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

Optimal. Leaf size=149 \[ \frac {a (4 b c-3 a d) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2}}+\frac {x \sqrt {a+b x^2} (4 b c-3 a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-1/4*d*x*(b*x^2+a)^(3/2)/c/(-a*d+b*c)/(d*x^2+c)^2+1/8*a*(-3*a*d+4*b*c)*arctanh(x*(-a*d+b*c)^(1/2)/c^(1/2)/(b*x
^2+a)^(1/2))/c^(5/2)/(-a*d+b*c)^(3/2)+1/8*(-3*a*d+4*b*c)*x*(b*x^2+a)^(1/2)/c^2/(-a*d+b*c)/(d*x^2+c)

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Rubi [A]  time = 0.09, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {382, 378, 377, 208} \[ \frac {x \sqrt {a+b x^2} (4 b c-3 a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)}+\frac {a (4 b c-3 a d) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2}}-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x^2]/(c + d*x^2)^3,x]

[Out]

-(d*x*(a + b*x^2)^(3/2))/(4*c*(b*c - a*d)*(c + d*x^2)^2) + ((4*b*c - 3*a*d)*x*Sqrt[a + b*x^2])/(8*c^2*(b*c - a
*d)*(c + d*x^2)) + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(b*c
- a*d)^(3/2))

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 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 378

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] - Dist[(c*q)/(a*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3} \, dx &=-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {(4 b c-3 a d) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {(4 b c-3 a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d) \left (c+d x^2\right )}+\frac {(a (4 b c-3 a d)) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {(4 b c-3 a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d) \left (c+d x^2\right )}+\frac {(a (4 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 c^2 (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {(4 b c-3 a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d) \left (c+d x^2\right )}+\frac {a (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.57, size = 176, normalized size = 1.18 \[ \frac {x \left (c \left (a^2 d \left (5 c+3 d x^2\right )+a b \left (-4 c^2+3 c d x^2+3 d^2 x^4\right )-2 b^2 c x^2 \left (2 c+d x^2\right )\right )+\frac {a \left (c+d x^2\right )^2 (3 a d-4 b c) \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}}\right )}{8 c^3 \sqrt {a+b x^2} \left (c+d x^2\right )^2 (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c*(-2*b^2*c*x^2*(2*c + d*x^2) + a^2*d*(5*c + 3*d*x^2) + a*b*(-4*c^2 + 3*c*d*x^2 + 3*d^2*x^4)) + (a*(-4*b*c
+ 3*a*d)*(c + d*x^2)^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2
))]))/(8*c^3*(-(b*c) + a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^2)

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fricas [B]  time = 1.41, size = 698, normalized size = 4.68 \[ \left [\frac {{\left (4 \, a b c^{3} - 3 \, a^{2} c^{2} d + {\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + 2 \, {\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left ({\left (2 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{3} + {\left (4 \, b^{2} c^{4} - 9 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{32 \, {\left (b^{2} c^{7} - 2 \, a b c^{6} d + a^{2} c^{5} d^{2} + {\left (b^{2} c^{5} d^{2} - 2 \, a b c^{4} d^{3} + a^{2} c^{3} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3}\right )} x^{2}\right )}}, -\frac {{\left (4 \, a b c^{3} - 3 \, a^{2} c^{2} d + {\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + 2 \, {\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left ({\left (2 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{3} + {\left (4 \, b^{2} c^{4} - 9 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (b^{2} c^{7} - 2 \, a b c^{6} d + a^{2} c^{5} d^{2} + {\left (b^{2} c^{5} d^{2} - 2 \, a b c^{4} d^{3} + a^{2} c^{3} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3}\right )} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/32*((4*a*b*c^3 - 3*a^2*c^2*d + (4*a*b*c*d^2 - 3*a^2*d^3)*x^4 + 2*(4*a*b*c^2*d - 3*a^2*c*d^2)*x^2)*sqrt(b*c^
2 - a*c*d)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 + 4*((2*b*c -
a*d)*x^3 + a*c*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + 4*((2*b^2*c^3*d - 5*a*b*
c^2*d^2 + 3*a^2*c*d^3)*x^3 + (4*b^2*c^4 - 9*a*b*c^3*d + 5*a^2*c^2*d^2)*x)*sqrt(b*x^2 + a))/(b^2*c^7 - 2*a*b*c^
6*d + a^2*c^5*d^2 + (b^2*c^5*d^2 - 2*a*b*c^4*d^3 + a^2*c^3*d^4)*x^4 + 2*(b^2*c^6*d - 2*a*b*c^5*d^2 + a^2*c^4*d
^3)*x^2), -1/16*((4*a*b*c^3 - 3*a^2*c^2*d + (4*a*b*c*d^2 - 3*a^2*d^3)*x^4 + 2*(4*a*b*c^2*d - 3*a^2*c*d^2)*x^2)
*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 + a)/((b^2*c^2 - a*
b*c*d)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - 2*((2*b^2*c^3*d - 5*a*b*c^2*d^2 + 3*a^2*c*d^3)*x^3 + (4*b^2*c^4 - 9*a*b
*c^3*d + 5*a^2*c^2*d^2)*x)*sqrt(b*x^2 + a))/(b^2*c^7 - 2*a*b*c^6*d + a^2*c^5*d^2 + (b^2*c^5*d^2 - 2*a*b*c^4*d^
3 + a^2*c^3*d^4)*x^4 + 2*(b^2*c^6*d - 2*a*b*c^5*d^2 + a^2*c^4*d^3)*x^2)]

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giac [B]  time = 3.75, size = 487, normalized size = 3.27 \[ -\frac {{\left (4 \, a b^{\frac {3}{2}} c - 3 \, a^{2} \sqrt {b} d\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{8 \, \sqrt {-b^{2} c^{2} + a b c d} {\left (b c^{3} - a c^{2} d\right )}} - \frac {4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {3}{2}} c d^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} \sqrt {b} d^{3} - 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {7}{2}} c^{3} + 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {5}{2}} c^{2} d - 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} c d^{2} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} \sqrt {b} d^{3} - 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {5}{2}} c^{2} d + 28 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} c d^{2} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} \sqrt {b} d^{3} - 2 \, a^{4} b^{\frac {3}{2}} c d^{2} + 3 \, a^{5} \sqrt {b} d^{3}}{4 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{2} {\left (b c^{3} d - a c^{2} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*a*b^(3/2)*c - 3*a^2*sqrt(b)*d)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*
c^2 + a*b*c*d))/(sqrt(-b^2*c^2 + a*b*c*d)*(b*c^3 - a*c^2*d)) - 1/4*(4*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2
)*c*d^2 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d^3 - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^3 +
40*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c^2*d - 30*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c*d^2 + 9
*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*sqrt(b)*d^3 - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c^2*d + 28*(
sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(3/2)*c*d^2 - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*sqrt(b)*d^3 - 2*a^4*b
^(3/2)*c*d^2 + 3*a^5*sqrt(b)*d^3)/(((sqrt(b)*x - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c
- 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)^2*(b*c^3*d - a*c^2*d^2))

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maple [B]  time = 0.02, size = 5101, normalized size = 34.23 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {b\,x^2+a}}{{\left (d\,x^2+c\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((b*x**2+a)**(1/2)/(d*x**2+c)**3,x)

[Out]

Timed out

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