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

Optimal. Leaf size=276 \[ -\frac{a^{5/4} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{13/4}}+\frac{a^{5/4} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{13/4}}-\frac{a^{5/4} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{13/4}}+\frac{a^{5/4} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{13/4}}+\frac{2 x^{5/2} (A b-a B)}{5 b^2}-\frac{2 a \sqrt{x} (A b-a B)}{b^3}+\frac{2 B x^{9/2}}{9 b} \]

[Out]

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

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Rubi [A]  time = 0.258758, antiderivative size = 276, 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 = {459, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{a^{5/4} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{13/4}}+\frac{a^{5/4} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{13/4}}-\frac{a^{5/4} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{13/4}}+\frac{a^{5/4} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{13/4}}+\frac{2 x^{5/2} (A b-a B)}{5 b^2}-\frac{2 a \sqrt{x} (A b-a B)}{b^3}+\frac{2 B x^{9/2}}{9 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 459

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

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

Mathematica [A]  time = 0.276573, size = 227, normalized size = 0.82 \[ \frac{\frac{45 \sqrt{2} a^{5/4} (a B-A b) \left (\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )\right )}{\sqrt [4]{b}}+\frac{90 \sqrt{2} a^{5/4} (a B-A b) \left (\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{\sqrt [4]{b}}+72 b x^{5/2} (A b-a B)+360 a \sqrt{x} (a B-A b)+40 b^2 B x^{9/2}}{180 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(360*a*(-(A*b) + a*B)*Sqrt[x] + 72*b*(A*b - a*B)*x^(5/2) + 40*b^2*B*x^(9/2) + (90*Sqrt[2]*a^(5/4)*(-(A*b) + a*
B)*(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]))/b^(1/4) +
(45*Sqrt[2]*a^(5/4)*(-(A*b) + a*B)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/b^(1/4))/(180*b^3)

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Maple [A]  time = 0.009, size = 330, normalized size = 1.2 \begin{align*}{\frac{2\,B}{9\,b}{x}^{{\frac{9}{2}}}}+{\frac{2\,A}{5\,b}{x}^{{\frac{5}{2}}}}-{\frac{2\,Ba}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}-2\,{\frac{aA\sqrt{x}}{{b}^{2}}}+2\,{\frac{{a}^{2}B\sqrt{x}}{{b}^{3}}}+{\frac{a\sqrt{2}A}{2\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{a\sqrt{2}A}{2\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{a\sqrt{2}A}{4\,{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{{a}^{2}\sqrt{2}B}{2\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{{a}^{2}\sqrt{2}B}{2\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{{a}^{2}\sqrt{2}B}{4\,{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),x)

[Out]

2/9*B*x^(9/2)/b+2/5/b*A*x^(5/2)-2/5/b^2*B*x^(5/2)*a-2/b^2*a*A*x^(1/2)+2/b^3*a^2*B*x^(1/2)+1/2*a/b^2*(1/b*a)^(1
/4)*2^(1/2)*A*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)+1/2*a/b^2*(1/b*a)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(1/b*a)
^(1/4)*x^(1/2)-1)+1/4*a/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)))-1/2*a^2/b^3*(1/b*a)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(
1/2)+1)-1/2*a^2/b^3*(1/b*a)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)-1/4*a^2/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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.932851, size = 1477, normalized size = 5.35 \begin{align*} \frac{180 \, b^{3} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{b^{6} \sqrt{-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}} +{\left (B^{2} a^{4} - 2 \, A B a^{3} b + A^{2} a^{2} b^{2}\right )} x} b^{10} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{3}{4}} +{\left (B a^{2} b^{10} - A a b^{11}\right )} \sqrt{x} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{3}{4}}}{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}\right ) + 45 \, b^{3} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{1}{4}} \log \left (b^{3} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{1}{4}} -{\left (B a^{2} - A a b\right )} \sqrt{x}\right ) - 45 \, b^{3} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{1}{4}} \log \left (-b^{3} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{1}{4}} -{\left (B a^{2} - A a b\right )} \sqrt{x}\right ) + 4 \,{\left (5 \, B b^{2} x^{4} + 45 \, B a^{2} - 45 \, A a b - 9 \,{\left (B a b - A b^{2}\right )} x^{2}\right )} \sqrt{x}}{90 \, b^{3}} \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),x, algorithm="fricas")

[Out]

1/90*(180*b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(1/4)*arct
an((sqrt(b^6*sqrt(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13) + (B^2*
a^4 - 2*A*B*a^3*b + A^2*a^2*b^2)*x)*b^10*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^
4*a^5*b^4)/b^13)^(3/4) + (B*a^2*b^10 - A*a*b^11)*sqrt(x)*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^
3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(3/4))/(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a
^5*b^4)) + 45*b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(1/4)*
log(b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(1/4) - (B*a^2 -
A*a*b)*sqrt(x)) - 45*b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13
)^(1/4)*log(-b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(1/4) -
(B*a^2 - A*a*b)*sqrt(x)) + 4*(5*B*b^2*x^4 + 45*B*a^2 - 45*A*a*b - 9*(B*a*b - A*b^2)*x^2)*sqrt(x))/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),x)

[Out]

Timed out

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Giac [A]  time = 1.14532, size = 402, normalized size = 1.46 \begin{align*} -\frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac{1}{4}} A 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 )}{2 \, b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac{1}{4}} A 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 )}{2 \, b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac{1}{4}} A a b\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac{1}{4}} A a b\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{4}} + \frac{2 \,{\left (5 \, B b^{8} x^{\frac{9}{2}} - 9 \, B a b^{7} x^{\frac{5}{2}} + 9 \, A b^{8} x^{\frac{5}{2}} + 45 \, B a^{2} b^{6} \sqrt{x} - 45 \, A a b^{7} \sqrt{x}\right )}}{45 \, b^{9}} \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),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/
(a/b)^(1/4))/b^4 - 1/2*sqrt(2)*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^
(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^4 - 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*log(sqrt(2)*sqrt
(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^4 + 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*log(-sqrt(2)*sq
rt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^4 + 2/45*(5*B*b^8*x^(9/2) - 9*B*a*b^7*x^(5/2) + 9*A*b^8*x^(5/2) + 45*B*a^
2*b^6*sqrt(x) - 45*A*a*b^7*sqrt(x))/b^9