3.20.55 \(\int (d+e x) (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\)
Optimal. Leaf size=297 \[ \frac {(2 c d-b e)^5 (-7 b e g+2 c d g+12 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{9/2} e^2}+\frac {(b+2 c x) (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+2 c d g+12 c e f)}{512 c^4 e}+\frac {(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+2 c d g+12 c e f)}{192 c^3 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-12 c (d g+e f)-10 c e g x)}{60 c^2 e^2} \]
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Rubi [A] time = 0.44, antiderivative size = 297, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used =
{779, 612, 621, 204} \begin {gather*} \frac {(b+2 c x) (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+2 c d g+12 c e f)}{512 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 )^{3/2} (-7 b e g+2 c d g+12 c e f)}{192 c^3 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-12 c (d g+e f)-10 c e g x)}{60 c^2 e^2}+\frac {(2 c d-b e)^5 (-7 b e g+2 c d g+12 c e f) \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 )}{1024 c^{9/2} 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)^(3/2),x]
[Out]
((2*c*d - b*e)^3*(12*c*e*f + 2*c*d*g - 7*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(512*c^
4*e) + ((2*c*d - b*e)*(12*c*e*f + 2*c*d*g - 7*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/
(192*c^3*e) + ((7*b*e*g - 12*c*(e*f + d*g) - 10*c*e*g*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(60*c^2*
e^2) + ((2*c*d - b*e)^5*(12*c*e*f + 2*c*d*g - 7*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2])])/(1024*c^(9/2)*e^2)
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 612
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 621
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 779
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 )^{3/2} \, dx &=\frac {(7 b e g-12 c (e f+d g)-10 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c^2 e^2}+\frac {((2 c d-b e) (12 c e f+2 c d g-7 b e g)) \int \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{24 c^2 e}\\ &=\frac {(2 c d-b e) (12 c e f+2 c d g-7 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}}{192 c^3 e}+\frac {(7 b e g-12 c (e f+d g)-10 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c^2 e^2}+\frac {\left ((2 c d-b e)^3 (12 c e f+2 c d g-7 b e g)\right ) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{128 c^3 e}\\ &=\frac {(2 c d-b e)^3 (12 c e f+2 c d g-7 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^4 e}+\frac {(2 c d-b e) (12 c e f+2 c d g-7 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}}{192 c^3 e}+\frac {(7 b e g-12 c (e f+d g)-10 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c^2 e^2}+\frac {\left ((2 c d-b e)^5 (12 c e f+2 c d g-7 b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{1024 c^4 e}\\ &=\frac {(2 c d-b e)^3 (12 c e f+2 c d g-7 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^4 e}+\frac {(2 c d-b e) (12 c e f+2 c d g-7 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}}{192 c^3 e}+\frac {(7 b e g-12 c (e f+d g)-10 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c^2 e^2}+\frac {\left ((2 c d-b e)^5 (12 c e f+2 c d g-7 b e g)\right ) \operatorname {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 )}{512 c^4 e}\\ &=\frac {(2 c d-b e)^3 (12 c e f+2 c d g-7 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^4 e}+\frac {(2 c d-b e) (12 c e f+2 c d g-7 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}}{192 c^3 e}+\frac {(7 b e g-12 c (e f+d g)-10 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c^2 e^2}+\frac {(2 c d-b e)^5 (12 c e f+2 c d g-7 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 )}{1024 c^{9/2} e^2}\\ \end {align*}
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Mathematica [A] time = 5.72, size = 504, normalized size = 1.70 \begin {gather*} \frac {(d+e x)^3 \sqrt {(d+e x) (c (d-e x)-b e)} \left (\frac {7 \sqrt {e (2 c d-b e)} \left (\frac {b e-c d+c e x}{b e-2 c d}\right )^{3/2} \left (c e (d g+6 e f)-\frac {7}{2} b e^2 g\right ) \left (16 c^4 e^{12} (d+e x)^4 \sqrt {e (2 c d-b e)} (b e-2 c d) \sqrt {\frac {b e-c d+c e x}{b e-2 c d}} (11 b e-14 c d+8 c e x)-e^6 (2 c d-b e)^3 \left (8 c^3 e^6 (d+e x)^3 \sqrt {e (2 c d-b e)} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}-10 c^2 e^6 (d+e x)^2 \sqrt {e (2 c d-b e)} (b e-2 c d) \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}+15 \sqrt {c} e^{13/2} \sqrt {d+e x} (b e-2 c d)^3 \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )+15 c e^6 (d+e x) \sqrt {e (2 c d-b e)} (b e-2 c d)^2 \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}\right )\right )}{640 c^4 e^{12} (d+e x)^4 (b e-c d+c e x)^2}-7 e^2 g (b e-c d+c e x)^2\right )}{42 c e^4} \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)^(3/2),x]
[Out]
((d + e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-7*e^2*g*(-(c*d) + b*e + c*e*x)^2 + (7*Sqrt[e*(2*c*d - b*
e)]*((-7*b*e^2*g)/2 + c*e*(6*e*f + d*g))*((-(c*d) + b*e + c*e*x)/(-2*c*d + b*e))^(3/2)*(16*c^4*e^12*Sqrt[e*(2*
c*d - b*e)]*(-2*c*d + b*e)*(d + e*x)^4*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]*(-14*c*d + 11*b*e + 8*c*e*x
) - e^6*(2*c*d - b*e)^3*(15*c*e^6*Sqrt[e*(2*c*d - b*e)]*(-2*c*d + b*e)^2*(d + e*x)*Sqrt[(-(c*d) + b*e + c*e*x)
/(-2*c*d + b*e)] - 10*c^2*e^6*Sqrt[e*(2*c*d - b*e)]*(-2*c*d + b*e)*(d + e*x)^2*Sqrt[(-(c*d) + b*e + c*e*x)/(-2
*c*d + b*e)] + 8*c^3*e^6*Sqrt[e*(2*c*d - b*e)]*(d + e*x)^3*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + 15*Sq
rt[c]*e^(13/2)*(-2*c*d + b*e)^3*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])))
/(640*c^4*e^12*(d + e*x)^4*(-(c*d) + b*e + c*e*x)^2)))/(42*c*e^4)
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IntegrateAlgebraic [F] time = 180.13, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(d + e*x)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]
[Out]
$Aborted
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fricas [B] time = 1.58, size = 1473, normalized size = 4.96
result too large to display
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)^(3/2),x, algorithm="fricas")
[Out]
[-1/30720*(15*(12*(32*c^6*d^5*e - 80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c^2*d*e^
5 - b^5*c*e^6)*f + (64*c^6*d^6 - 384*b*c^5*d^5*e + 720*b^2*c^4*d^4*e^2 - 640*b^3*c^3*d^3*e^3 + 300*b^4*c^2*d^2
*e^4 - 72*b^5*c*d*e^5 + 7*b^6*e^6)*g)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e
^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) + 4*(1280*c^6*e^5*g*x^5 + 128*(12*
c^6*e^5*f + (12*c^6*d*e^4 + 13*b*c^5*e^5)*g)*x^4 + 16*(12*(10*c^6*d*e^4 + 11*b*c^5*e^5)*f - (140*c^6*d^2*e^3 -
272*b*c^5*d*e^4 - 3*b^2*c^4*e^5)*g)*x^3 - 8*(12*(32*c^6*d^2*e^3 - 62*b*c^5*d*e^4 - b^2*c^4*e^5)*f + (384*c^6*
d^3*e^2 - 348*b*c^5*d^2*e^3 - 48*b^2*c^4*d*e^4 + 7*b^3*c^3*e^5)*g)*x^2 + 12*(128*c^6*d^4*e - 456*b*c^5*d^3*e^2
+ 428*b^2*c^4*d^2*e^3 - 130*b^3*c^3*d*e^4 + 15*b^4*c^2*e^5)*f + (1536*c^6*d^5 - 4368*b*c^5*d^4*e + 5328*b^2*c
^4*d^3*e^2 - 3256*b^3*c^3*d^2*e^3 + 940*b^4*c^2*d*e^4 - 105*b^5*c*e^5)*g - 2*(12*(200*c^6*d^3*e^2 - 172*b*c^5*
d^2*e^3 - 38*b^2*c^4*d*e^4 + 5*b^3*c^3*e^5)*f - (240*c^6*d^4*e - 816*b*c^5*d^3*e^2 + 792*b^2*c^4*d^2*e^3 - 276
*b^3*c^3*d*e^4 + 35*b^4*c^2*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^5*e^2), -1/15360*(15*(12
*(32*c^6*d^5*e - 80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c^2*d*e^5 - b^5*c*e^6)*f
+ (64*c^6*d^6 - 384*b*c^5*d^5*e + 720*b^2*c^4*d^4*e^2 - 640*b^3*c^3*d^3*e^3 + 300*b^4*c^2*d^2*e^4 - 72*b^5*c*d
*e^5 + 7*b^6*e^6)*g)*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^
2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) + 2*(1280*c^6*e^5*g*x^5 + 128*(12*c^6*e^5*f + (12*c^6*d*e^4 + 13*b
*c^5*e^5)*g)*x^4 + 16*(12*(10*c^6*d*e^4 + 11*b*c^5*e^5)*f - (140*c^6*d^2*e^3 - 272*b*c^5*d*e^4 - 3*b^2*c^4*e^5
)*g)*x^3 - 8*(12*(32*c^6*d^2*e^3 - 62*b*c^5*d*e^4 - b^2*c^4*e^5)*f + (384*c^6*d^3*e^2 - 348*b*c^5*d^2*e^3 - 48
*b^2*c^4*d*e^4 + 7*b^3*c^3*e^5)*g)*x^2 + 12*(128*c^6*d^4*e - 456*b*c^5*d^3*e^2 + 428*b^2*c^4*d^2*e^3 - 130*b^3
*c^3*d*e^4 + 15*b^4*c^2*e^5)*f + (1536*c^6*d^5 - 4368*b*c^5*d^4*e + 5328*b^2*c^4*d^3*e^2 - 3256*b^3*c^3*d^2*e^
3 + 940*b^4*c^2*d*e^4 - 105*b^5*c*e^5)*g - 2*(12*(200*c^6*d^3*e^2 - 172*b*c^5*d^2*e^3 - 38*b^2*c^4*d*e^4 + 5*b
^3*c^3*e^5)*f - (240*c^6*d^4*e - 816*b*c^5*d^3*e^2 + 792*b^2*c^4*d^2*e^3 - 276*b^3*c^3*d*e^4 + 35*b^4*c^2*e^5)
*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^5*e^2)]
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giac [B] time = 0.45, size = 708, normalized size = 2.38 \begin {gather*} -\frac {1}{7680} \, \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 (10 \, c g x e^{3} + \frac {{\left (12 \, c^{6} d g e^{10} + 12 \, c^{6} f e^{11} + 13 \, b c^{5} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x - \frac {{\left (140 \, c^{6} d^{2} g e^{9} - 120 \, c^{6} d f e^{10} - 272 \, b c^{5} d g e^{10} - 132 \, b c^{5} f e^{11} - 3 \, b^{2} c^{4} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x - \frac {{\left (384 \, c^{6} d^{3} g e^{8} + 384 \, c^{6} d^{2} f e^{9} - 348 \, b c^{5} d^{2} g e^{9} - 744 \, b c^{5} d f e^{10} - 48 \, b^{2} c^{4} d g e^{10} - 12 \, b^{2} c^{4} f e^{11} + 7 \, b^{3} c^{3} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x + \frac {{\left (240 \, c^{6} d^{4} g e^{7} - 2400 \, c^{6} d^{3} f e^{8} - 816 \, b c^{5} d^{3} g e^{8} + 2064 \, b c^{5} d^{2} f e^{9} + 792 \, b^{2} c^{4} d^{2} g e^{9} + 456 \, b^{2} c^{4} d f e^{10} - 276 \, b^{3} c^{3} d g e^{10} - 60 \, b^{3} c^{3} f e^{11} + 35 \, b^{4} c^{2} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x + \frac {{\left (1536 \, c^{6} d^{5} g e^{6} + 1536 \, c^{6} d^{4} f e^{7} - 4368 \, b c^{5} d^{4} g e^{7} - 5472 \, b c^{5} d^{3} f e^{8} + 5328 \, b^{2} c^{4} d^{3} g e^{8} + 5136 \, b^{2} c^{4} d^{2} f e^{9} - 3256 \, b^{3} c^{3} d^{2} g e^{9} - 1560 \, b^{3} c^{3} d f e^{10} + 940 \, b^{4} c^{2} d g e^{10} + 180 \, b^{4} c^{2} f e^{11} - 105 \, b^{5} c g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} + \frac {{\left (64 \, c^{6} d^{6} g + 384 \, c^{6} d^{5} f e - 384 \, b c^{5} d^{5} g e - 960 \, b c^{5} d^{4} f e^{2} + 720 \, b^{2} c^{4} d^{4} g e^{2} + 960 \, b^{2} c^{4} d^{3} f e^{3} - 640 \, b^{3} c^{3} d^{3} g e^{3} - 480 \, b^{3} c^{3} d^{2} f e^{4} + 300 \, b^{4} c^{2} d^{2} g e^{4} + 120 \, b^{4} c^{2} d f e^{5} - 72 \, b^{5} c d g e^{5} - 12 \, b^{5} c f e^{6} + 7 \, b^{6} g e^{6}\right )} \sqrt {-c e^{2}} e^{\left (-3\right )} \log \left ({\left | -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 - \sqrt {-c e^{2}} b \right |}\right )}{1024 \, c^{5}} \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)^(3/2),x, algorithm="giac")
[Out]
-1/7680*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(10*c*g*x*e^3 + (12*c^6*d*g*e^10 + 12*c^6*f*e^1
1 + 13*b*c^5*g*e^11)*e^(-8)/c^5)*x - (140*c^6*d^2*g*e^9 - 120*c^6*d*f*e^10 - 272*b*c^5*d*g*e^10 - 132*b*c^5*f*
e^11 - 3*b^2*c^4*g*e^11)*e^(-8)/c^5)*x - (384*c^6*d^3*g*e^8 + 384*c^6*d^2*f*e^9 - 348*b*c^5*d^2*g*e^9 - 744*b*
c^5*d*f*e^10 - 48*b^2*c^4*d*g*e^10 - 12*b^2*c^4*f*e^11 + 7*b^3*c^3*g*e^11)*e^(-8)/c^5)*x + (240*c^6*d^4*g*e^7
- 2400*c^6*d^3*f*e^8 - 816*b*c^5*d^3*g*e^8 + 2064*b*c^5*d^2*f*e^9 + 792*b^2*c^4*d^2*g*e^9 + 456*b^2*c^4*d*f*e^
10 - 276*b^3*c^3*d*g*e^10 - 60*b^3*c^3*f*e^11 + 35*b^4*c^2*g*e^11)*e^(-8)/c^5)*x + (1536*c^6*d^5*g*e^6 + 1536*
c^6*d^4*f*e^7 - 4368*b*c^5*d^4*g*e^7 - 5472*b*c^5*d^3*f*e^8 + 5328*b^2*c^4*d^3*g*e^8 + 5136*b^2*c^4*d^2*f*e^9
- 3256*b^3*c^3*d^2*g*e^9 - 1560*b^3*c^3*d*f*e^10 + 940*b^4*c^2*d*g*e^10 + 180*b^4*c^2*f*e^11 - 105*b^5*c*g*e^1
1)*e^(-8)/c^5) + 1/1024*(64*c^6*d^6*g + 384*c^6*d^5*f*e - 384*b*c^5*d^5*g*e - 960*b*c^5*d^4*f*e^2 + 720*b^2*c^
4*d^4*g*e^2 + 960*b^2*c^4*d^3*f*e^3 - 640*b^3*c^3*d^3*g*e^3 - 480*b^3*c^3*d^2*f*e^4 + 300*b^4*c^2*d^2*g*e^4 +
120*b^4*c^2*d*f*e^5 - 72*b^5*c*d*g*e^5 - 12*b^5*c*f*e^6 + 7*b^6*g*e^6)*sqrt(-c*e^2)*e^(-3)*log(abs(-2*(sqrt(-c
*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*c - sqrt(-c*e^2)*b))/c^5
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maple [B] time = 0.07, size = 2117, normalized size = 7.13
result too large to display
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)^(3/2),x)
[Out]
-1/6*g/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d-1/8*b/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e*f-5/8
*e^2*g*b^3/c/(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^3+7/256*
e^3*g*b^4/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+1/48/e*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^2
+15/16*b^2*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^3*f+9/
64*b^3/c^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e*f+1/4
*d*f*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+3/16*d^3*f*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b+1/16/e*g*c^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^6-11/128*e^2*g*b^4/c
^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d+1/16/e*g*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4-5/16*g*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b*d^3+1/24/e*g*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2+1/32/e*g*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^4-5/32*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d^3-1/12*g/c^2*(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^2*d+7/192*e*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+1/8*d*f/c
*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b+3/8*e*g*b^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2+15/128*b^
4/c^2*e^4/(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*f-15/32*b^3
/c*e^3/(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^2*f-15/16*b*c/
(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^4*e*f+75/256*e^3*g*b^
4/c^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^2+9/32*b^2/c*e^
2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*f-11/64*e^2*g*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d-
9/128*e^4*g*b^5/c^3/(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+3
/8*d^3*f*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+3/8*d^5*f*c^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))-1/16*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e*f-3/128*b^4/c
^3*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f+7/512*e^3*g*b^5/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+7/6
0/e*g*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-9/32*b^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*e*f-3
/8*g*c/(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))*b*d^5+7/1024*e^5
*g*b^6/c^4/(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))+45/64*e*g*b^
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^4-3/64*b^3/c^2*e^3*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*f-9/16*b*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2*e*f-3/256*b^5/c
^3*e^5/(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+7/96*e*g*b^2/c
^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+3/16*e*g*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2-1/6/e*
g*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2*d*g
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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)^(3/2),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(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?
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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 )}^{3/2} \,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)^(3/2),x)
[Out]
int((f + g*x)*(d + e*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2), x)
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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 {3}{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)**(3/2),x)
[Out]
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(d + e*x)*(f + g*x), x)
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