3.15.27 \(\int \frac {(b+2 c x) (d+e x)^{3/2}}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=224 \[ -\frac {3 e \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}+\frac {3 e \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}-\frac {(d+e x)^{3/2}}{a+b x+c x^2} \]

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Rubi [A]  time = 0.34, antiderivative size = 224, 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 = {768, 699, 1130, 208} \begin {gather*} -\frac {3 e \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\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 {2} \sqrt {c} \sqrt {b^2-4 a c}}+\frac {3 e \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\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 {2} \sqrt {c} \sqrt {b^2-4 a c}}-\frac {(d+e x)^{3/2}}{a+b x+c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 699

Int[Sqrt[(d_.) + (e_.)*(x_)]/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[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}, 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]

Rule 768

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

Rule 1130

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

Rubi steps

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

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Mathematica [A]  time = 0.42, size = 221, normalized size = 0.99 \begin {gather*} -\frac {3 e \sqrt {e \sqrt {b^2-4 a c}-b e+2 c d} \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 {2} \sqrt {c} \sqrt {b^2-4 a c}}+\frac {3 e \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\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 {2} \sqrt {c} \sqrt {b^2-4 a c}}-\frac {(d+e x)^{3/2}}{a+x (b+c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [C]  time = 3.40, size = 597, normalized size = 2.67 \begin {gather*} \frac {2 \sqrt {2} \left (e^2 \sqrt {b^2-4 a c}-2 b e^2+4 c d e\right ) \tan ^{-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 \sqrt {b^2-4 a c}+b e-2 c d}}+\frac {2 \sqrt {2} \left (e^2 \sqrt {b^2-4 a c}+2 b e^2-4 c d e\right ) \tan ^{-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 \sqrt {b^2-4 a c}+b e-2 c d}}-\frac {\left (e^2 \sqrt {4 a c-b^2}+5 i b e^2-10 i c d e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-i e \sqrt {4 a c-b^2}+b e-2 c d}}\right )}{\sqrt {2} \sqrt {c} \sqrt {4 a c-b^2} \sqrt {-i e \sqrt {4 a c-b^2}+b e-2 c d}}-\frac {\left (e^2 \sqrt {4 a c-b^2}-5 i b e^2+10 i c d e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {i e \sqrt {4 a c-b^2}+b e-2 c d}}\right )}{\sqrt {2} \sqrt {c} \sqrt {4 a c-b^2} \sqrt {i e \sqrt {4 a c-b^2}+b e-2 c d}}-\frac {e^2 (d+e x)^{3/2}}{a e^2+b e (d+e x)-b d e+c d^2-2 c d (d+e x)+c (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((e^2*(d + e*x)^(3/2))/(c*d^2 - b*d*e + a*e^2 - 2*c*d*(d + e*x) + b*e*(d + e*x) + c*(d + e*x)^2)) + (2*Sqrt[2
]*(4*c*d*e - 2*b*e^2 + Sqrt[b^2 - 4*a*c]*e^2)*ArcTan[(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*e - Sqrt[b^2 - 4*a*c]*e]) + (2*Sqrt[2]*(-4*c*d*e
+ 2*b*e^2 + Sqrt[b^2 - 4*a*c]*e^2)*ArcTan[(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*e + Sqrt[b^2 - 4*a*c]*e]) - (((-10*I)*c*d*e + (5*I)*b*e^2 +
Sqrt[-b^2 + 4*a*c]*e^2)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(
Sqrt[2]*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]) - (((10*I)*c*d*e - (5*I)*b*e^2
+ Sqrt[-b^2 + 4*a*c]*e^2)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]]
)/(Sqrt[2]*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e])

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fricas [B]  time = 0.45, size = 828, normalized size = 3.70 \begin {gather*} -\frac {3 \, \sqrt {\frac {1}{2}} {\left (c x^{2} + b x + a\right )} \sqrt {\frac {2 \, c d e^{2} - b e^{3} + \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )}}{b^{2} c - 4 \, a c^{2}}} \log \left (27 \, \sqrt {e x + d} e^{4} + 27 \, \sqrt {\frac {1}{2}} \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {2 \, c d e^{2} - b e^{3} + \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )}}{b^{2} c - 4 \, a c^{2}}}\right ) - 3 \, \sqrt {\frac {1}{2}} {\left (c x^{2} + b x + a\right )} \sqrt {\frac {2 \, c d e^{2} - b e^{3} + \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )}}{b^{2} c - 4 \, a c^{2}}} \log \left (27 \, \sqrt {e x + d} e^{4} - 27 \, \sqrt {\frac {1}{2}} \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {2 \, c d e^{2} - b e^{3} + \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )}}{b^{2} c - 4 \, a c^{2}}}\right ) - 3 \, \sqrt {\frac {1}{2}} {\left (c x^{2} + b x + a\right )} \sqrt {\frac {2 \, c d e^{2} - b e^{3} - \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )}}{b^{2} c - 4 \, a c^{2}}} \log \left (27 \, \sqrt {e x + d} e^{4} + 27 \, \sqrt {\frac {1}{2}} \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {2 \, c d e^{2} - b e^{3} - \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )}}{b^{2} c - 4 \, a c^{2}}}\right ) + 3 \, \sqrt {\frac {1}{2}} {\left (c x^{2} + b x + a\right )} \sqrt {\frac {2 \, c d e^{2} - b e^{3} - \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )}}{b^{2} c - 4 \, a c^{2}}} \log \left (27 \, \sqrt {e x + d} e^{4} - 27 \, \sqrt {\frac {1}{2}} \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {2 \, c d e^{2} - b e^{3} - \sqrt {\frac {e^{6}}{b^{2} c^{2} - 4 \, a c^{3}}} {\left (b^{2} c - 4 \, a c^{2}\right )}}{b^{2} c - 4 \, a c^{2}}}\right ) + 2 \, {\left (e x + d\right )}^{\frac {3}{2}}}{2 \, {\left (c x^{2} + b x + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))
/(b^2*c - 4*a*c^2))*log(27*sqrt(e*x + d)*e^4 + 27*sqrt(1/2)*sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2)*sq
rt((2*c*d*e^2 - b*e^3 + sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))) - 3*sqrt(1/2)*(c*
x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))*l
og(27*sqrt(e*x + d)*e^4 - 27*sqrt(1/2)*sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2)*sqrt((2*c*d*e^2 - b*e^3
+ sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))) - 3*sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((
2*c*d*e^2 - b*e^3 - sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))*log(27*sqrt(e*x + d)*e
^4 + 27*sqrt(1/2)*sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2)*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/(b^2*c^2
- 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))) + 3*sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 - s
qrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))*log(27*sqrt(e*x + d)*e^4 - 27*sqrt(1/2)*sqr
t(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2)*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c -
4*a*c^2))/(b^2*c - 4*a*c^2))) + 2*(e*x + d)^(3/2))/(c*x^2 + b*x + a)

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giac [A]  time = 1.11, size = 288, normalized size = 1.29 \begin {gather*} -\frac {{\left (x e + d\right )}^{\frac {3}{2}} e^{2}}{{\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e + a e^{2}} - \frac {3 \, \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right ) e}{2 \, \sqrt {b^{2} - 4 \, a c} {\left | c \right |}} + \frac {3 \, \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right ) e}{2 \, \sqrt {b^{2} - 4 \, a c} {\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.09, size = 590, normalized size = 2.63 \begin {gather*} \frac {3 \sqrt {2}\, b \,e^{3} \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 )}{2 \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 {3 \sqrt {2}\, b \,e^{3} \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 )}{2 \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 {3 \sqrt {2}\, c d \,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 {3 \sqrt {2}\, c d \,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 {3 \sqrt {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 )}{2 \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+\frac {3 \sqrt {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 )}{2 \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {\left (e x +d \right )^{\frac {3}{2}} e^{2}}{c \,e^{2} x^{2}+b \,e^{2} x +a \,e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^2*(e*x+d)^(3/2)/(c*e^2*x^2+b*e^2*x+a*e^2)+3/2*e^3/(-(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-3*e
^2*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)*d-3/2*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)+3/2*e^3/(-(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-3*e^2*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)*d+3/2*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.47, size = 841, normalized size = 3.75 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {d+e\,x}\,\left (-18\,b^2\,c\,e^6+36\,b\,c^2\,d\,e^5-36\,c^3\,d^2\,e^4+36\,a\,c^2\,e^6\right )+\frac {9\,\sqrt {d+e\,x}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (b^3\,e^3+e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c\,e^3+8\,a\,c^2\,d\,e^2-2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {-\frac {9\,\left (b^3\,e^3+e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c\,e^3+8\,a\,c^2\,d\,e^2-2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{54\,c^2\,d^2\,e^6-54\,b\,c\,d\,e^7+54\,a\,c\,e^8}\right )\,\sqrt {-\frac {9\,\left (b^3\,e^3+e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c\,e^3+8\,a\,c^2\,d\,e^2-2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}-2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {d+e\,x}\,\left (-18\,b^2\,c\,e^6+36\,b\,c^2\,d\,e^5-36\,c^3\,d^2\,e^4+36\,a\,c^2\,e^6\right )-\frac {9\,\sqrt {d+e\,x}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e^3+4\,a\,b\,c\,e^3-8\,a\,c^2\,d\,e^2+2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {\frac {9\,\left (e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e^3+4\,a\,b\,c\,e^3-8\,a\,c^2\,d\,e^2+2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{54\,c^2\,d^2\,e^6-54\,b\,c\,d\,e^7+54\,a\,c\,e^8}\right )\,\sqrt {\frac {9\,\left (e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e^3+4\,a\,b\,c\,e^3-8\,a\,c^2\,d\,e^2+2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}-\frac {e^2\,{\left (d+e\,x\right )}^{3/2}}{\left (b\,e-2\,c\,d\right )\,\left (d+e\,x\right )+c\,{\left (d+e\,x\right )}^2+a\,e^2+c\,d^2-b\,d\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- 2*atanh((2*((d + e*x)^(1/2)*(36*a*c^2*e^6 - 18*b^2*c*e^6 - 36*c^3*d^2*e^4 + 36*b*c^2*d*e^5) + (9*(d + e*x)^(
1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2)*(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1
/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))*(-(9*(b^3*e^3 + e^
3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^
2)))^(1/2))/(54*c^2*d^2*e^6 + 54*a*c*e^8 - 54*b*c*d*e^7))*(-(9*(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b
*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))^(1/2) - 2*atanh((2*((d + e*x)
^(1/2)*(36*a*c^2*e^6 - 18*b^2*c*e^6 - 36*c^3*d^2*e^4 + 36*b*c^2*d*e^5) - (9*(d + e*x)^(1/2)*(8*b^3*c^2*e^3 - 1
6*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2)*(e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a
*c^2*d*e^2 + 2*b^2*c*d*e^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))*((9*(e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e
^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))^(1/2))/(54*c^2*d^2*
e^6 + 54*a*c*e^8 - 54*b*c*d*e^7))*((9*(e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 +
2*b^2*c*d*e^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))^(1/2) - (e^2*(d + e*x)^(3/2))/((b*e - 2*c*d)*(d + e*x)
+ c*(d + e*x)^2 + a*e^2 + c*d^2 - b*d*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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