3.9.81 \(\int \frac {f+g x}{(d+e x) (a+b x+c x^2)^{3/2}} \, dx\) [881]

Optimal. Leaf size=187 \[ -\frac {2 \left (b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {e (e f-d g) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2}} \]

[Out]

e*(-d*g+e*f)*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*c*d*f-b^2*e*f+2*a*c*e*f-2*a*c*d*g+a*b*e*g+c*(2*c*d*f+2*a*e*g-b*(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 = 0.09, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {836, 12, 738, 212} \begin {gather*} \frac {e (e f-d g) \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}}-\frac {2 \left (c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f\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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + c*(2*c*d*f + 2*a*e*g - b*(e*f + 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*(e*f - d*g)*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 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {f+g x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 \left (b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) 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 (e f-d g)}{2 (d+e x) \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) 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 (e f-d g)) \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 c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) 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 e (e f-d g)) \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 c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) 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 (e f-d g) \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 = 0.83, size = 198, normalized size = 1.06 \begin {gather*} \frac {-2 b^2 e f+2 b (a e g-c e f x+c d (f-g x))+4 c (-a d g+c d f x+a e (f+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 {2 e \sqrt {-c d^2+b d e-a e^2} (-e f+d g) \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b^2*e*f + 2*b*(a*e*g - c*e*f*x + c*d*(f - g*x)) + 4*c*(-(a*d*g) + c*d*f*x + a*e*(f + g*x)))/((b^2 - 4*a*c)
*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + x*(b + c*x)]) - (2*e*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(-(e*f) + d*g)*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. \(444\) vs. \(2(177)=354\).
time = 0.13, size = 445, normalized size = 2.38

method	result	size
default	\(\frac {2 g \left (2 c x +b \right )}{e \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {\left (-d g +e f \right ) \left (\frac {e^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (e b -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {e b -2 c d}{e}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (e b -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{2}}\)	\(445\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*g/e*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+(-d*g+e*f)/e^2*(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)/(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. 793 vs. \(2 (187) = 374\).
time = 14.44, size = 1630, normalized size = 8.72 \begin {gather*} \left [\frac {\sqrt {c d^{2} - b d e + a e^{2}} {\left ({\left ({\left (b^{2} c - 4 \, a c^{2}\right )} f x^{2} + {\left (b^{3} - 4 \, a b c\right )} f x + {\left (a b^{2} - 4 \, a^{2} c\right )} f\right )} e^{2} - {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d g x^{2} + {\left (b^{3} - 4 \, a b c\right )} d g x + {\left (a b^{2} - 4 \, a^{2} c\right )} d g\right )} e\right )} \log \left (-\frac {8 \, c^{2} d^{2} x^{2} + 8 \, b c d^{2} x + {\left (b^{2} + 4 \, a c\right )} d^{2} + 4 \, \sqrt {c d^{2} - b d e + a e^{2}} {\left (2 \, c d x + b d - {\left (b x + 2 \, a\right )} e\right )} \sqrt {c x^{2} + b x + a} + {\left (8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} e^{2} - 2 \, {\left (4 \, b c d x^{2} + 4 \, a b d + {\left (3 \, b^{2} + 4 \, a c\right )} d x\right )} e}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - 4 \, {\left (b c^{2} d^{3} f - 2 \, a c^{2} d^{3} g + {\left (2 \, c^{3} d^{3} f - b c^{2} d^{3} g\right )} x + {\left (a^{2} b g - {\left (a b^{2} - 2 \, a^{2} c\right )} f - {\left (a b c f - 2 \, a^{2} c g\right )} x\right )} e^{3} + {\left ({\left (b^{3} - a b c\right )} d f - {\left (a b^{2} + 2 \, a^{2} c\right )} d g - {\left (3 \, a b c d g - {\left (b^{2} c + 2 \, a c^{2}\right )} d f\right )} x\right )} e^{2} + {\left (3 \, a b c d^{2} g - 2 \, {\left (b^{2} c - a c^{2}\right )} d^{2} f - {\left (3 \, b c^{2} d^{2} f - {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} g\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} + {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{2} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} e^{4} - 2 \, {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d x^{2} + {\left (a b^{4} - 4 \, a^{2} b^{2} c\right )} d x + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d\right )} e^{3} + {\left ({\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{2} x^{2} + {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} d^{2} x + {\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{2}\right )} e^{2} - 2 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{3} x^{2} + {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} d^{3} x + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{3}\right )} e\right )}}, \frac {\sqrt {-c d^{2} + b d e - a e^{2}} {\left ({\left ({\left (b^{2} c - 4 \, a c^{2}\right )} f x^{2} + {\left (b^{3} - 4 \, a b c\right )} f x + {\left (a b^{2} - 4 \, a^{2} c\right )} f\right )} e^{2} - {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d g x^{2} + {\left (b^{3} - 4 \, a b c\right )} d g x + {\left (a b^{2} - 4 \, a^{2} c\right )} d g\right )} e\right )} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e - a e^{2}} {\left (2 \, c d x + b d - {\left (b x + 2 \, a\right )} e\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} d^{2} x^{2} + b c d^{2} x + a c d^{2} + {\left (a c x^{2} + a b x + a^{2}\right )} e^{2} - {\left (b c d x^{2} + b^{2} d x + a b d\right )} e\right )}}\right ) - 2 \, {\left (b c^{2} d^{3} f - 2 \, a c^{2} d^{3} g + {\left (2 \, c^{3} d^{3} f - b c^{2} d^{3} g\right )} x + {\left (a^{2} b g - {\left (a b^{2} - 2 \, a^{2} c\right )} f - {\left (a b c f - 2 \, a^{2} c g\right )} x\right )} e^{3} + {\left ({\left (b^{3} - a b c\right )} d f - {\left (a b^{2} + 2 \, a^{2} c\right )} d g - {\left (3 \, a b c d g - {\left (b^{2} c + 2 \, a c^{2}\right )} d f\right )} x\right )} e^{2} + {\left (3 \, a b c d^{2} g - 2 \, {\left (b^{2} c - a c^{2}\right )} d^{2} f - {\left (3 \, b c^{2} d^{2} f - {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} g\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a}}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} + {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{2} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} e^{4} - 2 \, {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d x^{2} + {\left (a b^{4} - 4 \, a^{2} b^{2} c\right )} d x + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d\right )} e^{3} + {\left ({\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{2} x^{2} + {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} d^{2} x + {\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{2}\right )} e^{2} - 2 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{3} x^{2} + {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} d^{3} x + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{3}\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(c*d^2 - b*d*e + a*e^2)*(((b^2*c - 4*a*c^2)*f*x^2 + (b^3 - 4*a*b*c)*f*x + (a*b^2 - 4*a^2*c)*f)*e^2 -
((b^2*c - 4*a*c^2)*d*g*x^2 + (b^3 - 4*a*b*c)*d*g*x + (a*b^2 - 4*a^2*c)*d*g)*e)*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*a*c^2*d^3*g + (2*c^3*d^3*f - b*c^2*d^3*g)*x + (a^2*b*g - (a*b^2 - 2*a^2*c)*
f - (a*b*c*f - 2*a^2*c*g)*x)*e^3 + ((b^3 - a*b*c)*d*f - (a*b^2 + 2*a^2*c)*d*g - (3*a*b*c*d*g - (b^2*c + 2*a*c^
2)*d*f)*x)*e^2 + (3*a*b*c*d^2*g - 2*(b^2*c - a*c^2)*d^2*f - (3*b*c^2*d^2*f - (b^2*c + 2*a*c^2)*d^2*g)*x)*e)*sq
rt(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), (sqrt(-c*d^2 + b*d*e - a*e^2)*(((b
^2*c - 4*a*c^2)*f*x^2 + (b^3 - 4*a*b*c)*f*x + (a*b^2 - 4*a^2*c)*f)*e^2 - ((b^2*c - 4*a*c^2)*d*g*x^2 + (b^3 - 4
*a*b*c)*d*g*x + (a*b^2 - 4*a^2*c)*d*g)*e)*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*a*c^2*d^3*g + (2*c^3*d^3*f - b*c^2*d^3*g)*x + (a^2*b*g - (a*b^2 - 2*a^2
*c)*f - (a*b*c*f - 2*a^2*c*g)*x)*e^3 + ((b^3 - a*b*c)*d*f - (a*b^2 + 2*a^2*c)*d*g - (3*a*b*c*d*g - (b^2*c + 2*
a*c^2)*d*f)*x)*e^2 + (3*a*b*c*d^2*g - 2*(b^2*c - a*c^2)*d^2*f - (3*b*c^2*d^2*f - (b^2*c + 2*a*c^2)*d^2*g)*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 {f + g x}{\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)/(e*x+d)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral((f + g*x)/((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. 568 vs. \(2 (187) = 374\).
time = 4.50, size = 568, normalized size = 3.04 \begin {gather*} -\frac {2 \, {\left (\frac {{\left (2 \, c^{3} d^{3} f - b c^{2} d^{3} g - 3 \, b c^{2} d^{2} f e + b^{2} c d^{2} g e + 2 \, a c^{2} d^{2} g e + b^{2} c d f e^{2} + 2 \, a c^{2} d f e^{2} - 3 \, a b c d g e^{2} - a b c f e^{3} + 2 \, a^{2} c g 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 \, a c^{2} d^{3} g - 2 \, b^{2} c d^{2} f e + 2 \, a c^{2} d^{2} f e + 3 \, a b c d^{2} g e + b^{3} d f e^{2} - a b c d f e^{2} - a b^{2} d g e^{2} - 2 \, a^{2} c d g e^{2} - a b^{2} f e^{3} + 2 \, a^{2} c f e^{3} + a^{2} b g 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 g e - f 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)/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2*((2*c^3*d^3*f - b*c^2*d^3*g - 3*b*c^2*d^2*f*e + b^2*c*d^2*g*e + 2*a*c^2*d^2*g*e + b^2*c*d*f*e^2 + 2*a*c^2*d
*f*e^2 - 3*a*b*c*d*g*e^2 - a*b*c*f*e^3 + 2*a^2*c*g*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*a*c^2*d^3*g - 2*b^2*c*d^2*f*e + 2*a*c^2*d^2*f*e + 3*a*b*c*d^2*g*e + b^3*d*f
*e^2 - a*b*c*d*f*e^2 - a*b^2*d*g*e^2 - 2*a^2*c*d*g*e^2 - a*b^2*f*e^3 + 2*a^2*c*f*e^3 + a^2*b*g*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*g*e - f*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 = -0.01 \begin {gather*} \int \frac {f+g\,x}{\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)/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x)

[Out]

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

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