 c (d g+e f)-14 c e g x)}{112 c^2 e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]
[Out]
(5*(2*c*d - b*e)^5*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(1638
4*c^5*e) + (5*(2*c*d - b*e)^3*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2))/(6144*c^4*e) + ((2*c*d - b*e)*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(5/2))/(384*c^3*e) + ((9*b*e*g - 16*c*(e*f + d*g) - 14*c*e*g*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(7/2))/(112*c^2*e^2) + (5*(2*c*d - b*e)^7*(16*c*e*f + 2*c*d*g - 9*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sq
rt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(32768*c^(11/2)*e^2)
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 626
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]
Rule 635
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 793
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p +
3))), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(
a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Rubi steps
\begin {align*} \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx &=\frac {(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac {((2 c d-b e) (16 c e f+2 c d g-9 b e g)) \int \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{32 c^2 e}\\ &=\frac {(2 c d-b e) (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{384 c^3 e}+\frac {(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac {\left (5 (2 c d-b e)^3 (16 c e f+2 c d g-9 b e g)\right ) \int \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{768 c^3 e}\\ &=\frac {5 (2 c d-b e)^3 (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6144 c^4 e}+\frac {(2 c d-b e) (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{384 c^3 e}+\frac {(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac {\left (5 (2 c d-b e)^5 (16 c e f+2 c d g-9 b e g)\right ) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{4096 c^4 e}\\ &=\frac {5 (2 c d-b e)^5 (16 c e f+2 c d g-9 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^5 e}+\frac {5 (2 c d-b e)^3 (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6144 c^4 e}+\frac {(2 c d-b e) (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{384 c^3 e}+\frac {(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac {\left (5 (2 c d-b e)^7 (16 c e f+2 c d g-9 b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{32768 c^5 e}\\ &=\frac {5 (2 c d-b e)^5 (16 c e f+2 c d g-9 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^5 e}+\frac {5 (2 c d-b e)^3 (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6144 c^4 e}+\frac {(2 c d-b e) (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{384 c^3 e}+\frac {(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac {\left (5 (2 c d-b e)^7 (16 c e f+2 c d g-9 b e g)\right ) \text {Subst}\left (\int \frac {1}{-4 c e^2-x^2} \, dx,x,\frac {-b e^2-2 c e^2 x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{16384 c^5 e}\\ &=\frac {5 (2 c d-b e)^5 (16 c e f+2 c d g-9 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^5 e}+\frac {5 (2 c d-b e)^3 (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6144 c^4 e}+\frac {(2 c d-b e) (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{384 c^3 e}+\frac {(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac {5 (2 c d-b e)^7 (16 c e f+2 c d g-9 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{32768 c^{11/2} e^2}\\ \end {align*}
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Mathematica [A]
time = 2.85, size = 733, normalized size = 1.98 \begin {gather*} \frac {(2 c d-b e)^7 ((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {\sqrt {c} \left (945 b^7 e^7 g-210 b^6 c e^6 (8 e f+58 d g+3 e g x)+28 b^5 c^2 e^5 \left (2363 d^2 g+38 d e (20 f+7 g x)+2 e^2 x (20 f+9 g x)\right )-128 c^7 \left (384 d^7 g-64 d e^6 x^5 (7 f+6 g x)-48 e^7 x^6 (8 f+7 g x)+3 d^6 e (128 f+35 g x)-24 d^5 e^2 x (77 f+48 g x)+16 d^3 e^4 x^3 (91 f+72 g x)+8 d^2 e^5 x^4 (144 f+119 g x)-2 d^4 e^3 x^2 (576 f+413 g x)\right )+64 b c^6 e \left (2967 d^6 g-8 d^2 e^4 x^3 (30 f+19 g x)+16 e^6 x^5 (116 f+99 g x)+6 d^5 e (692 f+181 g x)+16 d e^5 x^4 (284 f+235 g x)-24 d^3 e^3 x^2 (374 f+269 g x)-6 d^4 e^2 x (1156 f+739 g x)\right )+16 b^3 c^4 e^3 \left (20779 d^4 g+24 e^4 x^3 (2 f+g x)+8 d e^3 x^2 (74 f+33 g x)+20 d^2 e^2 x (192 f+73 g x)+4 d^3 e (5024 f+1431 g x)\right )+32 b^2 c^5 e^2 \left (-10434 d^5 g+1224 d^3 e^2 x (4 f+3 g x)+8 e^5 x^4 (296 f+243 g x)+16 d e^4 x^3 (583 f+455 g x)-3 d^4 e (4616 f+1227 g x)+4 d^2 e^3 x^2 (3276 f+2375 g x)\right )-8 b^4 c^3 e^4 \left (24372 d^3 g+2 e^3 x^2 (56 f+27 g x)+8 d e^2 x (203 f+85 g x)+d^2 e (14112 f+4523 g x)\right )\right )}{(2 c d-b e)^7 (d+e x)^2 (-c d+b e+c e x)^2}+\frac {105 (9 b e g-2 c (8 e f+d g)) \tan ^{-1}\left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{(d+e x)^{5/2} (-b e+c (d-e x))^{5/2}}\right )}{344064 c^{11/2} e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]
[Out]
((2*c*d - b*e)^7*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((Sqrt[c]*(945*b^7*e^7*g - 210*b^6*c*e^6*(8*e*f + 58
*d*g + 3*e*g*x) + 28*b^5*c^2*e^5*(2363*d^2*g + 38*d*e*(20*f + 7*g*x) + 2*e^2*x*(20*f + 9*g*x)) - 128*c^7*(384*
d^7*g - 64*d*e^6*x^5*(7*f + 6*g*x) - 48*e^7*x^6*(8*f + 7*g*x) + 3*d^6*e*(128*f + 35*g*x) - 24*d^5*e^2*x*(77*f
+ 48*g*x) + 16*d^3*e^4*x^3*(91*f + 72*g*x) + 8*d^2*e^5*x^4*(144*f + 119*g*x) - 2*d^4*e^3*x^2*(576*f + 413*g*x)
) + 64*b*c^6*e*(2967*d^6*g - 8*d^2*e^4*x^3*(30*f + 19*g*x) + 16*e^6*x^5*(116*f + 99*g*x) + 6*d^5*e*(692*f + 18
1*g*x) + 16*d*e^5*x^4*(284*f + 235*g*x) - 24*d^3*e^3*x^2*(374*f + 269*g*x) - 6*d^4*e^2*x*(1156*f + 739*g*x)) +
16*b^3*c^4*e^3*(20779*d^4*g + 24*e^4*x^3*(2*f + g*x) + 8*d*e^3*x^2*(74*f + 33*g*x) + 20*d^2*e^2*x*(192*f + 73
*g*x) + 4*d^3*e*(5024*f + 1431*g*x)) + 32*b^2*c^5*e^2*(-10434*d^5*g + 1224*d^3*e^2*x*(4*f + 3*g*x) + 8*e^5*x^4
*(296*f + 243*g*x) + 16*d*e^4*x^3*(583*f + 455*g*x) - 3*d^4*e*(4616*f + 1227*g*x) + 4*d^2*e^3*x^2*(3276*f + 23
75*g*x)) - 8*b^4*c^3*e^4*(24372*d^3*g + 2*e^3*x^2*(56*f + 27*g*x) + 8*d*e^2*x*(203*f + 85*g*x) + d^2*e*(14112*
f + 4523*g*x))))/((2*c*d - b*e)^7*(d + e*x)^2*(-(c*d) + b*e + c*e*x)^2) + (105*(9*b*e*g - 2*c*(8*e*f + d*g))*A
rcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])])/((d + e*x)^(5/2)*(-(b*e) + c*(d - e*x))^(5/2))))/(3440
64*c^(11/2)*e^2)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1411\) vs.
\(2(345)=690\).
time = 0.04, size = 1412, normalized size = 3.81
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\(1412\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method=_RETURNVERBOSE)
[Out]
g*e*(-1/8*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e^2-9/16*b/c*(-1/7*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)
/c/e^2-1/2*b/c*(-1/12*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-5/24*(-4*c*e^2*(-b*d*e+c
*d^2)-b^2*e^4)/c/e^2*(-1/8*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-3/16*(-4*c*e^2*(-b*
d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*
(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2))))))+1/8*(-b*d*e+c*d^2)/c/e^2*(-1/12*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-5/2
4*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/8*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2
)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2))))))+(d*g+e*f)*(-1/7*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e^2-1/2*b/c*(-1/12*(-
2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-5/24*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1
/8*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^
2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/
c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))))+f*d*(-1/12*(-
2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-5/24*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1
/8*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^
2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/
c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)))))
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(2*c*d-%e*b>0)', see `assume?`
for more det
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1126 vs.
\(2 (351) = 702\).
time = 2.91, size = 2260, normalized size = 6.09 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")
[Out]
[-1/1376256*(105*(256*c^8*d^8*g - (16*b^7*c*f - 9*b^8*g)*e^8 + 32*(7*b^6*c^2*d*f - 4*b^7*c*d*g)*e^7 - 112*(12*
b^5*c^3*d^2*f - 7*b^6*c^2*d^2*g)*e^6 + 896*(5*b^4*c^4*d^3*f - 3*b^5*c^3*d^3*g)*e^5 - 1120*(8*b^3*c^5*d^4*f - 5
*b^4*c^4*d^4*g)*e^4 + 3584*(3*b^2*c^6*d^5*f - 2*b^3*c^5*d^5*g)*e^3 - 1792*(4*b*c^7*d^6*f - 3*b^2*c^6*d^6*g)*e^
2 + 2048*(c^8*d^7*f - b*c^7*d^7*g)*e)*sqrt(-c)*log(-4*c^2*d^2 + 4*b*c*d*e - 4*sqrt(c*d^2 - b*d*e - (c*x^2 + b*
x)*e^2)*(2*c*x + b)*sqrt(-c)*e + (8*c^2*x^2 + 8*b*c*x + b^2)*e^2) + 4*(49152*c^8*d^7*g - (43008*c^8*g*x^7 - 16
80*b^6*c^2*f + 945*b^7*c*g + 3072*(16*c^8*f + 33*b*c^7*g)*x^6 + 256*(464*b*c^7*f + 243*b^2*c^6*g)*x^5 + 128*(5
92*b^2*c^6*f + 3*b^3*c^5*g)*x^4 + 48*(16*b^3*c^5*f - 9*b^4*c^4*g)*x^3 - 56*(16*b^4*c^4*f - 9*b^5*c^3*g)*x^2 +
70*(16*b^5*c^3*f - 9*b^6*c^2*g)*x)*e^7 - 4*(12288*c^8*d*g*x^6 + 5320*b^5*c^3*d*f - 3045*b^6*c^2*d*g + 256*(56*
c^8*d*f + 235*b*c^7*d*g)*x^5 + 128*(568*b*c^7*d*f + 455*b^2*c^6*d*g)*x^4 + 352*(212*b^2*c^6*d*f + 3*b^3*c^5*d*
g)*x^3 + 16*(148*b^3*c^5*d*f - 85*b^4*c^4*d*g)*x^2 - 14*(232*b^4*c^4*d*f - 133*b^5*c^3*d*g)*x)*e^6 + 4*(30464*
c^8*d^2*g*x^5 + 28224*b^4*c^4*d^2*f - 16541*b^5*c^3*d^2*g + 128*(288*c^8*d^2*f + 19*b*c^7*d^2*g)*x^4 + 160*(24
*b*c^7*d^2*f - 475*b^2*c^6*d^2*g)*x^3 - 16*(6552*b^2*c^6*d^2*f + 365*b^3*c^5*d^2*g)*x^2 - 2*(7680*b^3*c^5*d^2*
f - 4523*b^4*c^4*d^2*g)*x)*e^5 + 32*(4608*c^8*d^3*g*x^4 - 10048*b^3*c^5*d^3*f + 6093*b^4*c^4*d^3*g + 16*(364*c
^8*d^3*f + 807*b*c^7*d^3*g)*x^3 + 408*(44*b*c^7*d^3*f - 9*b^2*c^6*d^3*g)*x^2 - 18*(272*b^2*c^6*d^3*f + 159*b^3
*c^5*d^3*g)*x)*e^4 - 16*(6608*c^8*d^4*g*x^3 - 27696*b^2*c^6*d^4*f + 20779*b^3*c^5*d^4*g + 24*(384*c^8*d^4*f -
739*b*c^7*d^4*g)*x^2 - 6*(4624*b*c^7*d^4*f + 1227*b^2*c^6*d^4*g)*x)*e^3 - 192*(768*c^8*d^5*g*x^2 + 1384*b*c^7*
d^5*f - 1739*b^2*c^6*d^5*g + 2*(616*c^8*d^5*f + 181*b*c^7*d^5*g)*x)*e^2 + 192*(70*c^8*d^6*g*x + 256*c^8*d^6*f
- 989*b*c^7*d^6*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2))*e^(-2)/c^6, -1/688128*(105*(256*c^8*d^8*g - (16
*b^7*c*f - 9*b^8*g)*e^8 + 32*(7*b^6*c^2*d*f - 4*b^7*c*d*g)*e^7 - 112*(12*b^5*c^3*d^2*f - 7*b^6*c^2*d^2*g)*e^6
+ 896*(5*b^4*c^4*d^3*f - 3*b^5*c^3*d^3*g)*e^5 - 1120*(8*b^3*c^5*d^4*f - 5*b^4*c^4*d^4*g)*e^4 + 3584*(3*b^2*c^6
*d^5*f - 2*b^3*c^5*d^5*g)*e^3 - 1792*(4*b*c^7*d^6*f - 3*b^2*c^6*d^6*g)*e^2 + 2048*(c^8*d^7*f - b*c^7*d^7*g)*e)
*sqrt(c)*arctan(-1/2*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*(2*c*x + b)*sqrt(c)*e/(c^2*d^2 - b*c*d*e - (c^2*x
^2 + b*c*x)*e^2)) + 2*(49152*c^8*d^7*g - (43008*c^8*g*x^7 - 1680*b^6*c^2*f + 945*b^7*c*g + 3072*(16*c^8*f + 33
*b*c^7*g)*x^6 + 256*(464*b*c^7*f + 243*b^2*c^6*g)*x^5 + 128*(592*b^2*c^6*f + 3*b^3*c^5*g)*x^4 + 48*(16*b^3*c^5
*f - 9*b^4*c^4*g)*x^3 - 56*(16*b^4*c^4*f - 9*b^5*c^3*g)*x^2 + 70*(16*b^5*c^3*f - 9*b^6*c^2*g)*x)*e^7 - 4*(1228
8*c^8*d*g*x^6 + 5320*b^5*c^3*d*f - 3045*b^6*c^2*d*g + 256*(56*c^8*d*f + 235*b*c^7*d*g)*x^5 + 128*(568*b*c^7*d*
f + 455*b^2*c^6*d*g)*x^4 + 352*(212*b^2*c^6*d*f + 3*b^3*c^5*d*g)*x^3 + 16*(148*b^3*c^5*d*f - 85*b^4*c^4*d*g)*x
^2 - 14*(232*b^4*c^4*d*f - 133*b^5*c^3*d*g)*x)*e^6 + 4*(30464*c^8*d^2*g*x^5 + 28224*b^4*c^4*d^2*f - 16541*b^5*
c^3*d^2*g + 128*(288*c^8*d^2*f + 19*b*c^7*d^2*g)*x^4 + 160*(24*b*c^7*d^2*f - 475*b^2*c^6*d^2*g)*x^3 - 16*(6552
*b^2*c^6*d^2*f + 365*b^3*c^5*d^2*g)*x^2 - 2*(7680*b^3*c^5*d^2*f - 4523*b^4*c^4*d^2*g)*x)*e^5 + 32*(4608*c^8*d^
3*g*x^4 - 10048*b^3*c^5*d^3*f + 6093*b^4*c^4*d^3*g + 16*(364*c^8*d^3*f + 807*b*c^7*d^3*g)*x^3 + 408*(44*b*c^7*
d^3*f - 9*b^2*c^6*d^3*g)*x^2 - 18*(272*b^2*c^6*d^3*f + 159*b^3*c^5*d^3*g)*x)*e^4 - 16*(6608*c^8*d^4*g*x^3 - 27
696*b^2*c^6*d^4*f + 20779*b^3*c^5*d^4*g + 24*(384*c^8*d^4*f - 739*b*c^7*d^4*g)*x^2 - 6*(4624*b*c^7*d^4*f + 122
7*b^2*c^6*d^4*g)*x)*e^3 - 192*(768*c^8*d^5*g*x^2 + 1384*b*c^7*d^5*f - 1739*b^2*c^6*d^5*g + 2*(616*c^8*d^5*f +
181*b*c^7*d^5*g)*x)*e^2 + 192*(70*c^8*d^6*g*x + 256*c^8*d^6*f - 989*b*c^7*d^6*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^
2 + b*x)*e^2))*e^(-2)/c^6]
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (d + e x\right ) \left (f + g x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)
[Out]
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(d + e*x)*(f + g*x), x)
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1142 vs.
\(2 (351) = 702\).
time = 1.50, size = 1142, normalized size = 3.08 \begin {gather*} \frac {1}{344064} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, c^{2} g x e^{5} + \frac {{\left (16 \, c^{9} d g e^{16} + 16 \, c^{9} f e^{17} + 33 \, b c^{8} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x - \frac {{\left (476 \, c^{9} d^{2} g e^{15} - 224 \, c^{9} d f e^{16} - 940 \, b c^{8} d g e^{16} - 464 \, b c^{8} f e^{17} - 243 \, b^{2} c^{7} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x - \frac {{\left (1152 \, c^{9} d^{3} g e^{14} + 1152 \, c^{9} d^{2} f e^{15} + 76 \, b c^{8} d^{2} g e^{15} - 2272 \, b c^{8} d f e^{16} - 1820 \, b^{2} c^{7} d g e^{16} - 592 \, b^{2} c^{7} f e^{17} - 3 \, b^{3} c^{6} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x + \frac {{\left (6608 \, c^{9} d^{4} g e^{13} - 11648 \, c^{9} d^{3} f e^{14} - 25824 \, b c^{8} d^{3} g e^{14} - 960 \, b c^{8} d^{2} f e^{15} + 19000 \, b^{2} c^{7} d^{2} g e^{15} + 18656 \, b^{2} c^{7} d f e^{16} + 264 \, b^{3} c^{6} d g e^{16} + 48 \, b^{3} c^{6} f e^{17} - 27 \, b^{4} c^{5} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x + \frac {{\left (18432 \, c^{9} d^{5} g e^{12} + 18432 \, c^{9} d^{4} f e^{13} - 35472 \, b c^{8} d^{4} g e^{13} - 71808 \, b c^{8} d^{3} f e^{14} + 14688 \, b^{2} c^{7} d^{3} g e^{14} + 52416 \, b^{2} c^{7} d^{2} f e^{15} + 2920 \, b^{3} c^{6} d^{2} g e^{15} + 1184 \, b^{3} c^{6} d f e^{16} - 680 \, b^{4} c^{5} d g e^{16} - 112 \, b^{4} c^{5} f e^{17} + 63 \, b^{5} c^{4} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x - \frac {{\left (6720 \, c^{9} d^{6} g e^{11} - 118272 \, c^{9} d^{5} f e^{12} - 34752 \, b c^{8} d^{5} g e^{12} + 221952 \, b c^{8} d^{4} f e^{13} + 58896 \, b^{2} c^{7} d^{4} g e^{13} - 78336 \, b^{2} c^{7} d^{3} f e^{14} - 45792 \, b^{3} c^{6} d^{3} g e^{14} - 30720 \, b^{3} c^{6} d^{2} f e^{15} + 18092 \, b^{4} c^{5} d^{2} g e^{15} + 6496 \, b^{4} c^{5} d f e^{16} - 3724 \, b^{5} c^{4} d g e^{16} - 560 \, b^{5} c^{4} f e^{17} + 315 \, b^{6} c^{3} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x - \frac {{\left (49152 \, c^{9} d^{7} g e^{10} + 49152 \, c^{9} d^{6} f e^{11} - 189888 \, b c^{8} d^{6} g e^{11} - 265728 \, b c^{8} d^{5} f e^{12} + 333888 \, b^{2} c^{7} d^{5} g e^{12} + 443136 \, b^{2} c^{7} d^{4} f e^{13} - 332464 \, b^{3} c^{6} d^{4} g e^{13} - 321536 \, b^{3} c^{6} d^{3} f e^{14} + 194976 \, b^{4} c^{5} d^{3} g e^{14} + 112896 \, b^{4} c^{5} d^{2} f e^{15} - 66164 \, b^{5} c^{4} d^{2} g e^{15} - 21280 \, b^{5} c^{4} d f e^{16} + 12180 \, b^{6} c^{3} d g e^{16} + 1680 \, b^{6} c^{3} f e^{17} - 945 \, b^{7} c^{2} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} + \frac {5 \, {\left (256 \, c^{8} d^{8} g + 2048 \, c^{8} d^{7} f e - 2048 \, b c^{7} d^{7} g e - 7168 \, b c^{7} d^{6} f e^{2} + 5376 \, b^{2} c^{6} d^{6} g e^{2} + 10752 \, b^{2} c^{6} d^{5} f e^{3} - 7168 \, b^{3} c^{5} d^{5} g e^{3} - 8960 \, b^{3} c^{5} d^{4} f e^{4} + 5600 \, b^{4} c^{4} d^{4} g e^{4} + 4480 \, b^{4} c^{4} d^{3} f e^{5} - 2688 \, b^{5} c^{3} d^{3} g e^{5} - 1344 \, b^{5} c^{3} d^{2} f e^{6} + 784 \, b^{6} c^{2} d^{2} g e^{6} + 224 \, b^{6} c^{2} d f e^{7} - 128 \, b^{7} c d g e^{7} - 16 \, b^{7} c f e^{8} + 9 \, b^{8} g e^{8}\right )} \sqrt {-c} e^{\left (-2\right )} \log \left ({\left | -b \sqrt {-c} e - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c \right |}\right )}{32768 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")
[Out]
1/344064*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(2*(12*(14*c^2*g*x*e^5 + (16*c^9*d*g*e^16 + 16
*c^9*f*e^17 + 33*b*c^8*g*e^17)*e^(-12)/c^7)*x - (476*c^9*d^2*g*e^15 - 224*c^9*d*f*e^16 - 940*b*c^8*d*g*e^16 -
464*b*c^8*f*e^17 - 243*b^2*c^7*g*e^17)*e^(-12)/c^7)*x - (1152*c^9*d^3*g*e^14 + 1152*c^9*d^2*f*e^15 + 76*b*c^8*
d^2*g*e^15 - 2272*b*c^8*d*f*e^16 - 1820*b^2*c^7*d*g*e^16 - 592*b^2*c^7*f*e^17 - 3*b^3*c^6*g*e^17)*e^(-12)/c^7)
*x + (6608*c^9*d^4*g*e^13 - 11648*c^9*d^3*f*e^14 - 25824*b*c^8*d^3*g*e^14 - 960*b*c^8*d^2*f*e^15 + 19000*b^2*c
^7*d^2*g*e^15 + 18656*b^2*c^7*d*f*e^16 + 264*b^3*c^6*d*g*e^16 + 48*b^3*c^6*f*e^17 - 27*b^4*c^5*g*e^17)*e^(-12)
/c^7)*x + (18432*c^9*d^5*g*e^12 + 18432*c^9*d^4*f*e^13 - 35472*b*c^8*d^4*g*e^13 - 71808*b*c^8*d^3*f*e^14 + 146
88*b^2*c^7*d^3*g*e^14 + 52416*b^2*c^7*d^2*f*e^15 + 2920*b^3*c^6*d^2*g*e^15 + 1184*b^3*c^6*d*f*e^16 - 680*b^4*c
^5*d*g*e^16 - 112*b^4*c^5*f*e^17 + 63*b^5*c^4*g*e^17)*e^(-12)/c^7)*x - (6720*c^9*d^6*g*e^11 - 118272*c^9*d^5*f
*e^12 - 34752*b*c^8*d^5*g*e^12 + 221952*b*c^8*d^4*f*e^13 + 58896*b^2*c^7*d^4*g*e^13 - 78336*b^2*c^7*d^3*f*e^14
- 45792*b^3*c^6*d^3*g*e^14 - 30720*b^3*c^6*d^2*f*e^15 + 18092*b^4*c^5*d^2*g*e^15 + 6496*b^4*c^5*d*f*e^16 - 37
24*b^5*c^4*d*g*e^16 - 560*b^5*c^4*f*e^17 + 315*b^6*c^3*g*e^17)*e^(-12)/c^7)*x - (49152*c^9*d^7*g*e^10 + 49152*
c^9*d^6*f*e^11 - 189888*b*c^8*d^6*g*e^11 - 265728*b*c^8*d^5*f*e^12 + 333888*b^2*c^7*d^5*g*e^12 + 443136*b^2*c^
7*d^4*f*e^13 - 332464*b^3*c^6*d^4*g*e^13 - 321536*b^3*c^6*d^3*f*e^14 + 194976*b^4*c^5*d^3*g*e^14 + 112896*b^4*
c^5*d^2*f*e^15 - 66164*b^5*c^4*d^2*g*e^15 - 21280*b^5*c^4*d*f*e^16 + 12180*b^6*c^3*d*g*e^16 + 1680*b^6*c^3*f*e
^17 - 945*b^7*c^2*g*e^17)*e^(-12)/c^7) + 5/32768*(256*c^8*d^8*g + 2048*c^8*d^7*f*e - 2048*b*c^7*d^7*g*e - 7168
*b*c^7*d^6*f*e^2 + 5376*b^2*c^6*d^6*g*e^2 + 10752*b^2*c^6*d^5*f*e^3 - 7168*b^3*c^5*d^5*g*e^3 - 8960*b^3*c^5*d^
4*f*e^4 + 5600*b^4*c^4*d^4*g*e^4 + 4480*b^4*c^4*d^3*f*e^5 - 2688*b^5*c^3*d^3*g*e^5 - 1344*b^5*c^3*d^2*f*e^6 +
784*b^6*c^2*d^2*g*e^6 + 224*b^6*c^2*d*f*e^7 - 128*b^7*c*d*g*e^7 - 16*b^7*c*f*e^8 + 9*b^8*g*e^8)*sqrt(-c)*e^(-2
)*log(abs(-b*sqrt(-c)*e - 2*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*c))/c^6
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (f+g\,x\right )\,\left (d+e\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f + g*x)*(d + e*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2),x)
[Out]
int((f + g*x)*(d + e*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2), x)
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