3.9.80 \(\int \frac {(f+g x)^2}{(d+e x) (a+b x+c x^2)^{3/2}} \, dx\) [880]

Optimal. Leaf size=240 \[ \frac {2 \left (b^2 e f^2+2 a \left (a e g^2-c f (e f-2 d g)\right )-b \left (c d f^2+a g (2 e f+d g)\right )-\left (2 c^2 d f^2+b (b d-a e) g^2+c (2 a g (2 e f-d g)-b f (e f+2 d g))\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {(e f-d g)^2 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2}} \]

[Out]

(-d*g+e*f)^2*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*
e+c*d^2)^(3/2)+2*(b^2*e*f^2+2*a*(a*e*g^2-c*f*(-2*d*g+e*f))-b*(c*d*f^2+a*g*(d*g+2*e*f))-(2*c^2*d*f^2+b*(-a*e+b*
d)*g^2+c*(2*a*g*(-d*g+2*e*f)-b*f*(2*d*g+e*f)))*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1660, 12, 738, 212} \begin {gather*} \frac {2 \left (-x \left (c (2 a g (2 e f-d g)-b f (2 d g+e f))+b g^2 (b d-a e)+2 c^2 d f^2\right )-b \left (a g (d g+2 e f)+c d f^2\right )+2 a \left (a e g^2-c f (e f-2 d g)\right )+b^2 e f^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac {(e f-d g)^2 \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^2/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(2*(b^2*e*f^2 + 2*a*(a*e*g^2 - c*f*(e*f - 2*d*g)) - b*(c*d*f^2 + a*g*(2*e*f + d*g)) - (2*c^2*d*f^2 + b*(b*d -
a*e)*g^2 + c*(2*a*g*(2*e*f - d*g) - b*f*(e*f + 2*d*g)))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*
x + c*x^2]) + ((e*f - d*g)^2*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b
*x + c*x^2])])/(c*d^2 - b*d*e + a*e^2)^(3/2)

Rule 12

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 1660

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*
c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(f+g x)^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac {2 \left (b^2 e f^2+2 a \left (a e g^2-c f (e f-2 d g)\right )-b \left (c d f^2+a g (2 e f+d g)\right )-\left (2 c^2 d f^2+b (b d-a e) g^2+c (2 a g (2 e f-d g)-b f (e f+2 d g))\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int -\frac {\left (b^2-4 a c\right ) (e f-d g)^2}{2 \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=\frac {2 \left (b^2 e f^2+2 a \left (a e g^2-c f (e f-2 d g)\right )-b \left (c d f^2+a g (2 e f+d g)\right )-\left (2 c^2 d f^2+b (b d-a e) g^2+c (2 a g (2 e f-d g)-b f (e f+2 d g))\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {(e f-d g)^2 \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{c d^2-b d e+a e^2}\\ &=\frac {2 \left (b^2 e f^2+2 a \left (a e g^2-c f (e f-2 d g)\right )-b \left (c d f^2+a g (2 e f+d g)\right )-\left (2 c^2 d f^2+b (b d-a e) g^2+c (2 a g (2 e f-d g)-b f (e f+2 d g))\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {\left (2 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{c d^2-b d e+a e^2}\\ &=\frac {2 \left (b^2 e f^2+2 a \left (a e g^2-c f (e f-2 d g)\right )-b \left (c d f^2+a g (2 e f+d g)\right )-\left (2 c^2 d f^2+b (b d-a e) g^2+c (2 a g (2 e f-d g)-b f (e f+2 d g))\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {(e f-d g)^2 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 1.20, size = 244, normalized size = 1.02 \begin {gather*} 2 \left (\frac {-2 a^2 e g^2+2 c^2 d f^2 x-2 a c d g (2 f+g x)+2 a c e f (f+2 g x)+a b g (2 e f+d g-e g x)+b^2 \left (-e f^2+d g^2 x\right )+b c f (-e f x+d (f-2 g x))}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \sqrt {a+x (b+c x)}}+\frac {\sqrt {-c d^2+b d e-a e^2} (e f-d g)^2 \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{\left (c d^2+e (-b d+a e)\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^2/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

2*((-2*a^2*e*g^2 + 2*c^2*d*f^2*x - 2*a*c*d*g*(2*f + g*x) + 2*a*c*e*f*(f + 2*g*x) + a*b*g*(2*e*f + d*g - e*g*x)
+ b^2*(-(e*f^2) + d*g^2*x) + b*c*f*(-(e*f*x) + d*(f - 2*g*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt
[a + x*(b + c*x)]) + (Sqrt[-(c*d^2) + b*d*e - a*e^2]*(e*f - d*g)^2*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b
+ c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/(c*d^2 + e*(-(b*d) + a*e))^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(554\) vs. \(2(230)=460\).
time = 0.13, size = 555, normalized size = 2.31

method	result	size
default	\(-\frac {g \left (-e g \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\frac {2 d g \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {4 e f \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{e^{2}}+\frac {\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \left (\frac {e^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (e b -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {e b -2 c d}{e}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (e b -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\)	\(555\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2/(e*x+d)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-g/e^2*(-e*g*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+2*d*g*(2*c*x+b)/(4*a*c-b
^2)/(c*x^2+b*x+a)^(1/2)-4*e*f*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+(d^2*g^2-2*d*e*f*g+e^2*f^2)/e^3*(1/(a
*e^2-b*d*e+c*d^2)*e^2/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b
*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*(x+d/e)^2+(b*e-2*c*
d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a
*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/
e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))

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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((g*x+f)^2/(e*x+d)/(c*x^2+b*x+a)^(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((%e^-1*b-2*%e^-2*c*d)^2>0)', s
ee `assume?`

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (239) = 478\).
time = 12.23, size = 2032, normalized size = 8.47 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[1/2*(((b^2*c - 4*a*c^2)*d^2*g^2*x^2 + (b^3 - 4*a*b*c)*d^2*g^2*x + (a*b^2 - 4*a^2*c)*d^2*g^2 + ((b^2*c - 4*a*c
^2)*f^2*x^2 + (b^3 - 4*a*b*c)*f^2*x + (a*b^2 - 4*a^2*c)*f^2)*e^2 - 2*((b^2*c - 4*a*c^2)*d*f*g*x^2 + (b^3 - 4*a
*b*c)*d*f*g*x + (a*b^2 - 4*a^2*c)*d*f*g)*e)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x + (b
^2 + 4*a*c)*d^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a*b
*x + (b^2 + 4*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e +
d^2)) - 4*(b*c^2*d^3*f^2 - 4*a*c^2*d^3*f*g + a*b*c*d^3*g^2 + (2*c^3*d^3*f^2 - 2*b*c^2*d^3*f*g + (b^2*c - 2*a*c
^2)*d^3*g^2)*x + (2*a^2*b*f*g - 2*a^3*g^2 - (a*b^2 - 2*a^2*c)*f^2 - (a*b*c*f^2 - 4*a^2*c*f*g + a^2*b*g^2)*x)*e
^3 + (3*a^2*b*d*g^2 + (b^3 - a*b*c)*d*f^2 - 2*(a*b^2 + 2*a^2*c)*d*f*g - (6*a*b*c*d*f*g - (b^2*c + 2*a*c^2)*d*f
^2 - 2*(a*b^2 - a^2*c)*d*g^2)*x)*e^2 + (6*a*b*c*d^2*f*g - 2*(b^2*c - a*c^2)*d^2*f^2 - (a*b^2 + 2*a^2*c)*d^2*g^
2 - (3*b*c^2*d^2*f^2 - 2*(b^2*c + 2*a*c^2)*d^2*f*g + (b^3 - a*b*c)*d^2*g^2)*x)*e)*sqrt(c*x^2 + b*x + a))/((b^2
*c^3 - 4*a*c^4)*d^4*x^2 + (b^3*c^2 - 4*a*b*c^3)*d^4*x + (a*b^2*c^2 - 4*a^2*c^3)*d^4 + (a^3*b^2 - 4*a^4*c + (a^
2*b^2*c - 4*a^3*c^2)*x^2 + (a^2*b^3 - 4*a^3*b*c)*x)*e^4 - 2*((a*b^3*c - 4*a^2*b*c^2)*d*x^2 + (a*b^4 - 4*a^2*b^
2*c)*d*x + (a^2*b^3 - 4*a^3*b*c)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*x^2 + (b^5 - 2*a*b^3*c - 8*a^
2*b*c^2)*d^2*x + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 - 4*a*b*c^3)*d^3*x^2 + (b^4*c - 4*a*
b^2*c^2)*d^3*x + (a*b^3*c - 4*a^2*b*c^2)*d^3)*e), (((b^2*c - 4*a*c^2)*d^2*g^2*x^2 + (b^3 - 4*a*b*c)*d^2*g^2*x
+ (a*b^2 - 4*a^2*c)*d^2*g^2 + ((b^2*c - 4*a*c^2)*f^2*x^2 + (b^3 - 4*a*b*c)*f^2*x + (a*b^2 - 4*a^2*c)*f^2)*e^2
- 2*((b^2*c - 4*a*c^2)*d*f*g*x^2 + (b^3 - 4*a*b*c)*d*f*g*x + (a*b^2 - 4*a^2*c)*d*f*g)*e)*sqrt(-c*d^2 + b*d*e -
a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)/(c^2*d^
2*x^2 + b*c*d^2*x + a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*x^2 + b^2*d*x + a*b*d)*e)) - 2*(b*c^2*d^3*f
^2 - 4*a*c^2*d^3*f*g + a*b*c*d^3*g^2 + (2*c^3*d^3*f^2 - 2*b*c^2*d^3*f*g + (b^2*c - 2*a*c^2)*d^3*g^2)*x + (2*a^
2*b*f*g - 2*a^3*g^2 - (a*b^2 - 2*a^2*c)*f^2 - (a*b*c*f^2 - 4*a^2*c*f*g + a^2*b*g^2)*x)*e^3 + (3*a^2*b*d*g^2 +
(b^3 - a*b*c)*d*f^2 - 2*(a*b^2 + 2*a^2*c)*d*f*g - (6*a*b*c*d*f*g - (b^2*c + 2*a*c^2)*d*f^2 - 2*(a*b^2 - a^2*c)
*d*g^2)*x)*e^2 + (6*a*b*c*d^2*f*g - 2*(b^2*c - a*c^2)*d^2*f^2 - (a*b^2 + 2*a^2*c)*d^2*g^2 - (3*b*c^2*d^2*f^2 -
2*(b^2*c + 2*a*c^2)*d^2*f*g + (b^3 - a*b*c)*d^2*g^2)*x)*e)*sqrt(c*x^2 + b*x + a))/((b^2*c^3 - 4*a*c^4)*d^4*x^
2 + (b^3*c^2 - 4*a*b*c^3)*d^4*x + (a*b^2*c^2 - 4*a^2*c^3)*d^4 + (a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x
^2 + (a^2*b^3 - 4*a^3*b*c)*x)*e^4 - 2*((a*b^3*c - 4*a^2*b*c^2)*d*x^2 + (a*b^4 - 4*a^2*b^2*c)*d*x + (a^2*b^3 -
4*a^3*b*c)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*x^2 + (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*d^2*x + (a*b^
4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 - 4*a*b*c^3)*d^3*x^2 + (b^4*c - 4*a*b^2*c^2)*d^3*x + (a*b^
3*c - 4*a^2*b*c^2)*d^3)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f + g x\right )^{2}}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**2/(e*x+d)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral((f + g*x)**2/((d + e*x)*(a + b*x + c*x**2)**(3/2)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 757 vs. \(2 (239) = 478\).
time = 6.14, size = 757, normalized size = 3.15 \begin {gather*} -\frac {2 \, {\left (\frac {{\left (2 \, c^{3} d^{3} f^{2} - 2 \, b c^{2} d^{3} f g + b^{2} c d^{3} g^{2} - 2 \, a c^{2} d^{3} g^{2} - 3 \, b c^{2} d^{2} f^{2} e + 2 \, b^{2} c d^{2} f g e + 4 \, a c^{2} d^{2} f g e - b^{3} d^{2} g^{2} e + a b c d^{2} g^{2} e + b^{2} c d f^{2} e^{2} + 2 \, a c^{2} d f^{2} e^{2} - 6 \, a b c d f g e^{2} + 2 \, a b^{2} d g^{2} e^{2} - 2 \, a^{2} c d g^{2} e^{2} - a b c f^{2} e^{3} + 4 \, a^{2} c f g e^{3} - a^{2} b g^{2} e^{3}\right )} x}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} + \frac {b c^{2} d^{3} f^{2} - 4 \, a c^{2} d^{3} f g + a b c d^{3} g^{2} - 2 \, b^{2} c d^{2} f^{2} e + 2 \, a c^{2} d^{2} f^{2} e + 6 \, a b c d^{2} f g e - a b^{2} d^{2} g^{2} e - 2 \, a^{2} c d^{2} g^{2} e + b^{3} d f^{2} e^{2} - a b c d f^{2} e^{2} - 2 \, a b^{2} d f g e^{2} - 4 \, a^{2} c d f g e^{2} + 3 \, a^{2} b d g^{2} e^{2} - a b^{2} f^{2} e^{3} + 2 \, a^{2} c f^{2} e^{3} + 2 \, a^{2} b f g e^{3} - 2 \, a^{3} g^{2} e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}\right )}}{\sqrt {c x^{2} + b x + a}} + \frac {2 \, {\left (d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2*((2*c^3*d^3*f^2 - 2*b*c^2*d^3*f*g + b^2*c*d^3*g^2 - 2*a*c^2*d^3*g^2 - 3*b*c^2*d^2*f^2*e + 2*b^2*c*d^2*f*g*e
+ 4*a*c^2*d^2*f*g*e - b^3*d^2*g^2*e + a*b*c*d^2*g^2*e + b^2*c*d*f^2*e^2 + 2*a*c^2*d*f^2*e^2 - 6*a*b*c*d*f*g*e
^2 + 2*a*b^2*d*g^2*e^2 - 2*a^2*c*d*g^2*e^2 - a*b*c*f^2*e^3 + 4*a^2*c*f*g*e^3 - a^2*b*g^2*e^3)*x/(b^2*c^2*d^4 -
4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3
*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*e^4) + (b*c^2*d^3*f^2 - 4*a*c^2*d^3*f*g + a*b*c*d^3*g^2 - 2*b
^2*c*d^2*f^2*e + 2*a*c^2*d^2*f^2*e + 6*a*b*c*d^2*f*g*e - a*b^2*d^2*g^2*e - 2*a^2*c*d^2*g^2*e + b^3*d*f^2*e^2 -
a*b*c*d*f^2*e^2 - 2*a*b^2*d*f*g*e^2 - 4*a^2*c*d*f*g*e^2 + 3*a^2*b*d*g^2*e^2 - a*b^2*f^2*e^3 + 2*a^2*c*f^2*e^3
+ 2*a^2*b*f*g*e^3 - 2*a^3*g^2*e^3)/(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2
- 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*e^4))/sqrt(
c*x^2 + b*x + a) + 2*(d^2*g^2 - 2*d*f*g*e + f^2*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*
d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c*d^2 - b*d*e + a*e^2)*sqrt(-c*d^2 + b*d*e - a*e^2))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f+g\,x\right )}^2}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)^2/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x)

[Out]

int((f + g*x)^2/((d + e*x)*(a + b*x + c*x^2)^(3/2)), x)

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