\(\int \frac {(c+d x^2)^{3/2}}{x^2 (a+b x^2)} \, dx\) [691]

Optimal result

Integrand size = 24, antiderivative size = 102 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c \sqrt {c+d x^2}}{a x}-\frac {(b c-a d)^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} b}+\frac {d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {485, 537, 223, 212, 385, 211} \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=-\frac {(b c-a d)^{3/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} b}-\frac {c \sqrt {c+d x^2}}{a x}+\frac {d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b} \]

[In]

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

[Out]

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

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 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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 485

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

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.26 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=\frac {(b c-a d)^{3/2} x \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )-\sqrt {a} \left (b c \sqrt {c+d x^2}+a d^{3/2} x \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )\right )}{a^{3/2} b x} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.00 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\frac {-x \left (a d -b c \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )+\left (\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) d^{\frac {3}{2}} a x -b c \sqrt {d \,x^{2}+c}\right ) \sqrt {\left (a d -b c \right ) a}}{\sqrt {\left (a d -b c \right ) a}\, a x b}\)	\(114\)
risch	\(-\frac {c \sqrt {d \,x^{2}+c}}{a x}+\frac {\frac {d^{\frac {3}{2}} a \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b}-\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{a}\)	\(403\)
default	\(\text {Expression too large to display}\)	\(1330\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 718, normalized size of antiderivative = 7.04 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=\left [\frac {2 \, a d^{\frac {3}{2}} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - {\left (b c - a d\right )} x \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 \, \sqrt {d x^{2} + c} b c}{4 \, a b x}, -\frac {4 \, a \sqrt {-d} d x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (b c - a d\right )} x \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 \, \sqrt {d x^{2} + c} b c}{4 \, a b x}, \frac {a d^{\frac {3}{2}} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - {\left (b c - a d\right )} x \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 \, \sqrt {d x^{2} + c} b c}{2 \, a b x}, -\frac {2 \, a \sqrt {-d} d x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (b c - a d\right )} x \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 \, \sqrt {d x^{2} + c} b c}{2 \, a b x}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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