3.24.39 \(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx\) [2339]

Optimal. Leaf size=153 \[ \frac {(b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{3/2}} \]

[Out]

-1/8*(-4*a*c+b^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)+1/4*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2

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Rubi [A]
time = 0.07, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {734, 738, 212} \begin {gather*} \frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \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 )}{8 \left (a e^2-b d e+c d^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - ((b^2 - 4*a*
c)*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])])/(8*(c*d^2 -
b*d*e + a*e^2)^(3/2))

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 734

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

Rubi steps

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

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Mathematica [A]
time = 10.12, size = 149, normalized size = 0.97 \begin {gather*} \frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1137\) vs. \(2(139)=278\).
time = 0.80, size = 1138, normalized size = 7.44

method	result	size
default	\(\frac {-\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{2 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {e \left (b e -2 c d \right ) \left (-\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {e \left (b e -2 c d \right ) \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{2 e^{2} a -2 b d e +2 c \,d^{2}}+\frac {2 c \,e^{2} \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{e^{2} a -b d e +c \,d^{2}}\right )}{4 \left (e^{2} a -b d e +c \,d^{2}\right )}+\frac {c \,e^{2} \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{2 e^{2} a -2 b d e +2 c \,d^{2}}}{e^{3}}\)	\(1138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(1/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/e^3*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3
/2)-1/4*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+
d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e
)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(
b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(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+1/e*(b*e-2*c*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*
(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))+2*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/4*(2*c*(x+d/e)+1/
e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d
^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+
d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))))+1/2*c/(a*e^2-b*d*e+c*d^2)*e^2*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*
e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2
*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(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+1/e*(b*e-2*c*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-
2*c*d)*(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((c*x^2+b*x+a)^(1/2)/(e*x+d)^3,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(c*d^2-%e*b*d+%e^2*a>0)', see `
assume?` for

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (144) = 288\).
time = 3.50, size = 913, normalized size = 5.97 \begin {gather*} \left [-\frac {{\left ({\left (b^{2} - 4 \, a c\right )} x^{2} e^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} d x e + {\left (b^{2} - 4 \, a c\right )} d^{2}\right )} \sqrt {c d^{2} - b d e + a e^{2}} \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 (2 \, c^{2} d^{3} x + b c d^{3} - {\left (a b x + 2 \, a^{2}\right )} e^{3} + {\left (3 \, a b d + {\left (b^{2} + 2 \, a c\right )} d x\right )} e^{2} - {\left (3 \, b c d^{2} x + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{16 \, {\left (c^{2} d^{6} + a^{2} x^{2} e^{6} - 2 \, {\left (a b d x^{2} - a^{2} d x\right )} e^{5} - {\left (4 \, a b d^{2} x - {\left (b^{2} + 2 \, a c\right )} d^{2} x^{2} - a^{2} d^{2}\right )} e^{4} - 2 \, {\left (b c d^{3} x^{2} + a b d^{3} - {\left (b^{2} + 2 \, a c\right )} d^{3} x\right )} e^{3} + {\left (c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} e^{2} + 2 \, {\left (c^{2} d^{5} x - b c d^{5}\right )} e\right )}}, -\frac {{\left ({\left (b^{2} - 4 \, a c\right )} x^{2} e^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} d x e + {\left (b^{2} - 4 \, a c\right )} d^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}} \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 (2 \, c^{2} d^{3} x + b c d^{3} - {\left (a b x + 2 \, a^{2}\right )} e^{3} + {\left (3 \, a b d + {\left (b^{2} + 2 \, a c\right )} d x\right )} e^{2} - {\left (3 \, b c d^{2} x + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{8 \, {\left (c^{2} d^{6} + a^{2} x^{2} e^{6} - 2 \, {\left (a b d x^{2} - a^{2} d x\right )} e^{5} - {\left (4 \, a b d^{2} x - {\left (b^{2} + 2 \, a c\right )} d^{2} x^{2} - a^{2} d^{2}\right )} e^{4} - 2 \, {\left (b c d^{3} x^{2} + a b d^{3} - {\left (b^{2} + 2 \, a c\right )} d^{3} x\right )} e^{3} + {\left (c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} e^{2} + 2 \, {\left (c^{2} d^{5} x - b c d^{5}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

[-1/16*(((b^2 - 4*a*c)*x^2*e^2 + 2*(b^2 - 4*a*c)*d*x*e + (b^2 - 4*a*c)*d^2)*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*(2*c^2*d^3*x + b*c*d^3 - (a*b*x + 2*a^2)*e^3 + (3*a*b*d + (b^2 + 2*a
*c)*d*x)*e^2 - (3*b*c*d^2*x + (b^2 + 2*a*c)*d^2)*e)*sqrt(c*x^2 + b*x + a))/(c^2*d^6 + a^2*x^2*e^6 - 2*(a*b*d*x
^2 - a^2*d*x)*e^5 - (4*a*b*d^2*x - (b^2 + 2*a*c)*d^2*x^2 - a^2*d^2)*e^4 - 2*(b*c*d^3*x^2 + a*b*d^3 - (b^2 + 2*
a*c)*d^3*x)*e^3 + (c^2*d^4*x^2 - 4*b*c*d^4*x + (b^2 + 2*a*c)*d^4)*e^2 + 2*(c^2*d^5*x - b*c*d^5)*e), -1/8*(((b^
2 - 4*a*c)*x^2*e^2 + 2*(b^2 - 4*a*c)*d*x*e + (b^2 - 4*a*c)*d^2)*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*(2*c^2*d^3*x + b*c*d^3 - (a*b*x + 2*a
^2)*e^3 + (3*a*b*d + (b^2 + 2*a*c)*d*x)*e^2 - (3*b*c*d^2*x + (b^2 + 2*a*c)*d^2)*e)*sqrt(c*x^2 + b*x + a))/(c^2
*d^6 + a^2*x^2*e^6 - 2*(a*b*d*x^2 - a^2*d*x)*e^5 - (4*a*b*d^2*x - (b^2 + 2*a*c)*d^2*x^2 - a^2*d^2)*e^4 - 2*(b*
c*d^3*x^2 + a*b*d^3 - (b^2 + 2*a*c)*d^3*x)*e^3 + (c^2*d^4*x^2 - 4*b*c*d^4*x + (b^2 + 2*a*c)*d^4)*e^2 + 2*(c^2*
d^5*x - b*c*d^5)*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**3,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (144) = 288\).
time = 1.37, size = 686, normalized size = 4.48 \begin {gather*} -\frac {{\left (b^{2} - 4 \, a c\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 )}{4 \, {\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} + \frac {8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} c^{2} d^{2} e + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {5}{2}} d^{3} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{2} d^{3} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b c d e^{2} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{2} c d^{2} e - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a c^{2} d^{2} e + 2 \, b^{2} c^{\frac {3}{2}} d^{3} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} \sqrt {c} d e^{2} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{\frac {3}{2}} d e^{2} - b^{3} \sqrt {c} d^{2} e - 4 \, a b c^{\frac {3}{2}} d^{2} e + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{2} e^{3} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a c e^{3} - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} d e^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c d e^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b \sqrt {c} e^{3} + a b^{2} \sqrt {c} d e^{2} + 4 \, a^{2} c^{\frac {3}{2}} d e^{2} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{2} e^{3} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} c e^{3}}{4 \, {\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} d + b d - a e\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

-1/4*(b^2 - 4*a*c)*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)) + 1/4*(8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*d^2*e
+ 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*d^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*d^3 - 8*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*d*e^2 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c*d^2*e - 8*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))*a*c^2*d^2*e + 2*b^2*c^(3/2)*d^3 - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*sqr
t(c)*d*e^2 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2)*d*e^2 - b^3*sqrt(c)*d^2*e - 4*a*b*c^(3/2)*d^2*e
+ (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*e^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*e^3 - (sqrt(c)*
x - sqrt(c*x^2 + b*x + a))*b^3*d*e^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c*d*e^2 + 8*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^2*a*b*sqrt(c)*e^3 + a*b^2*sqrt(c)*d*e^2 + 4*a^2*c^(3/2)*d*e^2 + (sqrt(c)*x - sqrt(c*x^2 + b
*x + a))*a*b^2*e^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*e^3)/((c*d^2*e^2 - b*d*e^3 + a*e^4)*((sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)*d + b*d - a*e)^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(1/2)/(d + e*x)^3,x)

[Out]

int((a + b*x + c*x^2)^(1/2)/(d + e*x)^3, x)

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