\(\int \frac {(c+d x^2)^3}{x^{15/2} (a+b x^2)} \, dx\) [451]

Optimal result

Integrand size = 24, antiderivative size = 325 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}-\frac {\sqrt [4]{b} (b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}-\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}} \]

[Out]

-2/13*c^3/a/x^(13/2)+2/9*c^2*(-3*a*d+b*c)/a^2/x^(9/2)-2/5*c*(3*a^2*d^2-3*a*b*c*d+b^2*c^2)/a^3/x^(5/2)-1/2*b^(1
/4)*(-a*d+b*c)^3*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(17/4)*2^(1/2)+1/2*b^(1/4)*(-a*d+b*c)^3*arctan(1+
b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(17/4)*2^(1/2)+1/4*b^(1/4)*(-a*d+b*c)^3*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4
)*2^(1/2)*x^(1/2))/a^(17/4)*2^(1/2)-1/4*b^(1/4)*(-a*d+b*c)^3*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1
/2))/a^(17/4)*2^(1/2)+2*(-a*d+b*c)^3/a^4/x^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {477, 472, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=-\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}-\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{5 a^3 x^{5/2}}-\frac {2 c^3}{13 a x^{13/2}} \]

[In]

Int[(c + d*x^2)^3/(x^(15/2)*(a + b*x^2)),x]

[Out]

(-2*c^3)/(13*a*x^(13/2)) + (2*c^2*(b*c - 3*a*d))/(9*a^2*x^(9/2)) - (2*c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2))/(5*
a^3*x^(5/2)) + (2*(b*c - a*d)^3)/(a^4*Sqrt[x]) - (b^(1/4)*(b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a
^(1/4)])/(Sqrt[2]*a^(17/4)) + (b^(1/4)*(b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a
^(17/4)) + (b^(1/4)*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(17
/4)) - (b^(1/4)*(b*c - a*d)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(17/4))

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 303

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

Rule 472

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

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

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}& = 2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^{14} \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {c^3}{a x^{14}}+\frac {c^2 (-b c+3 a d)}{a^2 x^{10}}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x^6}+\frac {(-b c+a d)^3}{a^4 x^2}-\frac {b (-b c+a d)^3 x^2}{a^4 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}+\frac {\left (2 b (b c-a d)^3\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^4} \\ & = -\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}-\frac {\left (\sqrt {b} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^4}+\frac {\left (\sqrt {b} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^4} \\ & = -\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^4}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^4}+\frac {\left (\sqrt [4]{b} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{17/4}}+\frac {\left (\sqrt [4]{b} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{17/4}} \\ & = -\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}+\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}-\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}+\frac {\left (\sqrt [4]{b} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{17/4}}-\frac {\left (\sqrt [4]{b} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{17/4}} \\ & = -\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}-\frac {\sqrt [4]{b} (b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}-\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.73 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\frac {1170 b^3 c^3 x^6-234 a b^2 c^2 x^4 \left (c+15 d x^2\right )+26 a^2 b c x^2 \left (5 c^2+27 c d x^2+135 d^2 x^4\right )-6 a^3 \left (15 c^3+65 c^2 d x^2+117 c d^2 x^4+195 d^3 x^6\right )}{585 a^4 x^{13/2}}+\frac {\sqrt [4]{b} (-b c+a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (-b c+a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{17/4}} \]

[In]

Integrate[(c + d*x^2)^3/(x^(15/2)*(a + b*x^2)),x]

[Out]

(1170*b^3*c^3*x^6 - 234*a*b^2*c^2*x^4*(c + 15*d*x^2) + 26*a^2*b*c*x^2*(5*c^2 + 27*c*d*x^2 + 135*d^2*x^4) - 6*a
^3*(15*c^3 + 65*c^2*d*x^2 + 117*c*d^2*x^4 + 195*d^3*x^6))/(585*a^4*x^(13/2)) + (b^(1/4)*(-(b*c) + a*d)^3*ArcTa
n[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(17/4)) + (b^(1/4)*(-(b*c) + a*d)^3*Arc
Tanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(17/4))

Maple [A] (verified)

Time = 2.75 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 c^{3}}{13 a \,x^{\frac {13}{2}}}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{a^{4} \sqrt {x}}-\frac {2 c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{9 a^{2} x^{\frac {9}{2}}}\)	\(249\)
default	\(-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 c^{3}}{13 a \,x^{\frac {13}{2}}}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{a^{4} \sqrt {x}}-\frac {2 c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{9 a^{2} x^{\frac {9}{2}}}\)	\(249\)
risch	\(-\frac {2 \left (585 a^{3} d^{3} x^{6}-1755 a^{2} b c \,d^{2} x^{6}+1755 a \,b^{2} c^{2} d \,x^{6}-585 b^{3} c^{3} x^{6}+351 a^{3} c \,d^{2} x^{4}-351 a^{2} b \,c^{2} d \,x^{4}+117 a \,b^{2} c^{3} x^{4}+195 a^{3} c^{2} d \,x^{2}-65 a^{2} b \,c^{3} x^{2}+45 c^{3} a^{3}\right )}{585 a^{4} x^{\frac {13}{2}}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(269\)

[In]

int((d*x^2+c)^3/x^(15/2)/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/a^4/(a/b)^(1/4)*2^(1/2)*(ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+
(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(
1/2)/(a/b)^(1/4)*x^(1/2)-1))-2/13*c^3/a/x^(13/2)-2*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/a^4/x^(1/2)-2
/5*c*(3*a^2*d^2-3*a*b*c*d+b^2*c^2)/a^3/x^(5/2)-2/9*c^2*(3*a*d-b*c)/a^2/x^(9/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 2059, normalized size of antiderivative = 6.34 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \]

[In]

integrate((d*x^2+c)^3/x^(15/2)/(b*x^2+a),x, algorithm="fricas")

[Out]

-1/1170*(585*a^4*x^7*(-(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b
^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b
^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12)/a^17)^(1/4)*log(a^13*(-(b^13*c^12 - 12*a
*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a
^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a
^11*b^2*c*d^11 + a^12*b*d^12)/a^17)^(3/4) - (b^10*c^9 - 9*a*b^9*c^8*d + 36*a^2*b^8*c^7*d^2 - 84*a^3*b^7*c^6*d^
3 + 126*a^4*b^6*c^5*d^4 - 126*a^5*b^5*c^4*d^5 + 84*a^6*b^4*c^3*d^6 - 36*a^7*b^3*c^2*d^7 + 9*a^8*b^2*c*d^8 - a^
9*b*d^9)*sqrt(x)) - 585*I*a^4*x^7*(-(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^
3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^
8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12)/a^17)^(1/4)*log(I*a^13*(-(b
^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*
c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*
c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12)/a^17)^(3/4) - (b^10*c^9 - 9*a*b^9*c^8*d + 36*a^2*b^8*c^7*d^2 - 84
*a^3*b^7*c^6*d^3 + 126*a^4*b^6*c^5*d^4 - 126*a^5*b^5*c^4*d^5 + 84*a^6*b^4*c^3*d^6 - 36*a^7*b^3*c^2*d^7 + 9*a^8
*b^2*c*d^8 - a^9*b*d^9)*sqrt(x)) + 585*I*a^4*x^7*(-(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*
a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495
*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12)/a^17)^(1/4)*
log(-I*a^13*(-(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^
4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^
9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12)/a^17)^(3/4) - (b^10*c^9 - 9*a*b^9*c^8*d + 36*a^2*
b^8*c^7*d^2 - 84*a^3*b^7*c^6*d^3 + 126*a^4*b^6*c^5*d^4 - 126*a^5*b^5*c^4*d^5 + 84*a^6*b^4*c^3*d^6 - 36*a^7*b^3
*c^2*d^7 + 9*a^8*b^2*c*d^8 - a^9*b*d^9)*sqrt(x)) - 585*a^4*x^7*(-(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c
^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6
*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12
)/a^17)^(1/4)*log(-a^13*(-(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^
4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^
9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12)/a^17)^(3/4) - (b^10*c^9 - 9*a*b^9*c^8
*d + 36*a^2*b^8*c^7*d^2 - 84*a^3*b^7*c^6*d^3 + 126*a^4*b^6*c^5*d^4 - 126*a^5*b^5*c^4*d^5 + 84*a^6*b^4*c^3*d^6
- 36*a^7*b^3*c^2*d^7 + 9*a^8*b^2*c*d^8 - a^9*b*d^9)*sqrt(x)) - 4*(585*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2
- a^3*d^3)*x^6 - 45*a^3*c^3 - 117*(a*b^2*c^3 - 3*a^2*b*c^2*d + 3*a^3*c*d^2)*x^4 + 65*(a^2*b*c^3 - 3*a^3*c^2*d
)*x^2)*sqrt(x))/(a^4*x^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \]

[In]

integrate((d*x**2+c)**3/x**(15/2)/(b*x**2+a),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.02 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\frac {{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, a^{4}} + \frac {2 \, {\left (585 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{6} - 45 \, a^{3} c^{3} - 117 \, {\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{4} + 65 \, {\left (a^{2} b c^{3} - 3 \, a^{3} c^{2} d\right )} x^{2}\right )}}{585 \, a^{4} x^{\frac {13}{2}}} \]

[In]

integrate((d*x^2+c)^3/x^(15/2)/(b*x^2+a),x, algorithm="maxima")

[Out]

1/4*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(
1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(
2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt
(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4
)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/a^4 + 2/585*(585*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*
b*c*d^2 - a^3*d^3)*x^6 - 45*a^3*c^3 - 117*(a*b^2*c^3 - 3*a^2*b*c^2*d + 3*a^3*c*d^2)*x^4 + 65*(a^2*b*c^3 - 3*a^
3*c^2*d)*x^2)/(a^4*x^(13/2))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (244) = 488\).

Time = 0.30 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.65 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{5} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{5} b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{5} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{5} b^{2}} + \frac {2 \, {\left (585 \, b^{3} c^{3} x^{6} - 1755 \, a b^{2} c^{2} d x^{6} + 1755 \, a^{2} b c d^{2} x^{6} - 585 \, a^{3} d^{3} x^{6} - 117 \, a b^{2} c^{3} x^{4} + 351 \, a^{2} b c^{2} d x^{4} - 351 \, a^{3} c d^{2} x^{4} + 65 \, a^{2} b c^{3} x^{2} - 195 \, a^{3} c^{2} d x^{2} - 45 \, a^{3} c^{3}\right )}}{585 \, a^{4} x^{\frac {13}{2}}} \]

[In]

integrate((d*x^2+c)^3/x^(15/2)/(b*x^2+a),x, algorithm="giac")

[Out]

1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)
*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^5*b^2) + 1/2*sqrt(2)*((a*b^3)^(
3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*arctan(-1/2*
sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^5*b^2) - 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b
^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) +
x + sqrt(a/b))/(a^5*b^2) + 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)
*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^5*b^2) + 2/585*(585
*b^3*c^3*x^6 - 1755*a*b^2*c^2*d*x^6 + 1755*a^2*b*c*d^2*x^6 - 585*a^3*d^3*x^6 - 117*a*b^2*c^3*x^4 + 351*a^2*b*c
^2*d*x^4 - 351*a^3*c*d^2*x^4 + 65*a^2*b*c^3*x^2 - 195*a^3*c^2*d*x^2 - 45*a^3*c^3)/(a^4*x^(13/2))

Mupad [B] (verification not implemented)

Time = 5.16 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.97 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (16\,a^{19}\,b^4\,d^6-96\,a^{18}\,b^5\,c\,d^5+240\,a^{17}\,b^6\,c^2\,d^4-320\,a^{16}\,b^7\,c^3\,d^3+240\,a^{15}\,b^8\,c^4\,d^2-96\,a^{14}\,b^9\,c^5\,d+16\,a^{13}\,b^{10}\,c^6\right )}{a^{17/4}\,\left (-16\,a^{18}\,b^4\,d^9+144\,a^{17}\,b^5\,c\,d^8-576\,a^{16}\,b^6\,c^2\,d^7+1344\,a^{15}\,b^7\,c^3\,d^6-2016\,a^{14}\,b^8\,c^4\,d^5+2016\,a^{13}\,b^9\,c^5\,d^4-1344\,a^{12}\,b^{10}\,c^6\,d^3+576\,a^{11}\,b^{11}\,c^7\,d^2-144\,a^{10}\,b^{12}\,c^8\,d+16\,a^9\,b^{13}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^{17/4}}-\frac {\frac {2\,c^3}{13\,a}+\frac {2\,x^6\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{a^4}+\frac {2\,c^2\,x^2\,\left (3\,a\,d-b\,c\right )}{9\,a^2}+\frac {2\,c\,x^4\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{5\,a^3}}{x^{13/2}}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (16\,a^{19}\,b^4\,d^6-96\,a^{18}\,b^5\,c\,d^5+240\,a^{17}\,b^6\,c^2\,d^4-320\,a^{16}\,b^7\,c^3\,d^3+240\,a^{15}\,b^8\,c^4\,d^2-96\,a^{14}\,b^9\,c^5\,d+16\,a^{13}\,b^{10}\,c^6\right )}{a^{17/4}\,\left (-16\,a^{18}\,b^4\,d^9+144\,a^{17}\,b^5\,c\,d^8-576\,a^{16}\,b^6\,c^2\,d^7+1344\,a^{15}\,b^7\,c^3\,d^6-2016\,a^{14}\,b^8\,c^4\,d^5+2016\,a^{13}\,b^9\,c^5\,d^4-1344\,a^{12}\,b^{10}\,c^6\,d^3+576\,a^{11}\,b^{11}\,c^7\,d^2-144\,a^{10}\,b^{12}\,c^8\,d+16\,a^9\,b^{13}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^{17/4}} \]

[In]

int((c + d*x^2)^3/(x^(15/2)*(a + b*x^2)),x)

[Out]

((-b)^(1/4)*atan(((-b)^(1/4)*x^(1/2)*(a*d - b*c)^3*(16*a^13*b^10*c^6 + 16*a^19*b^4*d^6 - 96*a^14*b^9*c^5*d - 9
6*a^18*b^5*c*d^5 + 240*a^15*b^8*c^4*d^2 - 320*a^16*b^7*c^3*d^3 + 240*a^17*b^6*c^2*d^4))/(a^(17/4)*(16*a^9*b^13
*c^9 - 16*a^18*b^4*d^9 - 144*a^10*b^12*c^8*d + 144*a^17*b^5*c*d^8 + 576*a^11*b^11*c^7*d^2 - 1344*a^12*b^10*c^6
*d^3 + 2016*a^13*b^9*c^5*d^4 - 2016*a^14*b^8*c^4*d^5 + 1344*a^15*b^7*c^3*d^6 - 576*a^16*b^6*c^2*d^7)))*(a*d -
b*c)^3)/a^(17/4) - ((2*c^3)/(13*a) + (2*x^6*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/a^4 + (2*c^2*
x^2*(3*a*d - b*c))/(9*a^2) + (2*c*x^4*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/(5*a^3))/x^(13/2) - ((-b)^(1/4)*atanh
(((-b)^(1/4)*x^(1/2)*(a*d - b*c)^3*(16*a^13*b^10*c^6 + 16*a^19*b^4*d^6 - 96*a^14*b^9*c^5*d - 96*a^18*b^5*c*d^5
+ 240*a^15*b^8*c^4*d^2 - 320*a^16*b^7*c^3*d^3 + 240*a^17*b^6*c^2*d^4))/(a^(17/4)*(16*a^9*b^13*c^9 - 16*a^18*b
^4*d^9 - 144*a^10*b^12*c^8*d + 144*a^17*b^5*c*d^8 + 576*a^11*b^11*c^7*d^2 - 1344*a^12*b^10*c^6*d^3 + 2016*a^13
*b^9*c^5*d^4 - 2016*a^14*b^8*c^4*d^5 + 1344*a^15*b^7*c^3*d^6 - 576*a^16*b^6*c^2*d^7)))*(a*d - b*c)^3)/a^(17/4)