\(\int \frac {(c+d x^2)^{3/2}}{x^2 (a+b x^2)^2} \, dx\) [746]

Optimal result

Integrand size = 24, antiderivative size = 128 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {(3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}-\frac {3 c \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {479, 597, 12, 385, 211} \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {3 c \sqrt {b c-a d} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2}}-\frac {\sqrt {c+d x^2} (3 b c-a d)}{2 a^2 b x}+\frac {\sqrt {c+d x^2} (b c-a d)}{2 a b x \left (a+b x^2\right )} \]

[In]

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

[Out]

-1/2*((3*b*c - a*d)*Sqrt[c + d*x^2])/(a^2*b*x) + ((b*c - a*d)*Sqrt[c + d*x^2])/(2*a*b*x*(a + b*x^2)) - (3*c*Sq
rt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

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

Rule 385

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 479

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

Rule 597

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

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}-\frac {\int \frac {-c (3 b c-a d)-2 b c d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a b} \\ & = -\frac {(3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}+\frac {\int -\frac {3 b c^2 (b c-a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^2 b c} \\ & = -\frac {(3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}-\frac {(3 c (b c-a d)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^2} \\ & = -\frac {(3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}-\frac {(3 c (b c-a d)) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a^2} \\ & = -\frac {(3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}-\frac {3 c \sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {c+d x^2} \left (-2 a c-3 b c x^2+a d x^2\right )}{2 a^2 x \left (a+b x^2\right )}+\frac {3 c \sqrt {b c-a d} \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{5/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(-\frac {c \left (\frac {2 \sqrt {d \,x^{2}+c}}{x}+\left (a d -b c \right ) \left (-\frac {\sqrt {d \,x^{2}+c}\, x}{c \left (b \,x^{2}+a \right )}-\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )\right )}{2 a^{2}}\)	\(97\)
risch	\(-\frac {c \sqrt {d \,x^{2}+c}}{a^{2} x}+\frac {\frac {\left (\frac {1}{4} a^{2} d^{2}-\frac {1}{2} a b c d +\frac {1}{4} b^{2} c^{2}\right ) \left (\frac {b \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{2}}+\frac {\left (\frac {1}{4} a^{2} d^{2}-\frac {1}{2} a b c d +\frac {1}{4} b^{2} c^{2}\right ) \left (\frac {b \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{2}}-\frac {\left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (-a^{2} d^{2}-2 a b c d +3 b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{a^{2}}\)	\(928\)
default	\(\text {Expression too large to display}\)	\(3462\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.74 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\left [\frac {3 \, {\left (b c x^{3} + a c x\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (3 \, b c - a d\right )} x^{2} + 2 \, a c\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{2} b x^{3} + a^{3} x\right )}}, -\frac {3 \, {\left (b c x^{3} + a c x\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (3 \, b c - a d\right )} x^{2} + 2 \, a c\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{2} b x^{3} + a^{3} x\right )}}\right ] \]

[In]

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

[Out]

[1/8*(3*(b*c*x^3 + a*c*x)*sqrt(-(b*c - a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b
*c^2 - 4*a^2*c*d)*x^2 + 4*(a^2*c*x - (a*b*c - 2*a^2*d)*x^3)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^4 + 2
*a*b*x^2 + a^2)) - 4*((3*b*c - a*d)*x^2 + 2*a*c)*sqrt(d*x^2 + c))/(a^2*b*x^3 + a^3*x), -1/4*(3*(b*c*x^3 + a*c*
x)*sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^
2)*x^3 + (b*c^2 - a*c*d)*x)) + 2*((3*b*c - a*d)*x^2 + 2*a*c)*sqrt(d*x^2 + c))/(a^2*b*x^3 + a^3*x)]

Sympy [F]

\[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}}}{x^{2} \left (a + b x^{2}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (108) = 216\).

Time = 0.84 (sec) , antiderivative size = 412, normalized size of antiderivative = 3.22 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {3 \, {\left (b c^{2} \sqrt {d} - a c d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a^{2}} + \frac {3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{2} \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c d^{\frac {3}{2}} + 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} d^{\frac {5}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{3} \sqrt {d} + 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{2} d^{\frac {3}{2}} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c d^{\frac {5}{2}} + 3 \, b^{2} c^{4} \sqrt {d} - a b c^{3} d^{\frac {3}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d - b c^{3}\right )} a^{2} b} \]

[In]

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

[Out]

3/2*(b*c^2*sqrt(d) - a*c*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d -
a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*a^2) + (3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^2*sqrt(d) - 3*(sqrt(d)*x -
sqrt(d*x^2 + c))^4*a*b*c*d^(3/2) + 2*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*d^(5/2) - 6*(sqrt(d)*x - sqrt(d*x^2 +
c))^2*b^2*c^3*sqrt(d) + 12*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^2*d^(3/2) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^
2*a^2*c*d^(5/2) + 3*b^2*c^4*sqrt(d) - a*b*c^3*d^(3/2))/(((sqrt(d)*x - sqrt(d*x^2 + c))^6*b - 3*(sqrt(d)*x - sq
rt(d*x^2 + c))^4*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*d + 3*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c^2 - 4*(sq
rt(d)*x - sqrt(d*x^2 + c))^2*a*c*d - b*c^3)*a^2*b)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}}{x^2\,{\left (b\,x^2+a\right )}^2} \,d x \]

[In]

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

[Out]

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