\(\int \frac {(c+d x^2)^{5/2}}{x^4 (a+b x^2)} \, dx\) [702]

Optimal result

Integrand size = 24, antiderivative size = 130 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\frac {c (b c-2 a d) \sqrt {c+d x^2}}{a^2 x}-\frac {c \left (c+d x^2\right )^{3/2}}{3 a x^3}+\frac {(b c-a d)^{5/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} b}+\frac {d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {485, 594, 537, 223, 212, 385, 211} \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\frac {(b c-a d)^{5/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} b}+\frac {c \sqrt {c+d x^2} (b c-2 a d)}{a^2 x}-\frac {c \left (c+d x^2\right )^{3/2}}{3 a x^3}+\frac {d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b} \]

[In]

Int[(c + d*x^2)^(5/2)/(x^4*(a + b*x^2)),x]

[Out]

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

Rule 594

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/(a*g*(m + 1))), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=-\frac {c \sqrt {c+d x^2} \left (-3 b c x^2+a \left (c+7 d x^2\right )\right )}{3 a^2 x^3}-\frac {(b c-a d)^{5/2} \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{5/2} b}-\frac {d^{5/2} \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{b} \]

[In]

Integrate[(c + d*x^2)^(5/2)/(x^4*(a + b*x^2)),x]

[Out]

-1/3*(c*Sqrt[c + d*x^2]*(-3*b*c*x^2 + a*(c + 7*d*x^2)))/(a^2*x^3) - ((b*c - a*d)^(5/2)*ArcTan[(a*Sqrt[d] + b*x
*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(a^(5/2)*b) - (d^(5/2)*Log[-(Sqrt[d]*x) + Sqrt[c +
d*x^2]])/b

Maple [A] (verified)

Time = 3.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(\frac {-x^{3} \left (a d -b c \right )^{3} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )+\sqrt {\left (a d -b c \right ) a}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) d^{\frac {5}{2}} a^{2} x^{3}-\frac {\left (\left (7 d \,x^{2}+c \right ) a -3 c b \,x^{2}\right ) b c \sqrt {d \,x^{2}+c}}{3}\right )}{\sqrt {\left (a d -b c \right ) a}\, a^{2} x^{3} b}\)	\(138\)
risch	\(-\frac {\sqrt {d \,x^{2}+c}\, c \left (7 a d \,x^{2}-3 c b \,x^{2}+a c \right )}{3 a^{2} x^{3}}+\frac {\frac {a^{2} d^{\frac {5}{2}} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b}-\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{a^{2}}\)	\(451\)
default	\(\text {Expression too large to display}\)	\(2302\)

[In]

int((d*x^2+c)^(5/2)/x^4/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

(-x^3*(a*d-b*c)^3*arctanh((d*x^2+c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2))+((a*d-b*c)*a)^(1/2)*(arctanh((d*x^2+c)^(1/2
)/x/d^(1/2))*d^(5/2)*a^2*x^3-1/3*((7*d*x^2+c)*a-3*c*b*x^2)*b*c*(d*x^2+c)^(1/2)))/((a*d-b*c)*a)^(1/2)/a^2/x^3/b

Fricas [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 901, normalized size of antiderivative = 6.93 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\left [\frac {6 \, a^{2} d^{\frac {5}{2}} x^{3} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \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 (a b c^{2} - {\left (3 \, b^{2} c^{2} - 7 \, a b c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, a^{2} b x^{3}}, -\frac {12 \, a^{2} \sqrt {-d} d^{2} x^{3} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \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 (a b c^{2} - {\left (3 \, b^{2} c^{2} - 7 \, a b c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, a^{2} b x^{3}}, \frac {3 \, a^{2} d^{\frac {5}{2}} x^{3} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \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 (a b c^{2} - {\left (3 \, b^{2} c^{2} - 7 \, a b c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, a^{2} b x^{3}}, -\frac {6 \, a^{2} \sqrt {-d} d^{2} x^{3} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \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 (a b c^{2} - {\left (3 \, b^{2} c^{2} - 7 \, a b c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, a^{2} b x^{3}}\right ] \]

[In]

integrate((d*x^2+c)^(5/2)/x^4/(b*x^2+a),x, algorithm="fricas")

[Out]

[1/12*(6*a^2*d^(5/2)*x^3*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x
^3*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*(a
*b*c^2 - (3*b^2*c^2 - 7*a*b*c*d)*x^2)*sqrt(d*x^2 + c))/(a^2*b*x^3), -1/12*(12*a^2*sqrt(-d)*d^2*x^3*arctan(sqrt
(-d)*x/sqrt(d*x^2 + c)) - 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^3*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*(a*b*c^2 - (3*b^2*c^2 - 7*a*b*c*d)*x^2)*sqrt(d*x^2
+ c))/(a^2*b*x^3), 1/6*(3*a^2*d^(5/2)*x^3*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 3*(b^2*c^2 - 2*a*b
*c*d + a^2*d^2)*x^3*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*(a*b*c^2 - (3*b^2*c^2 - 7*a*b*c*d)*x^2)*sqrt(d*x^2 + c))/(a^
2*b*x^3), -1/6*(6*a^2*sqrt(-d)*d^2*x^3*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*
x^3*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*(a*b*c^2 - (3*b^2*c^2 - 7*a*b*c*d)*x^2)*sqrt(d*x^2 + c))/(a^2*b*x^3)]

Sympy [F]

\[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{x^{4} \left (a + b x^{2}\right )}\, dx \]

[In]

integrate((d*x**2+c)**(5/2)/x**4/(b*x**2+a),x)

[Out]

Integral((c + d*x**2)**(5/2)/(x**4*(a + b*x**2)), x)

Maxima [F]

\[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )} x^{4}} \,d x } \]

[In]

integrate((d*x^2+c)^(5/2)/x^4/(b*x^2+a),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)*x^4), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((d*x^2+c)^(5/2)/x^4/(b*x^2+a),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{5/2}}{x^4\,\left (b\,x^2+a\right )} \,d x \]

[In]

int((c + d*x^2)^(5/2)/(x^4*(a + b*x^2)),x)

[Out]

int((c + d*x^2)^(5/2)/(x^4*(a + b*x^2)), x)