3.6.31 \(\int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx\) [531]

Optimal. Leaf size=103 \[ -\frac {(a e-c d x) \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {a c \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {735, 739, 212} \begin {gather*} -\frac {\sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac {a c \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^2]/(d + e*x)^3,x]

[Out]

-1/2*((a*e - c*d*x)*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x)^2) - (a*c*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 +
a*e^2]*Sqrt[a + c*x^2])])/(2*(c*d^2 + 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 735

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

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rubi steps

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

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Mathematica [A]
time = 0.65, size = 110, normalized size = 1.07 \begin {gather*} \frac {(-a e+c d x) \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}+\frac {a c \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(899\) vs. \(2(91)=182\).
time = 0.43, size = 900, normalized size = 8.74

method	result	size
default	\(\frac {-\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{2 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {c d e \left (-\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}+\frac {2 c \,e^{2} \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{e^{2} a +c \,d^{2}}\right )}{2 e^{2} a +2 c \,d^{2}}+\frac {c \,e^{2} \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 e^{2} a +2 c \,d^{2}}}{e^{3}}\)	\(900\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (92) = 184\).
time = 0.32, size = 242, normalized size = 2.35 \begin {gather*} -\frac {c^{2} d^{2} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-5\right )}}{2 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} + \frac {c \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-3\right )}}{2 \, \sqrt {c d^{2} e^{\left (-2\right )} + a}} - \frac {\sqrt {c x^{2} + a} c d}{2 \, {\left (c d^{2} x e^{2} + c d^{3} e + a x e^{4} + a d e^{3}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{2 \, {\left (c d^{2} x^{2} e + c d^{4} e^{\left (-1\right )} + 2 \, c d^{3} x + a x^{2} e^{3} + 2 \, a d x e^{2} + a d^{2} e\right )}} + \frac {\sqrt {c x^{2} + a} c}{2 \, {\left (c d^{2} e + a e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (92) = 184\).
time = 1.60, size = 511, normalized size = 4.96 \begin {gather*} \left [\frac {{\left (a c x^{2} e^{2} + 2 \, a c d x e + a c d^{2}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (c^{2} d^{3} x + a c d x e^{2} - a c d^{2} e - a^{2} e^{3}\right )} \sqrt {c x^{2} + a}}{4 \, {\left (2 \, c^{2} d^{5} x e + c^{2} d^{6} + 4 \, a c d^{3} x e^{3} + a^{2} x^{2} e^{6} + 2 \, a^{2} d x e^{5} + {\left (2 \, a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{4} + {\left (c^{2} d^{4} x^{2} + 2 \, a c d^{4}\right )} e^{2}\right )}}, \frac {{\left (a c x^{2} e^{2} + 2 \, a c d x e + a c d^{2}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) + {\left (c^{2} d^{3} x + a c d x e^{2} - a c d^{2} e - a^{2} e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (2 \, c^{2} d^{5} x e + c^{2} d^{6} + 4 \, a c d^{3} x e^{3} + a^{2} x^{2} e^{6} + 2 \, a^{2} d x e^{5} + {\left (2 \, a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{4} + {\left (c^{2} d^{4} x^{2} + 2 \, a c d^{4}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + 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+a)**(1/2)/(e*x+d)**3,x)

[Out]

Integral(sqrt(a + 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. 312 vs. \(2 (92) = 184\).
time = 1.27, size = 312, normalized size = 3.03 \begin {gather*} -\frac {a c \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt {-c d^{2} - a e^{2}}} + \frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{2} d^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} d^{3} - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} d^{2} e - {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} d e^{2} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c e^{3} + a^{2} c^{\frac {3}{2}} d e^{2} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c e^{3}}{{\left (c d^{2} e^{2} + a e^{4}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*c*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c*d^2 + a*e^2)*sqrt(-c*d^2 -
a*e^2)) + (2*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^2*d^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*d^3 - 2*(s
qrt(c)*x - sqrt(c*x^2 + a))*a*c^2*d^2*e - (sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*d*e^2 + (sqrt(c)*x - sqrt(
c*x^2 + a))^3*a*c*e^3 + a^2*c^(3/2)*d*e^2 + (sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*e^3)/((c*d^2*e^2 + a*e^4)*((sq
rt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*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+a}}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^(1/2)/(d + e*x)^3,x)

[Out]

int((a + c*x^2)^(1/2)/(d + e*x)^3, x)

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