3.458 \(\int \frac{(c+d x^2)^3}{x^{5/2} (a+b x^2)^2} \, dx\)
Optimal. Leaf size=367 \[ \frac{(b c-a d)^2 (5 a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (5 a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}+\frac{(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}-\frac{c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac{d^2 \sqrt{x} (b c-5 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{3/2} \left (a+b x^2\right )} \]
[Out]
-(c^2*(7*b*c - 3*a*d))/(6*a^2*b*x^(3/2)) - (d^2*(b*c - 5*a*d)*Sqrt[x])/(2*a*b^2) + ((b*c - a*d)*(c + d*x^2)^2)
/(2*a*b*x^(3/2)*(a + b*x^2)) + ((b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
4*Sqrt[2]*a^(11/4)*b^(9/4)) - ((b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4
*Sqrt[2]*a^(11/4)*b^(9/4)) + ((b*c - a*d)^2*(7*b*c + 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sq
rt[b]*x])/(8*Sqrt[2]*a^(11/4)*b^(9/4)) - ((b*c - a*d)^2*(7*b*c + 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*
Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*b^(9/4))
________________________________________________________________________________________
Rubi [A] time = 0.410948, antiderivative size = 367, normalized size of antiderivative = 1.,
number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used
= {466, 468, 570, 211, 1165, 628, 1162, 617, 204} \[ \frac{(b c-a d)^2 (5 a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (5 a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}+\frac{(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}-\frac{c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac{d^2 \sqrt{x} (b c-5 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{3/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)^2),x]
[Out]
-(c^2*(7*b*c - 3*a*d))/(6*a^2*b*x^(3/2)) - (d^2*(b*c - 5*a*d)*Sqrt[x])/(2*a*b^2) + ((b*c - a*d)*(c + d*x^2)^2)
/(2*a*b*x^(3/2)*(a + b*x^2)) + ((b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
4*Sqrt[2]*a^(11/4)*b^(9/4)) - ((b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4
*Sqrt[2]*a^(11/4)*b^(9/4)) + ((b*c - a*d)^2*(7*b*c + 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sq
rt[b]*x])/(8*Sqrt[2]*a^(11/4)*b^(9/4)) - ((b*c - a*d)^2*(7*b*c + 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*
Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*b^(9/4))
Rule 466
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 468
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[((c*b -
a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*e*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), I
nt[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p
+ 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 570
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
Rule 211
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 1165
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]
Rule 628
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 1162
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 617
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 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{\left (c+d x^4\right )^3}{x^4 \left (a+b x^4\right )^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{\left (c+d x^4\right ) \left (-c (7 b c-3 a d)+d (b c-5 a d) x^4\right )}{x^4 \left (a+b x^4\right )} \, dx,x,\sqrt{x}\right )}{2 a b}\\ &=\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac{\operatorname{Subst}\left (\int \left (\frac{d^2 (b c-5 a d)}{b}+\frac{c^2 (-7 b c+3 a d)}{a x^4}+\frac{(-b c+a d)^2 (7 b c+5 a d)}{a b \left (a+b x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{2 a b}\\ &=-\frac{c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac{d^2 (b c-5 a d) \sqrt{x}}{2 a b^2}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac{\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 a^2 b^2}\\ &=-\frac{c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac{d^2 (b c-5 a d) \sqrt{x}}{2 a b^2}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac{\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{5/2} b^2}-\frac{\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{5/2} b^2}\\ &=-\frac{c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac{d^2 (b c-5 a d) \sqrt{x}}{2 a b^2}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac{\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname{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 )}{8 a^{5/2} b^{5/2}}-\frac{\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname{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 )}{8 a^{5/2} b^{5/2}}+\frac{\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname{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 )}{8 \sqrt{2} a^{11/4} b^{9/4}}+\frac{\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname{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 )}{8 \sqrt{2} a^{11/4} b^{9/4}}\\ &=-\frac{c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac{d^2 (b c-5 a d) \sqrt{x}}{2 a b^2}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac{(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}-\frac{\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}+\frac{\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}\\ &=-\frac{c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac{d^2 (b c-5 a d) \sqrt{x}}{2 a b^2}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac{(b c-a d)^2 (7 b c+5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (7 b c+5 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}+\frac{(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}\\ \end{align*}
Mathematica [C] time = 2.1343, size = 350, normalized size = 0.95 \[ -\frac{32768 b^3 x^6 \left (c+d x^2\right )^3 \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},2,2,2,2\right \},\left \{1,1,1,\frac{17}{4}\right \},-\frac{b x^2}{a}\right )-585 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right ) \left (a^2 b x^2 \left (1875 c^2 d x^2+625 c^3+2259 c d^2 x^4+625 d^3 x^6\right )+a^3 \left (6561 c^2 d x^2+2187 c^3+6561 c d^2 x^4+1547 d^3 x^6\right )+a b^2 x^4 \left (1155 c^2 d x^2+c^3+3 c d^2 x^4+d^3 x^6\right )+b^3 x^6 \left (81 c^2 d x^2-869 c^3+81 c d^2 x^4+27 d^3 x^6\right )\right )+39 a \left (15 a^2 \left (6561 c^2 d x^2+2187 c^3+6561 c d^2 x^4+1547 d^3 x^6\right )+6 a b x^2 \left (1407 c^2 d x^2+469 c^3+2367 c d^2 x^4+789 d^3 x^6\right )-5 b^2 x^4 \left (81 c^2 d x^2-869 c^3+81 c d^2 x^4+27 d^3 x^6\right )\right )}{149760 a^3 b^2 x^{11/2}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)^2),x]
[Out]
-(39*a*(-5*b^2*x^4*(-869*c^3 + 81*c^2*d*x^2 + 81*c*d^2*x^4 + 27*d^3*x^6) + 6*a*b*x^2*(469*c^3 + 1407*c^2*d*x^2
+ 2367*c*d^2*x^4 + 789*d^3*x^6) + 15*a^2*(2187*c^3 + 6561*c^2*d*x^2 + 6561*c*d^2*x^4 + 1547*d^3*x^6)) - 585*(
a*b^2*x^4*(c^3 + 1155*c^2*d*x^2 + 3*c*d^2*x^4 + d^3*x^6) + b^3*x^6*(-869*c^3 + 81*c^2*d*x^2 + 81*c*d^2*x^4 + 2
7*d^3*x^6) + a^2*b*x^2*(625*c^3 + 1875*c^2*d*x^2 + 2259*c*d^2*x^4 + 625*d^3*x^6) + a^3*(2187*c^3 + 6561*c^2*d*
x^2 + 6561*c*d^2*x^4 + 1547*d^3*x^6))*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)] + 32768*b^3*x^6*(c + d*x^2)
^3*HypergeometricPFQ[{1/4, 2, 2, 2, 2}, {1, 1, 1, 17/4}, -((b*x^2)/a)])/(149760*a^3*b^2*x^(11/2))
________________________________________________________________________________________
Maple [B] time = 0.019, size = 682, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x^2+c)^3/x^(5/2)/(b*x^2+a)^2,x)
[Out]
2*d^3/b^2*x^(1/2)-2/3*c^3/a^2/x^(3/2)+1/2*a/b^2*x^(1/2)/(b*x^2+a)*d^3-3/2/b*x^(1/2)/(b*x^2+a)*c*d^2+3/2/a*x^(1
/2)/(b*x^2+a)*c^2*d-1/2/a^2*b*x^(1/2)/(b*x^2+a)*c^3-5/8/b^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)
*x^(1/2)+1)*d^3+3/8/a/b*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*c*d^2+9/8/a^2*(1/b*a)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*c^2*d-7/8/a^3*b*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*
a)^(1/4)*x^(1/2)+1)*c^3-5/8/b^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*d^3+3/8/a/b*(1/b
*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*c*d^2+9/8/a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
1/b*a)^(1/4)*x^(1/2)-1)*c^2*d-7/8/a^3*b*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*c^3-5/16
/b^2*(1/b*a)^(1/4)*2^(1/2)*ln((x+(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x-(1/b*a)^(1/4)*x^(1/2)*2^(1/2)
+(1/b*a)^(1/2)))*d^3+3/16/a/b*(1/b*a)^(1/4)*2^(1/2)*ln((x+(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x-(1/b
*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))*c*d^2+9/16/a^2*(1/b*a)^(1/4)*2^(1/2)*ln((x+(1/b*a)^(1/4)*x^(1/2)*2^(
1/2)+(1/b*a)^(1/2))/(x-(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))*c^2*d-7/16/a^3*b*(1/b*a)^(1/4)*2^(1/2)*ln
((x+(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x-(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))*c^3
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^2+c)^3/x^(5/2)/(b*x^2+a)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 1.37307, size = 4639, normalized size = 12.64 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^2+c)^3/x^(5/2)/(b*x^2+a)^2,x, algorithm="fricas")
[Out]
-1/24*(12*(a^2*b^3*x^4 + a^3*b^2*x^2)*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324
*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d
^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^1
2)/(a^11*b^9))^(1/4)*arctan((sqrt(a^6*b^4*sqrt(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^
2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b
^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*
a^12*d^12)/(a^11*b^9)) + (49*b^6*c^6 - 126*a*b^5*c^5*d + 39*a^2*b^4*c^4*d^2 + 124*a^3*b^3*c^3*d^3 - 81*a^4*b^2
*c^2*d^4 - 30*a^5*b*c*d^5 + 25*a^6*d^6)*x)*a^8*b^7*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^
10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*
a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 +
625*a^12*d^12)/(a^11*b^9))^(3/4) - (7*a^8*b^10*c^3 - 9*a^9*b^9*c^2*d - 3*a^10*b^8*c*d^2 + 5*a^11*b^7*d^3)*sqr
t(x)*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*
c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060
*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(3/4))/(2401*b^12*
c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^
5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 -
3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)) + 3*(a^2*b^3*x^4 + a^3*b^2*x^2)*(-(2401*b^12*c^
12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*
b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3
150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4)*log(a^3*b^2*(-(2401*b^12*c^12 -
12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c
^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a
^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4) + (7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*
b*c*d^2 + 5*a^3*d^3)*sqrt(x)) - 3*(a^2*b^3*x^4 + a^3*b^2*x^2)*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*
a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d
^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11
*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4)*log(-a^3*b^2*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*
b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 -
28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c
*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4) + (7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*sqrt(x)) -
4*(12*a^2*b*d^3*x^4 - 4*a*b^2*c^3 - (7*b^3*c^3 - 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 15*a^3*d^3)*x^2)*sqrt(x))/(a^
2*b^3*x^4 + a^3*b^2*x^2)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x**2+c)**3/x**(5/2)/(b*x**2+a)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [A] time = 1.18381, size = 676, normalized size = 1.84 \begin{align*} \frac{2 \, d^{3} \sqrt{x}}{b^{2}} - \frac{2 \, c^{3}}{3 \, a^{2} x^{\frac{3}{2}}} - \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac{1}{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 )}{8 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac{1}{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 )}{8 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac{1}{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 )}{16 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac{1}{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 )}{16 \, a^{3} b^{3}} - \frac{b^{3} c^{3} \sqrt{x} - 3 \, a b^{2} c^{2} d \sqrt{x} + 3 \, a^{2} b c d^{2} \sqrt{x} - a^{3} d^{3} \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} a^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^2+c)^3/x^(5/2)/(b*x^2+a)^2,x, algorithm="giac")
[Out]
2*d^3*sqrt(x)/b^2 - 2/3*c^3/(a^2*x^(3/2)) - 1/8*sqrt(2)*(7*(a*b^3)^(1/4)*b^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d
- 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5*(a*b^3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))
/(a/b)^(1/4))/(a^3*b^3) - 1/8*sqrt(2)*(7*(a*b^3)^(1/4)*b^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)
*a^2*b*c*d^2 + 5*(a*b^3)^(1/4)*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^
3*b^3) - 1/16*sqrt(2)*(7*(a*b^3)^(1/4)*b^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5
*(a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^3) + 1/16*sqrt(2)*(7*(a*b^3)^(
1/4)*b^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5*(a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(
2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^3) - 1/2*(b^3*c^3*sqrt(x) - 3*a*b^2*c^2*d*sqrt(x) + 3*a^2*b*c*d
^2*sqrt(x) - a^3*d^3*sqrt(x))/((b*x^2 + a)*a^2*b^2)