3.1614 \(\int \frac {(b+2 c x) \sqrt {d+e x}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=291 \[ -\frac {\sqrt {2} \left (2 c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \left (b e \left (\sqrt {b^2-4 a c}+b\right )-2 c \left (d \sqrt {b^2-4 a c}+2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+4 \sqrt {d+e x} \]

[Out]

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

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Rubi [A]  time = 0.89, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {824, 826, 1166, 208} \[ -\frac {\sqrt {2} \left (2 c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \left (b e \left (\sqrt {b^2-4 a c}+b\right )-2 c \left (d \sqrt {b^2-4 a c}+2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+4 \sqrt {d+e x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 208

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

Rule 824

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

Rule 826

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

Rule 1166

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

Rubi steps

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

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Mathematica [A]  time = 0.35, size = 288, normalized size = 0.99 \[ \frac {\sqrt {2} \left (c \left (4 a e-2 d \sqrt {b^2-4 a c}\right )+b e \left (\sqrt {b^2-4 a c}-b\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}+\frac {\sqrt {2} \left (b e \left (\sqrt {b^2-4 a c}+b\right )-2 c \left (d \sqrt {b^2-4 a c}+2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+4 \sqrt {d+e x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.94, size = 368, normalized size = 1.26 \[ -\frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} \log \left (\sqrt {2} \sqrt {\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} + 2 \, \sqrt {e x + d}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} \log \left (-\sqrt {2} \sqrt {\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} + 2 \, \sqrt {e x + d}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} \log \left (\sqrt {2} \sqrt {\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} + 2 \, \sqrt {e x + d}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} \log \left (-\sqrt {2} \sqrt {\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} + 2 \, \sqrt {e x + d}\right ) + 4 \, \sqrt {e x + d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.46, size = 614, normalized size = 2.11 \[ 4 \, \sqrt {x e + d} + \frac {{\left ({\left (2 \, \sqrt {b^{2} - 4 \, a c} c d e - \sqrt {b^{2} - 4 \, a c} b e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} - 4 \, {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} - {\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{3} d e - \sqrt {b^{2} - 4 \, a c} b c^{2} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{2} d - b c e + \sqrt {-4 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} c^{2} + {\left (2 \, c^{2} d - b c e\right )}^{2}}}{c^{2}}}}\right )}{4 \, {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} c^{2}} - \frac {{\left ({\left (2 \, \sqrt {b^{2} - 4 \, a c} c d e - \sqrt {b^{2} - 4 \, a c} b e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} + 4 \, {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} - {\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{3} d e - \sqrt {b^{2} - 4 \, a c} b c^{2} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{2} d - b c e - \sqrt {-4 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} c^{2} + {\left (2 \, c^{2} d - b c e\right )}^{2}}}{c^{2}}}}\right )}{4 \, {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*sqrt(x*e + d) + 1/4*((2*sqrt(b^2 - 4*a*c)*c*d*e - sqrt(b^2 - 4*a*c)*b*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2
- 4*a*c)*c)*e)*c^2 - 4*(c^3*d^2 - b*c^2*d*e + a*c^2*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs
(c) - (2*sqrt(b^2 - 4*a*c)*c^3*d*e - sqrt(b^2 - 4*a*c)*b*c^2*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c
)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^2*d - b*c*e + sqrt(-4*(c^2*d^2 - b*c*d*e + a*c*e^2)*c^2 + (2
*c^2*d - b*c*e)^2))/c^2))/((c^3*d^2 - b*c^2*d*e + a*c^2*e^2)*c^2) - 1/4*((2*sqrt(b^2 - 4*a*c)*c*d*e - sqrt(b^2
- 4*a*c)*b*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c^2 + 4*(c^3*d^2 - b*c^2*d*e + a*c^2*e^2)*sq
rt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(c) - (2*sqrt(b^2 - 4*a*c)*c^3*d*e - sqrt(b^2 - 4*a*c)*b*c^2
*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^2*d - b*c*
e - sqrt(-4*(c^2*d^2 - b*c*d*e + a*c*e^2)*c^2 + (2*c^2*d - b*c*e)^2))/c^2))/((c^3*d^2 - b*c^2*d*e + a*c^2*e^2)
*c^2)

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maple [B]  time = 0.09, size = 724, normalized size = 2.49 \[ \frac {4 \sqrt {2}\, a c \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+\frac {4 \sqrt {2}\, a c \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, b^{2} e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, b^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+\frac {\sqrt {2}\, b e \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, b e \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {2 \sqrt {2}\, c d \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+\frac {2 \sqrt {2}\, c d \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+4 \sqrt {e x +d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, c x + b\right )} \sqrt {e x + d}}{c x^{2} + b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*c*x + b)*sqrt(e*x + d)/(c*x^2 + b*x + a), x)

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mupad [B]  time = 2.14, size = 2611, normalized size = 8.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(d + e*x)^(1/2) + 2*atanh((128*a^2*c^3*e^4*(d + e*x)^(1/2)*(d - (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c))
^(1/2))/(64*a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 - 4
*a*c)^(1/2) + 16*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(b^
2 - 4*a*c)^(1/2)) + (8*b^4*c*e^4*(d + e*x)^(1/2)*(d - (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c))^(1/2))/(64*
a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2)
+ 16*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(b^2 - 4*a*c)^(
1/2)) - (8*b^3*c*e^4*(b^2 - 4*a*c)^(1/2)*(d + e*x)^(1/2)*(d - (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c))^(1/
2))/(64*a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 - 4*a*c
)^(1/2) + 16*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(b^2 -
4*a*c)^(1/2)) - (64*a*b^2*c^2*e^4*(d + e*x)^(1/2)*(d - (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c))^(1/2))/(64
*a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2)
+ 16*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(b^2 - 4*a*c)^
(1/2)) + (16*b^2*c^2*d*e^3*(b^2 - 4*a*c)^(1/2)*(d + e*x)^(1/2)*(d - (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c
))^(1/2))/(64*a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 -
4*a*c)^(1/2) + 16*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(
b^2 - 4*a*c)^(1/2)) + (32*a*b*c^2*e^4*(b^2 - 4*a*c)^(1/2)*(d + e*x)^(1/2)*(d - (e*(b^2 - 4*a*c)^(1/2))/(2*c) -
(b*e)/(2*c))^(1/2))/(64*a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c
*e^5*(b^2 - 4*a*c)^(1/2) + 16*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*
c^2*d*e^4*(b^2 - 4*a*c)^(1/2)) - (64*a*c^3*d*e^3*(b^2 - 4*a*c)^(1/2)*(d + e*x)^(1/2)*(d - (e*(b^2 - 4*a*c)^(1/
2))/(2*c) - (b*e)/(2*c))^(1/2))/(64*a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) -
16*a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2) + 16*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2
) - 64*a*b*c^2*d*e^4*(b^2 - 4*a*c)^(1/2)))*(-(b*e - 2*c*d + e*(b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 2*atanh((128
*a^2*c^3*e^4*(d + e*x)^(1/2)*(d + (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c))^(1/2))/(64*a^2*c^2*e^5*(b^2 - 4
*a*c)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2) + 16*b^3*c*d*e^4*(b^
2 - 4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(b^2 - 4*a*c)^(1/2)) + (8*b^4*c*e^4
*(d + e*x)^(1/2)*(d + (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c))^(1/2))/(64*a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2)
- 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2) + 16*b^3*c*d*e^4*(b^2 - 4*a*c)^(
1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(b^2 - 4*a*c)^(1/2)) + (8*b^3*c*e^4*(b^2 - 4*a*
c)^(1/2)*(d + e*x)^(1/2)*(d + (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c))^(1/2))/(64*a^2*c^2*e^5*(b^2 - 4*a*c
)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2) + 16*b^3*c*d*e^4*(b^2 -
4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(b^2 - 4*a*c)^(1/2)) - (64*a*b^2*c^2*e^
4*(d + e*x)^(1/2)*(d + (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c))^(1/2))/(64*a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2)
- 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2) + 16*b^3*c*d*e^4*(b^2 - 4*a*c)^
(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(b^2 - 4*a*c)^(1/2)) - (16*b^2*c^2*d*e^3*(b^2
- 4*a*c)^(1/2)*(d + e*x)^(1/2)*(d + (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c))^(1/2))/(64*a^2*c^2*e^5*(b^2 -
4*a*c)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2) + 16*b^3*c*d*e^4*(
b^2 - 4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(b^2 - 4*a*c)^(1/2)) - (32*a*b*c^
2*e^4*(b^2 - 4*a*c)^(1/2)*(d + e*x)^(1/2)*(d + (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c))^(1/2))/(64*a^2*c^2
*e^5*(b^2 - 4*a*c)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2) + 16*b^
3*c*d*e^4*(b^2 - 4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(b^2 - 4*a*c)^(1/2)) +
(64*a*c^3*d*e^3*(b^2 - 4*a*c)^(1/2)*(d + e*x)^(1/2)*(d + (e*(b^2 - 4*a*c)^(1/2))/(2*c) - (b*e)/(2*c))^(1/2))/
(64*a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2) - 16*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 16*a*b^2*c*e^5*(b^2 - 4*a*c)^(1
/2) + 16*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) + 64*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 64*a*b*c^2*d*e^4*(b^2 - 4*a*
c)^(1/2)))*((2*c*d - b*e + e*(b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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