3.8.1 \(\int \frac {(c+d x^2)^{5/2}}{x^3 (a+b x^2)} \, dx\) [701]

Optimal. Leaf size=144 \[ \frac {d (b c+2 a d) \sqrt {c+d x^2}}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}+\frac {c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2}-\frac {(b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 100, 159, 162, 65, 214} \begin {gather*} -\frac {(b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2}}+\frac {c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2}+\frac {d \sqrt {c+d x^2} (2 a d+b c)}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^(5/2)/(x^3*(a + b*x^2)),x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 100

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

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 214

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

Rule 457

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

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^{5/2}}{x^3 \left (a+b x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^{5/2}}{x^2 (a+b x)} \, dx,x,x^2\right )\\ &=-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}-\frac {\text {Subst}\left (\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (2 b c-5 a d)-\frac {1}{2} d (b c+2 a d) x\right )}{x (a+b x)} \, dx,x,x^2\right )}{2 a}\\ &=\frac {d (b c+2 a d) \sqrt {c+d x^2}}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}-\frac {\text {Subst}\left (\int \frac {\frac {1}{4} b c^2 (2 b c-5 a d)+\frac {1}{4} d \left (b^2 c^2-6 a b c d+2 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{a b}\\ &=\frac {d (b c+2 a d) \sqrt {c+d x^2}}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}-\frac {\left (c^2 (2 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^2}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a^2 b}\\ &=\frac {d (b c+2 a d) \sqrt {c+d x^2}}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}-\frac {\left (c^2 (2 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^2 d}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a^2 b d}\\ &=\frac {d (b c+2 a d) \sqrt {c+d x^2}}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}+\frac {c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2}-\frac {(b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.24, size = 125, normalized size = 0.87 \begin {gather*} \frac {\frac {a \sqrt {c+d x^2} \left (-b c^2+2 a d^2 x^2\right )}{b x^2}-\frac {2 (-b c+a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{b^{3/2}}+c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^(5/2)/(x^3*(a + b*x^2)),x]

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2235\) vs. \(2(118)=236\).
time = 0.14, size = 2236, normalized size = 15.53

method	result	size
risch	\(-\frac {c^{2} \sqrt {d \,x^{2}+c}}{2 a \,x^{2}}+\frac {d^{2} \sqrt {d \,x^{2}+c}}{b}-\frac {5 c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) d}{2 a}+\frac {c^{\frac {5}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) b}{a^{2}}+\frac {a \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 ) d^{3}}{2 b^{2} \sqrt {-\frac {a d -b c}{b}}}-\frac {3 \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 ) c \,d^{2}}{2 b \sqrt {-\frac {a d -b c}{b}}}+\frac {3 \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 ) c^{2} d}{2 a \sqrt {-\frac {a d -b c}{b}}}-\frac {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 ) c^{3}}{2 a^{2} \sqrt {-\frac {a d -b c}{b}}}+\frac {a \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 ) d^{3}}{2 b^{2} \sqrt {-\frac {a d -b c}{b}}}-\frac {3 \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 ) c \,d^{2}}{2 b \sqrt {-\frac {a d -b c}{b}}}+\frac {3 \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 ) c^{2} d}{2 a \sqrt {-\frac {a d -b c}{b}}}-\frac {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 ) c^{3}}{2 a^{2} \sqrt {-\frac {a d -b c}{b}}}\)	\(1325\)
default	\(\text {Expression too large to display}\)	\(2236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^(5/2)/x^3/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

1/2*b/a^2*(1/5*(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(5/2)+d*(-a*b)^(
1/2)/b*(1/8*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/d*(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/
b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+3/16*(-4*d*(a*d-b*c)/b+4*d^2*a/b)/d*(1/4*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*
b)^(1/2)/b)/d*(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/8*(-4*d*(
a*d-b*c)/b+4*d^2*a/b)/d^(3/2)*ln((d*(-a*b)^(1/2)/b+d*(x-1/b*(-a*b)^(1/2)))/d^(1/2)+(d*(x-1/b*(-a*b)^(1/2))^2+2
*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))))-(a*d-b*c)/b*(1/3*(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-
a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+d*(-a*b)^(1/2)/b*(1/4*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b
)^(1/2)/b)/d*(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/8*(-4*d*(a
*d-b*c)/b+4*d^2*a/b)/d^(3/2)*ln((d*(-a*b)^(1/2)/b+d*(x-1/b*(-a*b)^(1/2)))/d^(1/2)+(d*(x-1/b*(-a*b)^(1/2))^2+2*
d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)))-(a*d-b*c)/b*((d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(
1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+d^(1/2)*(-a*b)^(1/2)/b*ln((d*(-a*b)^(1/2)/b+d*(x-1/b*(-a*b)^(1/
2)))/d^(1/2)+(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+(a*d-b*c)/b
/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*(d*(x-
1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2))))))+1/2*b
/a^2*(1/5*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(5/2)-d*(-a*b)^(1/2)/
b*(1/8*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2)/b)/d*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a
*b)^(1/2))-(a*d-b*c)/b)^(3/2)+3/16*(-4*d*(a*d-b*c)/b+4*d^2*a/b)/d*(1/4*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1
/2)/b)/d*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/8*(-4*d*(a*d-b
*c)/b+4*d^2*a/b)/d^(3/2)*ln((-d*(-a*b)^(1/2)/b+d*(x+1/b*(-a*b)^(1/2)))/d^(1/2)+(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(
-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))))-(a*d-b*c)/b*(1/3*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)
^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-d*(-a*b)^(1/2)/b*(1/4*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1
/2)/b)/d*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/8*(-4*d*(a*d-b
*c)/b+4*d^2*a/b)/d^(3/2)*ln((-d*(-a*b)^(1/2)/b+d*(x+1/b*(-a*b)^(1/2)))/d^(1/2)+(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(
-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)))-(a*d-b*c)/b*((d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2
)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-d^(1/2)*(-a*b)^(1/2)/b*ln((-d*(-a*b)^(1/2)/b+d*(x+1/b*(-a*b)^(1/2)
))/d^(1/2)+(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+(a*d-b*c)/b/(
-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*(d*(x+1/
b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2))))))+1/a*(-1
/2/c/x^2*(d*x^2+c)^(7/2)+5/2*d/c*(1/5*(d*x^2+c)^(5/2)+c*(1/3*(d*x^2+c)^(3/2)+c*((d*x^2+c)^(1/2)-c^(1/2)*ln((2*
c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)))))-b/a^2*(1/5*(d*x^2+c)^(5/2)+c*(1/3*(d*x^2+c)^(3/2)+c*((d*x^2+c)^(1/2)-c^(1/
2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x))))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(5/2)/x^3/(b*x^2+a),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)*x^3), x)

________________________________________________________________________________________

Fricas [A]
time = 3.24, size = 891, normalized size = 6.19 \begin {gather*} \left [\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (2 \, a^{2} d^{2} x^{2} - a b c^{2}\right )} \sqrt {d x^{2} + c}}{4 \, a^{2} b x^{2}}, -\frac {2 \, {\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 2 \, {\left (2 \, a^{2} d^{2} x^{2} - a b c^{2}\right )} \sqrt {d x^{2} + c}}{4 \, a^{2} b x^{2}}, -\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, {\left (2 \, a^{2} d^{2} x^{2} - a b c^{2}\right )} \sqrt {d x^{2} + c}}{4 \, a^{2} b x^{2}}, -\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - {\left (2 \, a^{2} d^{2} x^{2} - a b c^{2}\right )} \sqrt {d x^{2} + c}}{2 \, a^{2} b x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(5/2)/x^3/(b*x^2+a),x, algorithm="fricas")

[Out]

[1/4*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*sqrt((b*c - a*d)/b)*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d
^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(b^2*d*x^2 + 2*b^2*c - a*b*d)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/b))/(b^2
*x^4 + 2*a*b*x^2 + a^2)) - (2*b^2*c^2 - 5*a*b*c*d)*sqrt(c)*x^2*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/
x^2) + 2*(2*a^2*d^2*x^2 - a*b*c^2)*sqrt(d*x^2 + c))/(a^2*b*x^2), -1/4*(2*(2*b^2*c^2 - 5*a*b*c*d)*sqrt(-c)*x^2*
arctan(sqrt(-c)/sqrt(d*x^2 + c)) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*sqrt((b*c - a*d)/b)*log((b^2*d^2*x^4 +
8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(b^2*d*x^2 + 2*b^2*c - a*b*d)*sqrt(d*x^2 +
c)*sqrt((b*c - a*d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 2*(2*a^2*d^2*x^2 - a*b*c^2)*sqrt(d*x^2 + c))/(a^2*b*x^
2), -1/4*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*sqrt(-(b*c - a*d)/b)*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(
d*x^2 + c)*sqrt(-(b*c - a*d)/b)/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)) + (2*b^2*c^2 - 5*a*b*c*d)*sqrt(c)*x^2*l
og(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*(2*a^2*d^2*x^2 - a*b*c^2)*sqrt(d*x^2 + c))/(a^2*b*x^2),
-1/2*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*sqrt(-(b*c - a*d)/b)*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2
+ c)*sqrt(-(b*c - a*d)/b)/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)) + (2*b^2*c^2 - 5*a*b*c*d)*sqrt(-c)*x^2*arcta
n(sqrt(-c)/sqrt(d*x^2 + c)) - (2*a^2*d^2*x^2 - a*b*c^2)*sqrt(d*x^2 + c))/(a^2*b*x^2)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{x^{3} \left (a + b x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**(5/2)/x**3/(b*x**2+a),x)

[Out]

Integral((c + d*x**2)**(5/2)/(x**3*(a + b*x**2)), x)

________________________________________________________________________________________

Giac [A]
time = 0.65, size = 158, normalized size = 1.10 \begin {gather*} \frac {\sqrt {d x^{2} + c} d^{2}}{b} - \frac {\sqrt {d x^{2} + c} c^{2}}{2 \, a x^{2}} - \frac {{\left (2 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{2} \sqrt {-c}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(5/2)/x^3/(b*x^2+a),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.74, size = 1428, normalized size = 9.92 \begin {gather*} \frac {d^2\,\sqrt {d\,x^2+c}}{b}-\frac {b\,c^2\,d\,\sqrt {d\,x^2+c}}{2\,a\,\left (b\,\left (d\,x^2+c\right )-b\,c\right )}+\frac {\mathrm {atan}\left (\frac {a^3\,d^9\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,5{}\mathrm {i}}{5\,a^3\,c^2\,d^9-\frac {395\,b^3\,c^5\,d^6}{4}+87\,a\,b^2\,c^4\,d^7-32\,a^2\,b\,c^3\,d^8+\frac {185\,b^4\,c^6\,d^5}{4\,a}-\frac {15\,b^5\,c^7\,d^4}{2\,a^2}}+\frac {a^2\,c\,d^8\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,32{}\mathrm {i}}{32\,a^2\,c^3\,d^8+\frac {395\,b^2\,c^5\,d^6}{4}-\frac {185\,b^3\,c^6\,d^5}{4\,a}-\frac {5\,a^3\,c^2\,d^9}{b}+\frac {15\,b^4\,c^7\,d^4}{2\,a^2}-87\,a\,b\,c^4\,d^7}+\frac {b^2\,c^3\,d^6\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,395{}\mathrm {i}}{4\,\left (32\,a^2\,c^3\,d^8+\frac {395\,b^2\,c^5\,d^6}{4}-\frac {185\,b^3\,c^6\,d^5}{4\,a}-\frac {5\,a^3\,c^2\,d^9}{b}+\frac {15\,b^4\,c^7\,d^4}{2\,a^2}-87\,a\,b\,c^4\,d^7\right )}-\frac {b^3\,c^4\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,185{}\mathrm {i}}{4\,\left (32\,a^3\,c^3\,d^8-\frac {185\,b^3\,c^6\,d^5}{4}+\frac {395\,a\,b^2\,c^5\,d^6}{4}-87\,a^2\,b\,c^4\,d^7+\frac {15\,b^4\,c^7\,d^4}{2\,a}-\frac {5\,a^4\,c^2\,d^9}{b}\right )}+\frac {b^4\,c^5\,d^4\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,15{}\mathrm {i}}{2\,\left (32\,a^4\,c^3\,d^8+\frac {15\,b^4\,c^7\,d^4}{2}-\frac {185\,a\,b^3\,c^6\,d^5}{4}-87\,a^3\,b\,c^4\,d^7+\frac {395\,a^2\,b^2\,c^5\,d^6}{4}-\frac {5\,a^5\,c^2\,d^9}{b}\right )}-\frac {a\,b\,c^2\,d^7\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,87{}\mathrm {i}}{32\,a^2\,c^3\,d^8+\frac {395\,b^2\,c^5\,d^6}{4}-\frac {185\,b^3\,c^6\,d^5}{4\,a}-\frac {5\,a^3\,c^2\,d^9}{b}+\frac {15\,b^4\,c^7\,d^4}{2\,a^2}-87\,a\,b\,c^4\,d^7}\right )\,\left (5\,a\,d-2\,b\,c\right )\,\sqrt {c^3}\,1{}\mathrm {i}}{2\,a^2}-\frac {\mathrm {atan}\left (\frac {c^3\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {-a^5\,b^3\,d^5+5\,a^4\,b^4\,c\,d^4-10\,a^3\,b^5\,c^2\,d^3+10\,a^2\,b^6\,c^3\,d^2-5\,a\,b^7\,c^4\,d+b^8\,c^5}\,20{}\mathrm {i}}{\frac {185\,a\,b^3\,c^5\,d^6}{2}-\frac {85\,b^4\,c^6\,d^5}{2}-16\,a^4\,c^2\,d^9+56\,a^3\,b\,c^3\,d^8+\frac {2\,a^5\,c\,d^{10}}{b}-\frac {199\,a^2\,b^2\,c^4\,d^7}{2}+\frac {15\,b^5\,c^7\,d^4}{2\,a}}-\frac {c^2\,d^6\,\sqrt {d\,x^2+c}\,\sqrt {-a^5\,b^3\,d^5+5\,a^4\,b^4\,c\,d^4-10\,a^3\,b^5\,c^2\,d^3+10\,a^2\,b^6\,c^3\,d^2-5\,a\,b^7\,c^4\,d+b^8\,c^5}\,10{}\mathrm {i}}{2\,a^4\,c\,d^{10}+\frac {185\,b^4\,c^5\,d^6}{2}-\frac {199\,a\,b^3\,c^4\,d^7}{2}-16\,a^3\,b\,c^2\,d^9+56\,a^2\,b^2\,c^3\,d^8-\frac {85\,b^5\,c^6\,d^5}{2\,a}+\frac {15\,b^6\,c^7\,d^4}{2\,a^2}}-\frac {c^4\,d^4\,\sqrt {d\,x^2+c}\,\sqrt {-a^5\,b^3\,d^5+5\,a^4\,b^4\,c\,d^4-10\,a^3\,b^5\,c^2\,d^3+10\,a^2\,b^6\,c^3\,d^2-5\,a\,b^7\,c^4\,d+b^8\,c^5}\,15{}\mathrm {i}}{2\,\left (56\,a^4\,c^3\,d^8+\frac {15\,b^4\,c^7\,d^4}{2}-\frac {85\,a\,b^3\,c^6\,d^5}{2}-\frac {199\,a^3\,b\,c^4\,d^7}{2}+\frac {2\,a^6\,c\,d^{10}}{b^2}+\frac {185\,a^2\,b^2\,c^5\,d^6}{2}-\frac {16\,a^5\,c^2\,d^9}{b}\right )}+\frac {a\,c\,d^7\,\sqrt {d\,x^2+c}\,\sqrt {-a^5\,b^3\,d^5+5\,a^4\,b^4\,c\,d^4-10\,a^3\,b^5\,c^2\,d^3+10\,a^2\,b^6\,c^3\,d^2-5\,a\,b^7\,c^4\,d+b^8\,c^5}\,2{}\mathrm {i}}{\frac {185\,b^5\,c^5\,d^6}{2}-\frac {199\,a\,b^4\,c^4\,d^7}{2}+56\,a^2\,b^3\,c^3\,d^8-16\,a^3\,b^2\,c^2\,d^9-\frac {85\,b^6\,c^6\,d^5}{2\,a}+\frac {15\,b^7\,c^7\,d^4}{2\,a^2}+2\,a^4\,b\,c\,d^{10}}\right )\,\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^5}\,1{}\mathrm {i}}{a^2\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^(5/2)/(x^3*(a + b*x^2)),x)

[Out]

(d^2*(c + d*x^2)^(1/2))/b + (atan((a^3*d^9*(c + d*x^2)^(1/2)*(c^3)^(1/2)*5i)/(5*a^3*c^2*d^9 - (395*b^3*c^5*d^6
)/4 + 87*a*b^2*c^4*d^7 - 32*a^2*b*c^3*d^8 + (185*b^4*c^6*d^5)/(4*a) - (15*b^5*c^7*d^4)/(2*a^2)) + (a^2*c*d^8*(
c + d*x^2)^(1/2)*(c^3)^(1/2)*32i)/(32*a^2*c^3*d^8 + (395*b^2*c^5*d^6)/4 - (185*b^3*c^6*d^5)/(4*a) - (5*a^3*c^2
*d^9)/b + (15*b^4*c^7*d^4)/(2*a^2) - 87*a*b*c^4*d^7) + (b^2*c^3*d^6*(c + d*x^2)^(1/2)*(c^3)^(1/2)*395i)/(4*(32
*a^2*c^3*d^8 + (395*b^2*c^5*d^6)/4 - (185*b^3*c^6*d^5)/(4*a) - (5*a^3*c^2*d^9)/b + (15*b^4*c^7*d^4)/(2*a^2) -
87*a*b*c^4*d^7)) - (b^3*c^4*d^5*(c + d*x^2)^(1/2)*(c^3)^(1/2)*185i)/(4*(32*a^3*c^3*d^8 - (185*b^3*c^6*d^5)/4 +
(395*a*b^2*c^5*d^6)/4 - 87*a^2*b*c^4*d^7 + (15*b^4*c^7*d^4)/(2*a) - (5*a^4*c^2*d^9)/b)) + (b^4*c^5*d^4*(c + d
*x^2)^(1/2)*(c^3)^(1/2)*15i)/(2*(32*a^4*c^3*d^8 + (15*b^4*c^7*d^4)/2 - (185*a*b^3*c^6*d^5)/4 - 87*a^3*b*c^4*d^
7 + (395*a^2*b^2*c^5*d^6)/4 - (5*a^5*c^2*d^9)/b)) - (a*b*c^2*d^7*(c + d*x^2)^(1/2)*(c^3)^(1/2)*87i)/(32*a^2*c^
3*d^8 + (395*b^2*c^5*d^6)/4 - (185*b^3*c^6*d^5)/(4*a) - (5*a^3*c^2*d^9)/b + (15*b^4*c^7*d^4)/(2*a^2) - 87*a*b*
c^4*d^7))*(5*a*d - 2*b*c)*(c^3)^(1/2)*1i)/(2*a^2) - (atan((c^3*d^5*(c + d*x^2)^(1/2)*(b^8*c^5 - a^5*b^3*d^5 +
5*a^4*b^4*c*d^4 + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 - 5*a*b^7*c^4*d)^(1/2)*20i)/((185*a*b^3*c^5*d^6)/2 -
(85*b^4*c^6*d^5)/2 - 16*a^4*c^2*d^9 + 56*a^3*b*c^3*d^8 + (2*a^5*c*d^10)/b - (199*a^2*b^2*c^4*d^7)/2 + (15*b^5
*c^7*d^4)/(2*a)) - (c^2*d^6*(c + d*x^2)^(1/2)*(b^8*c^5 - a^5*b^3*d^5 + 5*a^4*b^4*c*d^4 + 10*a^2*b^6*c^3*d^2 -
10*a^3*b^5*c^2*d^3 - 5*a*b^7*c^4*d)^(1/2)*10i)/(2*a^4*c*d^10 + (185*b^4*c^5*d^6)/2 - (199*a*b^3*c^4*d^7)/2 - 1
6*a^3*b*c^2*d^9 + 56*a^2*b^2*c^3*d^8 - (85*b^5*c^6*d^5)/(2*a) + (15*b^6*c^7*d^4)/(2*a^2)) - (c^4*d^4*(c + d*x^
2)^(1/2)*(b^8*c^5 - a^5*b^3*d^5 + 5*a^4*b^4*c*d^4 + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 - 5*a*b^7*c^4*d)^(
1/2)*15i)/(2*(56*a^4*c^3*d^8 + (15*b^4*c^7*d^4)/2 - (85*a*b^3*c^6*d^5)/2 - (199*a^3*b*c^4*d^7)/2 + (2*a^6*c*d^
10)/b^2 + (185*a^2*b^2*c^5*d^6)/2 - (16*a^5*c^2*d^9)/b)) + (a*c*d^7*(c + d*x^2)^(1/2)*(b^8*c^5 - a^5*b^3*d^5 +
5*a^4*b^4*c*d^4 + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 - 5*a*b^7*c^4*d)^(1/2)*2i)/((185*b^5*c^5*d^6)/2 - (
199*a*b^4*c^4*d^7)/2 + 56*a^2*b^3*c^3*d^8 - 16*a^3*b^2*c^2*d^9 - (85*b^6*c^6*d^5)/(2*a) + (15*b^7*c^7*d^4)/(2*
a^2) + 2*a^4*b*c*d^10))*(-b^3*(a*d - b*c)^5)^(1/2)*1i)/(a^2*b^3) - (b*c^2*d*(c + d*x^2)^(1/2))/(2*a*(b*(c + d*
x^2) - b*c))

________________________________________________________________________________________