3.20.80 \(\int \frac {(d+e x)^2 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=265 \[ \frac {(2 c d-b e)^2 (-5 b e g+4 c d g+6 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 c^{7/2} e^2}-\frac {(2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+4 c d g+6 c e f)}{8 c^3 e^2}-\frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+4 c d g+6 c e f)}{12 c^2 e^2}-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2} \]

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Rubi [A]  time = 0.35, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1638, 12, 670, 640, 621, 204} \begin {gather*} -\frac {(2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+4 c d g+6 c e f)}{8 c^3 e^2}-\frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+4 c d g+6 c e f)}{12 c^2 e^2}+\frac {(2 c d-b e)^2 (-5 b e g+4 c d g+6 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 )}{16 c^{7/2} e^2}-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^2*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]

[Out]

-((2*c*d - b*e)*(6*c*e*f + 4*c*d*g - 5*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(8*c^3*e^2) - ((6*c*e
*f + 4*c*d*g - 5*b*e*g)*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(12*c^2*e^2) - (g*(d + e*x)^2*Sqr
t[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*c*e^2) + ((2*c*d - b*e)^2*(6*c*e*f + 4*c*d*g - 5*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])])/(16*c^(7/2)*e^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 1638

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e +
(2*c*d - b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] &&
NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^2 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}-\frac {\int -\frac {e^2 (6 c e f+4 c d g-5 b e g) (d+e x)^2}{2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{3 c e^3}\\ &=-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}+\frac {(6 c e f+4 c d g-5 b e g) \int \frac {(d+e x)^2}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{6 c e}\\ &=-\frac {(6 c e f+4 c d g-5 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{12 c^2 e^2}-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}+\frac {((2 c d-b e) (6 c e f+4 c d g-5 b e g)) \int \frac {d+e x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 c^2 e}\\ &=-\frac {(2 c d-b e) (6 c e f+4 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c^3 e^2}-\frac {(6 c e f+4 c d g-5 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{12 c^2 e^2}-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}+\frac {\left ((2 c d-b e)^2 (6 c e f+4 c d g-5 b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{16 c^3 e}\\ &=-\frac {(2 c d-b e) (6 c e f+4 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c^3 e^2}-\frac {(6 c e f+4 c d g-5 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{12 c^2 e^2}-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}+\frac {\left ((2 c d-b e)^2 (6 c e f+4 c d g-5 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 )}{8 c^3 e}\\ &=-\frac {(2 c d-b e) (6 c e f+4 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c^3 e^2}-\frac {(6 c e f+4 c d g-5 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{12 c^2 e^2}-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}+\frac {(2 c d-b e)^2 (6 c e f+4 c d g-5 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 )}{16 c^{7/2} e^2}\\ \end {align*}

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Mathematica [A]  time = 1.28, size = 251, normalized size = 0.95 \begin {gather*} \frac {\sqrt {(d+e x) (c (d-e x)-b e)} \left (\frac {e^7 (-5 b e g+4 c d g+6 c e f) \left (3 \sqrt {c} \sqrt {e} \sqrt {d+e x} (b e-2 c d)^2 \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )-c (d+e x) \sqrt {e (2 c d-b e)} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}} (-3 b e+8 c d+2 c e x)\right )}{\sqrt {e (2 c d-b e)} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}}-8 c^3 e^7 g (d+e x)^3\right )}{24 c^4 e^9 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^2*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]

[Out]

(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-8*c^3*e^7*g*(d + e*x)^3 + (e^7*(6*c*e*f + 4*c*d*g - 5*b*e*g)*(-(c*Sq
rt[e*(2*c*d - b*e)]*(d + e*x)*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]*(8*c*d - 3*b*e + 2*c*e*x)) + 3*Sqrt[
c]*Sqrt[e]*(-2*c*d + b*e)^2*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]]))/(Sqr
t[e*(2*c*d - b*e)]*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])))/(24*c^4*e^9*(d + e*x))

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IntegrateAlgebraic [B]  time = 26.40, size = 22293, normalized size = 84.12 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((d + e*x)^2*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]

[Out]

Result too large to show

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fricas [A]  time = 0.87, size = 585, normalized size = 2.21 \begin {gather*} \left [\frac {3 \, {\left (6 \, {\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + {\left (16 \, c^{3} d^{3} - 36 \, b c^{2} d^{2} e + 24 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g\right )} \sqrt {-c} \log \left (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} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (8 \, c^{3} e^{2} g x^{2} + 6 \, {\left (8 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f + {\left (40 \, c^{3} d^{2} - 52 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g + 2 \, {\left (6 \, c^{3} e^{2} f + {\left (12 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{96 \, c^{4} e^{2}}, -\frac {3 \, {\left (6 \, {\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + {\left (16 \, c^{3} d^{3} - 36 \, b c^{2} d^{2} e + 24 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \, {\left (8 \, c^{3} e^{2} g x^{2} + 6 \, {\left (8 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f + {\left (40 \, c^{3} d^{2} - 52 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g + 2 \, {\left (6 \, c^{3} e^{2} f + {\left (12 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{48 \, c^{4} e^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

[1/96*(3*(6*(4*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^3)*f + (16*c^3*d^3 - 36*b*c^2*d^2*e + 24*b^2*c*d*e^2 - 5*b^
3*e^3)*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*(8*c^3*e^2*g*x^2 + 6*(8*c^3*d*e - 3*b*c^2*e^2)*f + (40*c^3
*d^2 - 52*b*c^2*d*e + 15*b^2*c*e^2)*g + 2*(6*c^3*e^2*f + (12*c^3*d*e - 5*b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*
e^2*x + c*d^2 - b*d*e))/(c^4*e^2), -1/48*(3*(6*(4*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^3)*f + (16*c^3*d^3 - 36*
b*c^2*d^2*e + 24*b^2*c*d*e^2 - 5*b^3*e^3)*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*(8*c^3*e^2*g*x^2 + 6*(8*c^3*d*e - 3*b*
c^2*e^2)*f + (40*c^3*d^2 - 52*b*c^2*d*e + 15*b^2*c*e^2)*g + 2*(6*c^3*e^2*f + (12*c^3*d*e - 5*b*c^2*e^2)*g)*x)*
sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^4*e^2)]

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giac [A]  time = 0.55, size = 264, normalized size = 1.00 \begin {gather*} -\frac {1}{24} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, {\left (\frac {4 \, g x}{c} + \frac {{\left (12 \, c^{2} d g e^{2} + 6 \, c^{2} f e^{3} - 5 \, b c g e^{3}\right )} e^{\left (-3\right )}}{c^{3}}\right )} x + \frac {{\left (40 \, c^{2} d^{2} g e + 48 \, c^{2} d f e^{2} - 52 \, b c d g e^{2} - 18 \, b c f e^{3} + 15 \, b^{2} g e^{3}\right )} e^{\left (-3\right )}}{c^{3}}\right )} + \frac {{\left (16 \, c^{3} d^{3} g + 24 \, c^{3} d^{2} f e - 36 \, b c^{2} d^{2} g e - 24 \, b c^{2} d f e^{2} + 24 \, b^{2} c d g e^{2} + 6 \, b^{2} c f e^{3} - 5 \, b^{3} g e^{3}\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 )}{16 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")

[Out]

-1/24*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*g*x/c + (12*c^2*d*g*e^2 + 6*c^2*f*e^3 - 5*b*c*g*e^3)*e^
(-3)/c^3)*x + (40*c^2*d^2*g*e + 48*c^2*d*f*e^2 - 52*b*c*d*g*e^2 - 18*b*c*f*e^3 + 15*b^2*g*e^3)*e^(-3)/c^3) + 1
/16*(16*c^3*d^3*g + 24*c^3*d^2*f*e - 36*b*c^2*d^2*g*e - 24*b*c^2*d*f*e^2 + 24*b^2*c*d*g*e^2 + 6*b^2*c*f*e^3 -
5*b^3*g*e^3)*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^4

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maple [B]  time = 0.06, size = 786, normalized size = 2.97 \begin {gather*} -\frac {5 b^{3} e^{2} g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{16 \sqrt {c \,e^{2}}\, c^{3}}+\frac {3 b^{2} d e g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{2 \sqrt {c \,e^{2}}\, c^{2}}+\frac {3 b^{2} e^{2} f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{8 \sqrt {c \,e^{2}}\, c^{2}}-\frac {9 b \,d^{2} g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{4 \sqrt {c \,e^{2}}\, c}-\frac {3 b d e f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{2 \sqrt {c \,e^{2}}\, c}+\frac {d^{3} g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{\sqrt {c \,e^{2}}\, e}+\frac {3 d^{2} f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{2 \sqrt {c \,e^{2}}}-\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, g \,x^{2}}{3 c}+\frac {5 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b g x}{12 c^{2}}-\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, d g x}{c e}-\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, f x}{2 c}-\frac {5 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b^{2} g}{8 c^{3}}+\frac {13 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b d g}{6 c^{2} e}+\frac {3 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b f}{4 c^{2}}-\frac {5 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, d^{2} g}{3 c \,e^{2}}-\frac {2 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, d f}{c e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

-1/3*g*x^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+5/12*g*b/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-5/8*
g*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-5/16*e^2*g*b^3/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))+3/2*e*g*b^2/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-9/4*g*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^2+13/6/e*g/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d-5/3/e^2*g/c*(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2-x/c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*g-1/2*x/c*(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(1/2)*f+3/4*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f+3/8*b^2/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))*e^2*f-3/2/c*e/(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*f+1/e/(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*g+3/2*d^2*f/(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))-2/c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f

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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)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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 \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^2}{\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(d + e*x)^2)/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2),x)

[Out]

int(((f + g*x)*(d + e*x)^2)/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{2} \left (f + g x\right )}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

Integral((d + e*x)**2*(f + g*x)/sqrt(-(d + e*x)*(b*e - c*d + c*e*x)), x)

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