\(\int \frac {(a+b x^2)^2}{x^{5/2} (c+d x^2)^2} \, dx\) [431]

Optimal result

Integrand size = 24, antiderivative size = 332 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) \sqrt {x}}{6 c^2 d \left (c+d x^2\right )}-\frac {(b c-a d) (b c+7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} d^{5/4}}+\frac {(b c-a d) (b c+7 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} d^{5/4}}-\frac {(b c-a d) (b c+7 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} d^{5/4}}+\frac {(b c-a d) (b c+7 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} d^{5/4}} \]

[Out]

-2/3*a^2/c/x^(3/2)/(d*x^2+c)-1/8*(-a*d+b*c)*(7*a*d+b*c)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/d^(
5/4)*2^(1/2)+1/8*(-a*d+b*c)*(7*a*d+b*c)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/d^(5/4)*2^(1/2)-1/1
6*(-a*d+b*c)*(7*a*d+b*c)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(11/4)/d^(5/4)*2^(1/2)+1/16*(
-a*d+b*c)*(7*a*d+b*c)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(11/4)/d^(5/4)*2^(1/2)-1/6*(7*a^
2*d^2-6*a*b*c*d+3*b^2*c^2)*x^(1/2)/c^2/d/(d*x^2+c)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {473, 468, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=\frac {\sqrt {x} \left (-\frac {7 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{6 c \left (c+d x^2\right )}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}-\frac {(b c-a d) (7 a d+b c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} d^{5/4}}+\frac {(b c-a d) (7 a d+b c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{11/4} d^{5/4}}-\frac {(b c-a d) (7 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} d^{5/4}}+\frac {(b c-a d) (7 a d+b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} d^{5/4}} \]

[In]

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

[Out]

(-2*a^2)/(3*c*x^(3/2)*(c + d*x^2)) + ((6*a*b - (3*b^2*c)/d - (7*a^2*d)/c)*Sqrt[x])/(6*c*(c + d*x^2)) - ((b*c -
a*d)*(b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(11/4)*d^(5/4)) + ((b*c - a*d)
*(b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(11/4)*d^(5/4)) - ((b*c - a*d)*(b*c
+ 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*d^(5/4)) + ((b*c - a
*d)*(b*c + 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*d^(5/4))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

Rule 473

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}+\frac {2 \int \frac {\frac {1}{2} a (6 b c-7 a d)+\frac {3}{2} b^2 c x^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx}{3 c} \\ & = -\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {7 a^2 d}{c}\right ) \sqrt {x}}{6 c \left (c+d x^2\right )}+\frac {((b c-a d) (b c+7 a d)) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{4 c^2 d} \\ & = -\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {7 a^2 d}{c}\right ) \sqrt {x}}{6 c \left (c+d x^2\right )}+\frac {((b c-a d) (b c+7 a d)) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c^2 d} \\ & = -\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {7 a^2 d}{c}\right ) \sqrt {x}}{6 c \left (c+d x^2\right )}+\frac {((b c-a d) (b c+7 a d)) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^{5/2} d}+\frac {((b c-a d) (b c+7 a d)) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^{5/2} d} \\ & = -\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {7 a^2 d}{c}\right ) \sqrt {x}}{6 c \left (c+d x^2\right )}+\frac {((b c-a d) (b c+7 a d)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 c^{5/2} d^{3/2}}+\frac {((b c-a d) (b c+7 a d)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 c^{5/2} d^{3/2}}-\frac {((b c-a d) (b c+7 a d)) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{11/4} d^{5/4}}-\frac {((b c-a d) (b c+7 a d)) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{11/4} d^{5/4}} \\ & = -\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {7 a^2 d}{c}\right ) \sqrt {x}}{6 c \left (c+d x^2\right )}-\frac {(b c-a d) (b c+7 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} d^{5/4}}+\frac {(b c-a d) (b c+7 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} d^{5/4}}+\frac {((b c-a d) (b c+7 a d)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} d^{5/4}}-\frac {((b c-a d) (b c+7 a d)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} d^{5/4}} \\ & = -\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}+\frac {\left (6 a b-\frac {3 b^2 c}{d}-\frac {7 a^2 d}{c}\right ) \sqrt {x}}{6 c \left (c+d x^2\right )}-\frac {(b c-a d) (b c+7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} d^{5/4}}+\frac {(b c-a d) (b c+7 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} d^{5/4}}-\frac {(b c-a d) (b c+7 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} d^{5/4}}+\frac {(b c-a d) (b c+7 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} d^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.63 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=\frac {-\frac {4 c^{3/4} \sqrt [4]{d} \left (3 b^2 c^2 x^2-6 a b c d x^2+a^2 d \left (4 c+7 d x^2\right )\right )}{x^{3/2} \left (c+d x^2\right )}-3 \sqrt {2} \left (b^2 c^2+6 a b c d-7 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )+3 \sqrt {2} \left (b^2 c^2+6 a b c d-7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{24 c^{11/4} d^{5/4}} \]

[In]

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

[Out]

((-4*c^(3/4)*d^(1/4)*(3*b^2*c^2*x^2 - 6*a*b*c*d*x^2 + a^2*d*(4*c + 7*d*x^2)))/(x^(3/2)*(c + d*x^2)) - 3*Sqrt[2
]*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + 3*Sqrt[2
]*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(24*c^(1
1/4)*d^(5/4))

Maple [A] (verified)

Time = 2.76 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.51

method	result	size
risch	\(-\frac {2 a^{2}}{3 c^{2} x^{\frac {3}{2}}}-\frac {\left (2 a d -2 b c \right ) \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 d \left (d \,x^{2}+c \right )}+\frac {\left (7 a d +b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 d c}\right )}{c^{2}}\)	\(169\)
derivativedivides	\(-\frac {2 \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {x}}{4 d \left (d \,x^{2}+c \right )}+\frac {\left (7 a^{2} d^{2}-6 a b c d -b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 d c}\right )}{c^{2}}-\frac {2 a^{2}}{3 c^{2} x^{\frac {3}{2}}}\)	\(188\)
default	\(-\frac {2 \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {x}}{4 d \left (d \,x^{2}+c \right )}+\frac {\left (7 a^{2} d^{2}-6 a b c d -b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 d c}\right )}{c^{2}}-\frac {2 a^{2}}{3 c^{2} x^{\frac {3}{2}}}\)	\(188\)

[In]

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

[Out]

-2/3*a^2/c^2/x^(3/2)-1/c^2*(2*a*d-2*b*c)*(1/4/d*(a*d-b*c)*x^(1/2)/(d*x^2+c)+1/32*(7*a*d+b*c)/d*(c/d)^(1/4)/c*2
^(1/2)*(ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2
^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 1219, normalized size of antiderivative = 3.67 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/24*(3*(c^2*d^2*x^4 + c^3*d*x^2)*(-(b^8*c^8 + 24*a*b^7*c^7*d + 188*a^2*b^6*c^6*d^2 + 360*a^3*b^5*c^5*d^3 - 1
434*a^4*b^4*c^4*d^4 - 2520*a^5*b^3*c^3*d^5 + 9212*a^6*b^2*c^2*d^6 - 8232*a^7*b*c*d^7 + 2401*a^8*d^8)/(c^11*d^5
))^(1/4)*log(c^3*d*(-(b^8*c^8 + 24*a*b^7*c^7*d + 188*a^2*b^6*c^6*d^2 + 360*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*
d^4 - 2520*a^5*b^3*c^3*d^5 + 9212*a^6*b^2*c^2*d^6 - 8232*a^7*b*c*d^7 + 2401*a^8*d^8)/(c^11*d^5))^(1/4) - (b^2*
c^2 + 6*a*b*c*d - 7*a^2*d^2)*sqrt(x)) + 3*(I*c^2*d^2*x^4 + I*c^3*d*x^2)*(-(b^8*c^8 + 24*a*b^7*c^7*d + 188*a^2*
b^6*c^6*d^2 + 360*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 - 2520*a^5*b^3*c^3*d^5 + 9212*a^6*b^2*c^2*d^6 - 8232*
a^7*b*c*d^7 + 2401*a^8*d^8)/(c^11*d^5))^(1/4)*log(I*c^3*d*(-(b^8*c^8 + 24*a*b^7*c^7*d + 188*a^2*b^6*c^6*d^2 +
360*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 - 2520*a^5*b^3*c^3*d^5 + 9212*a^6*b^2*c^2*d^6 - 8232*a^7*b*c*d^7 +
2401*a^8*d^8)/(c^11*d^5))^(1/4) - (b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*sqrt(x)) + 3*(-I*c^2*d^2*x^4 - I*c^3*d*x^2
)*(-(b^8*c^8 + 24*a*b^7*c^7*d + 188*a^2*b^6*c^6*d^2 + 360*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 - 2520*a^5*b^
3*c^3*d^5 + 9212*a^6*b^2*c^2*d^6 - 8232*a^7*b*c*d^7 + 2401*a^8*d^8)/(c^11*d^5))^(1/4)*log(-I*c^3*d*(-(b^8*c^8
+ 24*a*b^7*c^7*d + 188*a^2*b^6*c^6*d^2 + 360*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 - 2520*a^5*b^3*c^3*d^5 + 9
212*a^6*b^2*c^2*d^6 - 8232*a^7*b*c*d^7 + 2401*a^8*d^8)/(c^11*d^5))^(1/4) - (b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*s
qrt(x)) - 3*(c^2*d^2*x^4 + c^3*d*x^2)*(-(b^8*c^8 + 24*a*b^7*c^7*d + 188*a^2*b^6*c^6*d^2 + 360*a^3*b^5*c^5*d^3
- 1434*a^4*b^4*c^4*d^4 - 2520*a^5*b^3*c^3*d^5 + 9212*a^6*b^2*c^2*d^6 - 8232*a^7*b*c*d^7 + 2401*a^8*d^8)/(c^11*
d^5))^(1/4)*log(-c^3*d*(-(b^8*c^8 + 24*a*b^7*c^7*d + 188*a^2*b^6*c^6*d^2 + 360*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*
c^4*d^4 - 2520*a^5*b^3*c^3*d^5 + 9212*a^6*b^2*c^2*d^6 - 8232*a^7*b*c*d^7 + 2401*a^8*d^8)/(c^11*d^5))^(1/4) - (
b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*sqrt(x)) + 4*(4*a^2*c*d + (3*b^2*c^2 - 6*a*b*c*d + 7*a^2*d^2)*x^2)*sqrt(x))/(
c^2*d^2*x^4 + c^3*d*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1418 vs. \(2 (313) = 626\).

Time = 112.77 (sec) , antiderivative size = 1418, normalized size of antiderivative = 4.27 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*(-2*a**2/(11*x**(11/2)) - 4*a*b/(7*x**(7/2)) - 2*b**2/(3*x**(3/2))), Eq(c, 0) & Eq(d, 0)), ((-2
*a**2/(3*x**(3/2)) + 4*a*b*sqrt(x) + 2*b**2*x**(5/2)/5)/c**2, Eq(d, 0)), ((-2*a**2/(11*x**(11/2)) - 4*a*b/(7*x
**(7/2)) - 2*b**2/(3*x**(3/2)))/d**2, Eq(c, 0)), (-16*a**2*c**2*d/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2))
+ 21*a**2*c*d**2*x**(3/2)*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7
/2)) - 21*a**2*c*d**2*x**(3/2)*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x
**(7/2)) - 42*a**2*c*d**2*x**(3/2)*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**
2*x**(7/2)) - 28*a**2*c*d**2*x**2/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) + 21*a**2*d**3*x**(7/2)*(-c/d)*
*(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) - 21*a**2*d**3*x**(7/2)*(-c/d
)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) - 42*a**2*d**3*x**(7/2)*(-c
/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) - 18*a*b*c**2*d*x**(3/2)*(
-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) + 18*a*b*c**2*d*x**(3/2
)*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) + 36*a*b*c**2*d*x**(
3/2)*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) + 24*a*b*c**2*d*x*
*2/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) - 18*a*b*c*d**2*x**(7/2)*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(
1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) + 18*a*b*c*d**2*x**(7/2)*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)
**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) + 36*a*b*c*d**2*x**(7/2)*(-c/d)**(1/4)*atan(sqrt(x)/(-c/
d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) - 3*b**2*c**3*x**(3/2)*(-c/d)**(1/4)*log(sqrt(x) - (-c
/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) + 3*b**2*c**3*x**(3/2)*(-c/d)**(1/4)*log(sqrt(x) + (-
c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) + 6*b**2*c**3*x**(3/2)*(-c/d)**(1/4)*atan(sqrt(x)/(-
c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) - 12*b**2*c**3*x**2/(24*c**4*d*x**(3/2) + 24*c**3*d*
*2*x**(7/2)) - 3*b**2*c**2*d*x**(7/2)*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3
*d**2*x**(7/2)) + 3*b**2*c**2*d*x**(7/2)*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c
**3*d**2*x**(7/2)) + 6*b**2*c**2*d*x**(7/2)*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24
*c**3*d**2*x**(7/2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=-\frac {4 \, a^{2} c d + {\left (3 \, b^{2} c^{2} - 6 \, a b c d + 7 \, a^{2} d^{2}\right )} x^{2}}{6 \, {\left (c^{2} d^{2} x^{\frac {7}{2}} + c^{3} d x^{\frac {3}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (b^{2} c^{2} + 6 \, a b c d - 7 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b^{2} c^{2} + 6 \, a b c d - 7 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b^{2} c^{2} + 6 \, a b c d - 7 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{2} c^{2} + 6 \, a b c d - 7 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{16 \, c^{2} d} \]

[In]

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

[Out]

-1/6*(4*a^2*c*d + (3*b^2*c^2 - 6*a*b*c*d + 7*a^2*d^2)*x^2)/(c^2*d^2*x^(7/2) + c^3*d*x^(3/2)) + 1/16*(2*sqrt(2)
*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(
c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*arctan(-1/2*sqrt(2)
*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2
)*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4
)) - sqrt(2)*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^
(3/4)*d^(1/4)))/(c^2*d)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=-\frac {2 \, a^{2}}{3 \, c^{2} x^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{3} d^{2}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{3} d^{2}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{3} d^{2}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{3} d^{2}} - \frac {b^{2} c^{2} \sqrt {x} - 2 \, a b c d \sqrt {x} + a^{2} d^{2} \sqrt {x}}{2 \, {\left (d x^{2} + c\right )} c^{2} d} \]

[In]

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

[Out]

-2/3*a^2/(c^2*x^(3/2)) + 1/8*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d - 7*(c*d^3)^(1/4)*a^2*d^
2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c^3*d^2) + 1/8*sqrt(2)*((c*d^3)^(1/4)*b^
2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d - 7*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x
))/(c/d)^(1/4))/(c^3*d^2) + 1/16*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d - 7*(c*d^3)^(1/4)*a^
2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^3*d^2) - 1/16*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 + 6*(c
*d^3)^(1/4)*a*b*c*d - 7*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^3*d^2) - 1
/2*(b^2*c^2*sqrt(x) - 2*a*b*c*d*sqrt(x) + a^2*d^2*sqrt(x))/((d*x^2 + c)*c^2*d)

Mupad [B] (verification not implemented)

Time = 5.39 (sec) , antiderivative size = 1340, normalized size of antiderivative = 4.04 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan((((x^(1/2)*(1568*a^4*c^6*d^10 + 32*b^4*c^10*d^6 + 384*a*b^3*c^9*d^7 - 2688*a^3*b*c^7*d^9 + 704*a^2*b^2*c
^8*d^8) - ((a*d - b*c)*(7*a*d + b*c)*(256*b^2*c^11*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b*c^10*d^8))/(8*(-c)^(11/4)
*d^(5/4)))*(a*d - b*c)*(7*a*d + b*c)*1i)/(8*(-c)^(11/4)*d^(5/4)) + ((x^(1/2)*(1568*a^4*c^6*d^10 + 32*b^4*c^10*
d^6 + 384*a*b^3*c^9*d^7 - 2688*a^3*b*c^7*d^9 + 704*a^2*b^2*c^8*d^8) + ((a*d - b*c)*(7*a*d + b*c)*(256*b^2*c^11
*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b*c^10*d^8))/(8*(-c)^(11/4)*d^(5/4)))*(a*d - b*c)*(7*a*d + b*c)*1i)/(8*(-c)^(
11/4)*d^(5/4)))/(((x^(1/2)*(1568*a^4*c^6*d^10 + 32*b^4*c^10*d^6 + 384*a*b^3*c^9*d^7 - 2688*a^3*b*c^7*d^9 + 704
*a^2*b^2*c^8*d^8) - ((a*d - b*c)*(7*a*d + b*c)*(256*b^2*c^11*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b*c^10*d^8))/(8*(
-c)^(11/4)*d^(5/4)))*(a*d - b*c)*(7*a*d + b*c))/(8*(-c)^(11/4)*d^(5/4)) - ((x^(1/2)*(1568*a^4*c^6*d^10 + 32*b^
4*c^10*d^6 + 384*a*b^3*c^9*d^7 - 2688*a^3*b*c^7*d^9 + 704*a^2*b^2*c^8*d^8) + ((a*d - b*c)*(7*a*d + b*c)*(256*b
^2*c^11*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b*c^10*d^8))/(8*(-c)^(11/4)*d^(5/4)))*(a*d - b*c)*(7*a*d + b*c))/(8*(-
c)^(11/4)*d^(5/4))))*(a*d - b*c)*(7*a*d + b*c)*1i)/(4*(-c)^(11/4)*d^(5/4)) - ((2*a^2)/(3*c) + (x^2*(7*a^2*d^2
+ 3*b^2*c^2 - 6*a*b*c*d))/(6*c^2*d))/(c*x^(3/2) + d*x^(7/2)) + (atan((((x^(1/2)*(1568*a^4*c^6*d^10 + 32*b^4*c^
10*d^6 + 384*a*b^3*c^9*d^7 - 2688*a^3*b*c^7*d^9 + 704*a^2*b^2*c^8*d^8) - ((a*d - b*c)*(7*a*d + b*c)*(256*b^2*c
^11*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b*c^10*d^8)*1i)/(8*(-c)^(11/4)*d^(5/4)))*(a*d - b*c)*(7*a*d + b*c))/(8*(-c
)^(11/4)*d^(5/4)) + ((x^(1/2)*(1568*a^4*c^6*d^10 + 32*b^4*c^10*d^6 + 384*a*b^3*c^9*d^7 - 2688*a^3*b*c^7*d^9 +
704*a^2*b^2*c^8*d^8) + ((a*d - b*c)*(7*a*d + b*c)*(256*b^2*c^11*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b*c^10*d^8)*1i
)/(8*(-c)^(11/4)*d^(5/4)))*(a*d - b*c)*(7*a*d + b*c))/(8*(-c)^(11/4)*d^(5/4)))/(((x^(1/2)*(1568*a^4*c^6*d^10 +
32*b^4*c^10*d^6 + 384*a*b^3*c^9*d^7 - 2688*a^3*b*c^7*d^9 + 704*a^2*b^2*c^8*d^8) - ((a*d - b*c)*(7*a*d + b*c)*
(256*b^2*c^11*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b*c^10*d^8)*1i)/(8*(-c)^(11/4)*d^(5/4)))*(a*d - b*c)*(7*a*d + b*
c)*1i)/(8*(-c)^(11/4)*d^(5/4)) - ((x^(1/2)*(1568*a^4*c^6*d^10 + 32*b^4*c^10*d^6 + 384*a*b^3*c^9*d^7 - 2688*a^3
*b*c^7*d^9 + 704*a^2*b^2*c^8*d^8) + ((a*d - b*c)*(7*a*d + b*c)*(256*b^2*c^11*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b
*c^10*d^8)*1i)/(8*(-c)^(11/4)*d^(5/4)))*(a*d - b*c)*(7*a*d + b*c)*1i)/(8*(-c)^(11/4)*d^(5/4))))*(a*d - b*c)*(7
*a*d + b*c))/(4*(-c)^(11/4)*d^(5/4))