3.7.27 \(\int \frac {\sqrt {d+e x}}{(a-c x^2)^2} \, dx\) [627]

Optimal. Leaf size=194 \[ \frac {x \sqrt {d+e x}}{2 a \left (a-c x^2\right )}-\frac {\left (\frac {2 \sqrt {c} d}{\sqrt {a}}-e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (\frac {2 \sqrt {c} d}{\sqrt {a}}+e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a c^{3/4} \sqrt {\sqrt {c} d+\sqrt {a} e}} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {751, 841, 1180, 214} \begin {gather*} -\frac {\left (\frac {2 \sqrt {c} d}{\sqrt {a}}-e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (\frac {2 \sqrt {c} d}{\sqrt {a}}+e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a c^{3/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}+\frac {x \sqrt {d+e x}}{2 a \left (a-c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(a - c*x^2)^2,x]

[Out]

(x*Sqrt[d + e*x])/(2*a*(a - c*x^2)) - (((2*Sqrt[c]*d)/Sqrt[a] - e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c
]*d - Sqrt[a]*e]])/(4*a*c^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + (((2*Sqrt[c]*d)/Sqrt[a] + e)*ArcTanh[(c^(1/4)*S
qrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e])

Rule 214

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

Rule 751

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x)^m*((a + c*x^2)^(p +
1)/(2*a*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(d*(2*p + 3) + e*(m + 2*p + 3)*x)*(a + c*x
^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[m, 1
] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 841

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

Rule 1180

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

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Mathematica [A]
time = 0.51, size = 226, normalized size = 1.16 \begin {gather*} \frac {\frac {2 \sqrt {a} x \sqrt {d+e x}}{a-c x^2}+\frac {\left (2 \sqrt {c} d+\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {c} \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {\left (2 \sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {c} \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{4 a^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.47, size = 251, normalized size = 1.29

method	result	size
derivativedivides	\(2 e^{3} c^{2} \left (\frac {\frac {\sqrt {a c \,e^{2}}\, \sqrt {e x +d}}{2 c^{2} \left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}+\frac {\left (\sqrt {a c \,e^{2}}+2 c d \right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 c a \,e^{2} \sqrt {a c \,e^{2}}}+\frac {\frac {\sqrt {a c \,e^{2}}\, \sqrt {e x +d}}{2 c^{2} \left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}-\frac {\left (\sqrt {a c \,e^{2}}-2 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 c a \,e^{2} \sqrt {a c \,e^{2}}}\right )\)	\(251\)
default	\(2 e^{3} c^{2} \left (\frac {\frac {\sqrt {a c \,e^{2}}\, \sqrt {e x +d}}{2 c^{2} \left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}+\frac {\left (\sqrt {a c \,e^{2}}+2 c d \right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 c a \,e^{2} \sqrt {a c \,e^{2}}}+\frac {\frac {\sqrt {a c \,e^{2}}\, \sqrt {e x +d}}{2 c^{2} \left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}-\frac {\left (\sqrt {a c \,e^{2}}-2 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 c a \,e^{2} \sqrt {a c \,e^{2}}}\right )\)	\(251\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1274 vs. \(2 (152) = 304\).
time = 3.77, size = 1274, normalized size = 6.57 \begin {gather*} -\frac {{\left (a c x^{2} - a^{2}\right )} \sqrt {\frac {4 \, c d^{3} - 3 \, a d e^{2} + \frac {{\left (a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}}{a^{3} c^{2} d^{2} - a^{4} c e^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt {x e + d} + {\left (a^{2} c d e^{4} - \frac {{\left (2 \, a^{3} c^{4} d^{4} - 3 \, a^{4} c^{3} d^{2} e^{2} + a^{5} c^{2} e^{4}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}\right )} \sqrt {\frac {4 \, c d^{3} - 3 \, a d e^{2} + \frac {{\left (a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}}{a^{3} c^{2} d^{2} - a^{4} c e^{2}}}\right ) - {\left (a c x^{2} - a^{2}\right )} \sqrt {\frac {4 \, c d^{3} - 3 \, a d e^{2} + \frac {{\left (a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}}{a^{3} c^{2} d^{2} - a^{4} c e^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt {x e + d} - {\left (a^{2} c d e^{4} - \frac {{\left (2 \, a^{3} c^{4} d^{4} - 3 \, a^{4} c^{3} d^{2} e^{2} + a^{5} c^{2} e^{4}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}\right )} \sqrt {\frac {4 \, c d^{3} - 3 \, a d e^{2} + \frac {{\left (a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}}{a^{3} c^{2} d^{2} - a^{4} c e^{2}}}\right ) + {\left (a c x^{2} - a^{2}\right )} \sqrt {\frac {4 \, c d^{3} - 3 \, a d e^{2} - \frac {{\left (a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}}{a^{3} c^{2} d^{2} - a^{4} c e^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt {x e + d} + {\left (a^{2} c d e^{4} + \frac {{\left (2 \, a^{3} c^{4} d^{4} - 3 \, a^{4} c^{3} d^{2} e^{2} + a^{5} c^{2} e^{4}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}\right )} \sqrt {\frac {4 \, c d^{3} - 3 \, a d e^{2} - \frac {{\left (a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}}{a^{3} c^{2} d^{2} - a^{4} c e^{2}}}\right ) - {\left (a c x^{2} - a^{2}\right )} \sqrt {\frac {4 \, c d^{3} - 3 \, a d e^{2} - \frac {{\left (a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}}{a^{3} c^{2} d^{2} - a^{4} c e^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt {x e + d} - {\left (a^{2} c d e^{4} + \frac {{\left (2 \, a^{3} c^{4} d^{4} - 3 \, a^{4} c^{3} d^{2} e^{2} + a^{5} c^{2} e^{4}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}\right )} \sqrt {\frac {4 \, c d^{3} - 3 \, a d e^{2} - \frac {{\left (a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} e^{3}}{\sqrt {a^{3} c^{5} d^{4} - 2 \, a^{4} c^{4} d^{2} e^{2} + a^{5} c^{3} e^{4}}}}{a^{3} c^{2} d^{2} - a^{4} c e^{2}}}\right ) + 4 \, \sqrt {x e + d} x}{8 \, {\left (a c x^{2} - a^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
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((e*x+d)**(1/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (152) = 304\).
time = 3.64, size = 342, normalized size = 1.76 \begin {gather*} \frac {{\left (2 \, a c d^{2} {\left | c \right |} - \sqrt {a c} d {\left | a \right |} {\left | c \right |} e - a^{2} {\left | c \right |} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c d + \sqrt {a^{2} c^{2} d^{2} - {\left (a c d^{2} - a^{2} e^{2}\right )} a c}}{a c}}}\right )}{4 \, {\left (a^{2} c e - \sqrt {a c} a c d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |}} + \frac {{\left (a c d {\left | a \right |} {\left | c \right |} e + 2 \, \sqrt {a c} a c d^{2} {\left | c \right |} - \sqrt {a c} a^{2} {\left | c \right |} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c d - \sqrt {a^{2} c^{2} d^{2} - {\left (a c d^{2} - a^{2} e^{2}\right )} a c}}{a c}}}\right )}{4 \, {\left (a^{2} c^{2} d + \sqrt {a c} a^{2} c e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |}} - \frac {{\left (x e + d\right )}^{\frac {3}{2}} e - \sqrt {x e + d} d e}{2 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 2.19, size = 2332, normalized size = 12.02 \begin {gather*} -\frac {\frac {e\,{\left (d+e\,x\right )}^{3/2}}{2\,a}-\frac {d\,e\,\sqrt {d+e\,x}}{2\,a}}{c\,{\left (d+e\,x\right )}^2-a\,e^2+c\,d^2-2\,c\,d\,\left (d+e\,x\right )}-\mathrm {atan}\left (\frac {\left (\left (8\,c^3\,d\,e^3-64\,a\,c^4\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\right )\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}+\frac {\left (4\,c^3\,d^2\,e^2+a\,c^2\,e^4\right )\,\sqrt {d+e\,x}}{a^2}\right )\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\,1{}\mathrm {i}-\left (\left (8\,c^3\,d\,e^3+64\,a\,c^4\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\right )\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}-\frac {\left (4\,c^3\,d^2\,e^2+a\,c^2\,e^4\right )\,\sqrt {d+e\,x}}{a^2}\right )\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\,1{}\mathrm {i}}{\left (\left (8\,c^3\,d\,e^3-64\,a\,c^4\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\right )\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}+\frac {\left (4\,c^3\,d^2\,e^2+a\,c^2\,e^4\right )\,\sqrt {d+e\,x}}{a^2}\right )\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}-\frac {4\,c^2\,d^2\,e^3-a\,c\,e^5}{4\,a^3}+\left (\left (8\,c^3\,d\,e^3+64\,a\,c^4\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\right )\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}-\frac {\left (4\,c^3\,d^2\,e^2+a\,c^2\,e^4\right )\,\sqrt {d+e\,x}}{a^2}\right )\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}}\right )\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^3}-4\,a^3\,c^3\,d^3+3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\left (\left (8\,c^3\,d\,e^3-64\,a\,c^4\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\right )\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}+\frac {\left (4\,c^3\,d^2\,e^2+a\,c^2\,e^4\right )\,\sqrt {d+e\,x}}{a^2}\right )\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\,1{}\mathrm {i}-\left (\left (8\,c^3\,d\,e^3+64\,a\,c^4\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\right )\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}-\frac {\left (4\,c^3\,d^2\,e^2+a\,c^2\,e^4\right )\,\sqrt {d+e\,x}}{a^2}\right )\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\,1{}\mathrm {i}}{\left (\left (8\,c^3\,d\,e^3-64\,a\,c^4\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\right )\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}+\frac {\left (4\,c^3\,d^2\,e^2+a\,c^2\,e^4\right )\,\sqrt {d+e\,x}}{a^2}\right )\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}-\frac {4\,c^2\,d^2\,e^3-a\,c\,e^5}{4\,a^3}+\left (\left (8\,c^3\,d\,e^3+64\,a\,c^4\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\right )\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}-\frac {\left (4\,c^3\,d^2\,e^2+a\,c^2\,e^4\right )\,\sqrt {d+e\,x}}{a^2}\right )\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}}\right )\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^3}+4\,a^3\,c^3\,d^3-3\,a^4\,c^2\,d\,e^2}{64\,\left (a^6\,c^4\,d^2-a^7\,c^3\,e^2\right )}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)/(a - c*x^2)^2,x)

[Out]

- atan((((8*c^3*d*e^3 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^
2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6
*c^4*d^2 - a^7*c^3*e^2)))^(1/2) + ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-(e^3*(a^9*c^3)^(1/2) -
4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)*1i - ((8*c^3*d*e^3 + 64*a*c^4*d*e^2*(
d + e*x)^(1/2)*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/
2))*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) - ((a*c^
2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^
6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)*1i)/(((8*c^3*d*e^3 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(e^3*(a^9*c^3)^(1/2) -
4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d
^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) + ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/
a^2)*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) - (4*c^
2*d^2*e^3 - a*c*e^5)/(4*a^3) + ((8*c^3*d*e^3 + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c
^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*
a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) - ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-
(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)))*(-(e^3*(a^9*
c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)*2i - atan((((8*c^3*d*e^3
- 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 -
a^7*c^3*e^2)))^(1/2))*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)
))^(1/2) + ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^
2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)*1i - ((8*c^3*d*e^3 + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((e^3*(a^
9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*((e^3*(a^9*c^3)^(1/2)
+ 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) - ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d
+ e*x)^(1/2))/a^2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^
(1/2)*1i)/(((8*c^3*d*e^3 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*
e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^
6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) + ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*((e^3*(a^9*c^3)^(1/2) +
4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) - (4*c^2*d^2*e^3 - a*c*e^5)/(4*a^3) +
((8*c^3*d*e^3 + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(
a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2
- a^7*c^3*e^2)))^(1/2) - ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*
d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)))*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4
*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)*2i - ((e*(d + e*x)^(3/2))/(2*a) - (d*e*(d + e*x)^(1/2))/(2
*a))/(c*(d + e*x)^2 - a*e^2 + c*d^2 - 2*c*d*(d + e*x))

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