3.375 \(\int \frac{x^{7/2} (A+B x^2)}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=310 \[ -\frac{x^{5/2} (5 A b-9 a B)}{10 a b^2}+\frac{\sqrt{x} (5 A b-9 a B)}{2 b^3}+\frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}+\frac{\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{13/4}}+\frac{x^{9/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

[Out]

((5*A*b - 9*a*B)*Sqrt[x])/(2*b^3) - ((5*A*b - 9*a*B)*x^(5/2))/(10*a*b^2) + ((A*b - a*B)*x^(9/2))/(2*a*b*(a + b
*x^2)) + (a^(1/4)*(5*A*b - 9*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(13/4)) - (a^(1/
4)*(5*A*b - 9*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(13/4)) + (a^(1/4)*(5*A*b - 9*a
*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(13/4)) - (a^(1/4)*(5*A*b - 9*a*B
)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(13/4))

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Rubi [A]  time = 0.243083, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {457, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{x^{5/2} (5 A b-9 a B)}{10 a b^2}+\frac{\sqrt{x} (5 A b-9 a B)}{2 b^3}+\frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}+\frac{\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{13/4}}+\frac{x^{9/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

((5*A*b - 9*a*B)*Sqrt[x])/(2*b^3) - ((5*A*b - 9*a*B)*x^(5/2))/(10*a*b^2) + ((A*b - a*B)*x^(9/2))/(2*a*b*(a + b
*x^2)) + (a^(1/4)*(5*A*b - 9*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(13/4)) - (a^(1/
4)*(5*A*b - 9*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(13/4)) + (a^(1/4)*(5*A*b - 9*a
*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(13/4)) - (a^(1/4)*(5*A*b - 9*a*B
)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(13/4))

Rule 457

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 321

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

Rule 329

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 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{x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac{(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}+\frac{\left (-\frac{5 A b}{2}+\frac{9 a B}{2}\right ) \int \frac{x^{7/2}}{a+b x^2} \, dx}{2 a b}\\ &=-\frac{(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac{(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}+\frac{(5 A b-9 a B) \int \frac{x^{3/2}}{a+b x^2} \, dx}{4 b^2}\\ &=\frac{(5 A b-9 a B) \sqrt{x}}{2 b^3}-\frac{(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac{(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}-\frac{(a (5 A b-9 a B)) \int \frac{1}{\sqrt{x} \left (a+b x^2\right )} \, dx}{4 b^3}\\ &=\frac{(5 A b-9 a B) \sqrt{x}}{2 b^3}-\frac{(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac{(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}-\frac{(a (5 A b-9 a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 b^3}\\ &=\frac{(5 A b-9 a B) \sqrt{x}}{2 b^3}-\frac{(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac{(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}-\frac{\left (\sqrt{a} (5 A b-9 a B)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 b^3}-\frac{\left (\sqrt{a} (5 A b-9 a B)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 b^3}\\ &=\frac{(5 A b-9 a B) \sqrt{x}}{2 b^3}-\frac{(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac{(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}-\frac{\left (\sqrt{a} (5 A b-9 a B)\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 b^{7/2}}-\frac{\left (\sqrt{a} (5 A b-9 a B)\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 b^{7/2}}+\frac{\left (\sqrt [4]{a} (5 A b-9 a B)\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} b^{13/4}}+\frac{\left (\sqrt [4]{a} (5 A b-9 a B)\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} b^{13/4}}\\ &=\frac{(5 A b-9 a B) \sqrt{x}}{2 b^3}-\frac{(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac{(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}+\frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}-\frac{\left (\sqrt [4]{a} (5 A b-9 a B)\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} b^{13/4}}+\frac{\left (\sqrt [4]{a} (5 A b-9 a B)\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} b^{13/4}}\\ &=\frac{(5 A b-9 a B) \sqrt{x}}{2 b^3}-\frac{(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac{(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}+\frac{\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{13/4}}+\frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}\\ \end{align*}

Mathematica [A]  time = 0.429392, size = 385, normalized size = 1.24 \[ \frac{-\frac{40 a^2 \sqrt [4]{b} B \sqrt{x}}{a+b x^2}-45 \sqrt{2} a^{5/4} B \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+45 \sqrt{2} a^{5/4} B \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+\frac{40 a A b^{5/4} \sqrt{x}}{a+b x^2}-10 \sqrt{2} \sqrt [4]{a} (9 a B-5 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+10 \sqrt{2} \sqrt [4]{a} (9 a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+25 \sqrt{2} \sqrt [4]{a} A b \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-25 \sqrt{2} \sqrt [4]{a} A b \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-320 a \sqrt [4]{b} B \sqrt{x}+160 A b^{5/4} \sqrt{x}+32 b^{5/4} B x^{5/2}}{80 b^{13/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

(160*A*b^(5/4)*Sqrt[x] - 320*a*b^(1/4)*B*Sqrt[x] + 32*b^(5/4)*B*x^(5/2) + (40*a*A*b^(5/4)*Sqrt[x])/(a + b*x^2)
- (40*a^2*b^(1/4)*B*Sqrt[x])/(a + b*x^2) - 10*Sqrt[2]*a^(1/4)*(-5*A*b + 9*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sq
rt[x])/a^(1/4)] + 10*Sqrt[2]*a^(1/4)*(-5*A*b + 9*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 25*Sqrt[
2]*a^(1/4)*A*b*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 45*Sqrt[2]*a^(5/4)*B*Log[Sqrt[a] -
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 25*Sqrt[2]*a^(1/4)*A*b*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[x] + Sqrt[b]*x] + 45*Sqrt[2]*a^(5/4)*B*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(80*b^(
13/4))

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Maple [A]  time = 0.014, size = 339, normalized size = 1.1 \begin{align*}{\frac{2\,B}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}+2\,{\frac{A\sqrt{x}}{{b}^{2}}}-4\,{\frac{Ba\sqrt{x}}{{b}^{3}}}+{\frac{Aa}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }\sqrt{x}}-{\frac{{a}^{2}B}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }\sqrt{x}}-{\frac{5\,\sqrt{2}A}{8\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}A}{8\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{5\,\sqrt{2}A}{16\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{9\,a\sqrt{2}B}{8\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{9\,a\sqrt{2}B}{8\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{9\,a\sqrt{2}B}{16\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x^2+A)/(b*x^2+a)^2,x)

[Out]

2/5/b^2*B*x^(5/2)+2/b^2*A*x^(1/2)-4/b^3*B*a*x^(1/2)+1/2*a/b^2*x^(1/2)/(b*x^2+a)*A-1/2*a^2/b^3*x^(1/2)/(b*x^2+a
)*B-5/8/b^2*(1/b*a)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)-5/8/b^2*(1/b*a)^(1/4)*2^(1/2)*A*ar
ctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)-5/16/b^2*(1/b*a)^(1/4)*2^(1/2)*A*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)))+9/8*a/b^3*(1/b*a)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/
(1/b*a)^(1/4)*x^(1/2)+1)+9/8*a/b^3*(1/b*a)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)+9/16*a/b^3*
(1/b*a)^(1/4)*2^(1/2)*B*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)))

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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(x^(7/2)*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.00914, size = 1775, normalized size = 5.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

-1/40*(20*(b^4*x^2 + a*b^3)*(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 +
625*A^4*a*b^4)/b^13)^(1/4)*arctan((sqrt(b^6*sqrt(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 -
4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^13) + (81*B^2*a^2 - 90*A*B*a*b + 25*A^2*b^2)*x)*b^10*(-(6561*B^4*a^5 -
14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^13)^(3/4) + (9*B*a*b^10 - 5*
A*b^11)*sqrt(x)*(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b
^4)/b^13)^(3/4))/(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^
4)) + 5*(b^4*x^2 + a*b^3)*(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 6
25*A^4*a*b^4)/b^13)^(1/4)*log(b^3*(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2
*b^3 + 625*A^4*a*b^4)/b^13)^(1/4) - (9*B*a - 5*A*b)*sqrt(x)) - 5*(b^4*x^2 + a*b^3)*(-(6561*B^4*a^5 - 14580*A*B
^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^13)^(1/4)*log(-b^3*(-(6561*B^4*a^5 -
14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^13)^(1/4) - (9*B*a - 5*A*b)*
sqrt(x)) - 4*(4*B*b^2*x^4 - 45*B*a^2 + 25*A*a*b - 4*(9*B*a*b - 5*A*b^2)*x^2)*sqrt(x))/(b^4*x^2 + a*b^3)

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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(x**(7/2)*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.16991, size = 402, normalized size = 1.3 \begin{align*} \frac{\sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\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 \, b^{4}} + \frac{\sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\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 \, b^{4}} + \frac{\sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, b^{4}} - \frac{\sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, b^{4}} - \frac{B a^{2} \sqrt{x} - A a b \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} b^{3}} + \frac{2 \,{\left (B b^{8} x^{\frac{5}{2}} - 10 \, B a b^{7} \sqrt{x} + 5 \, A b^{8} \sqrt{x}\right )}}{5 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

1/8*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - 5*(a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(
a/b)^(1/4))/b^4 + 1/8*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - 5*(a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(
1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^4 + 1/16*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - 5*(a*b^3)^(1/4)*A*b)*log(sqrt(2)*sqrt
(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^4 - 1/16*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - 5*(a*b^3)^(1/4)*A*b)*log(-sqrt(2)*s
qrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^4 - 1/2*(B*a^2*sqrt(x) - A*a*b*sqrt(x))/((b*x^2 + a)*b^3) + 2/5*(B*b^8*x
^(5/2) - 10*B*a*b^7*sqrt(x) + 5*A*b^8*sqrt(x))/b^10