3.5.51 \(\int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx\)

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

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Rubi [A]  time = 0.04, 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 = {721, 725, 206} \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]

-((a*e - c*d*x)*Sqrt[a + c*x^2])/(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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 721

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 725

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) \operatorname {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 = 0.11, size = 127, normalized size = 1.23 \begin {gather*} \frac {\sqrt {a+c x^2} \sqrt {a e^2+c d^2} (c d x-a e)-a c (d+e x)^2 \log \left (\sqrt {a+c x^2} \sqrt {a e^2+c d^2}+a e-c d x\right )+a c (d+e x)^2 \log (d+e x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c*d^2 + a*e^2]*(-(a*e) + c*d*x)*Sqrt[a + c*x^2] + a*c*(d + e*x)^2*Log[d + e*x] - a*c*(d + e*x)^2*Log[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)*(d + e*x)^2)

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IntegrateAlgebraic [A]  time = 0.90, size = 160, normalized size = 1.55 \begin {gather*} \frac {\sqrt {a+c x^2} (c d x-a e)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}+\frac {a c \sqrt {-a e^2-c d^2} \tan ^{-1}\left (-\frac {e \sqrt {a+c x^2}}{\sqrt {-a e^2-c d^2}}+\frac {\sqrt {c} e x}{\sqrt {-a e^2-c d^2}}+\frac {\sqrt {c} d}{\sqrt {-a e^2-c d^2}}\right )}{\left (a e^2+c d^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [B]  time = 0.57, size = 519, normalized size = 5.04 \begin {gather*} \left [\frac {{\left (a c e^{2} x^{2} + 2 \, a c d e x + a c d^{2}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (a c d^{2} e + a^{2} e^{3} - {\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{2} d^{6} + 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} + {\left (c^{2} d^{4} e^{2} + 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e + 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )}}, -\frac {{\left (a c e^{2} x^{2} + 2 \, a c d e x + 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}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (a c d^{2} e + a^{2} e^{3} - {\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{2} d^{6} + 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} + {\left (c^{2} d^{4} e^{2} + 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e + 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\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*e^2*x^2 + 2*a*c*d*e*x + a*c*d^2)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^
2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(a*
c*d^2*e + a^2*e^3 - (c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*x^2 + a))/(c^2*d^6 + 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^
4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e + 2*a*c*d^3*e^3 + a^2*d*e^5)*x), -1/2*((a*c*e^2*x^2 + 2*a*
c*d*e*x + 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)/(a*c*d^2 + a
^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) + (a*c*d^2*e + a^2*e^3 - (c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*x^2 + a))/(c^2*d^6
+ 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e + 2*a*c*d^3*e^3 +
a^2*d*e^5)*x)]

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giac [B]  time = 0.24, 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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maple [B]  time = 0.06, size = 1174, normalized size = 11.40

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.66, size = 243, normalized size = 2.36 \begin {gather*} -\frac {\sqrt {c x^{2} + a} c d}{2 \, {\left (c d^{2} e^{2} x + a e^{4} x + c d^{3} e + a d e^{3}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{2 \, {\left (c d^{2} e x^{2} + a e^{3} x^{2} + 2 \, c d^{3} x + 2 \, a d e^{2} x + \frac {c d^{4}}{e} + a d^{2} e\right )}} + \frac {\sqrt {c x^{2} + a} c}{2 \, {\left (c d^{2} e + a e^{3}\right )}} - \frac {c^{2} d^{2} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{2 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {3}{2}} e^{5}} + \frac {c \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{2 \, \sqrt {a + \frac {c d^{2}}{e^{2}}} e^{3}} \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*sqrt(c*x^2 + a)*c*d/(c*d^2*e^2*x + a*e^4*x + c*d^3*e + a*d*e^3) - 1/2*(c*x^2 + a)^(3/2)/(c*d^2*e*x^2 + a*
e^3*x^2 + 2*c*d^3*x + 2*a*d*e^2*x + c*d^4/e + a*d^2*e) + 1/2*sqrt(c*x^2 + a)*c/(c*d^2*e + a*e^3) - 1/2*c^2*d^2
*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(3/2)*e^5) + 1/2*c*ar
csinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/(sqrt(a + c*d^2/e^2)*e^3)

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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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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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