3.373 \(\int \frac {A+B x^2}{x^{5/2} (a+b x^2)} \, dx\)

Optimal. Leaf size=237 \[ \frac {(A b-a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(A b-a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {(A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(A b-a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {2 A}{3 a x^{3/2}} \]

[Out]

-2/3*A/a/x^(3/2)+1/2*(A*b-B*a)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/b^(1/4)*2^(1/2)-1/2*(A*b-B*a)
*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/b^(1/4)*2^(1/2)+1/4*(A*b-B*a)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*
b^(1/4)*2^(1/2)*x^(1/2))/a^(7/4)/b^(1/4)*2^(1/2)-1/4*(A*b-B*a)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^
(1/2))/a^(7/4)/b^(1/4)*2^(1/2)

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Rubi [A]  time = 0.18, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {453, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {(A b-a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(A b-a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {(A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(A b-a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {2 A}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^(5/2)*(a + b*x^2)),x]

[Out]

(-2*A)/(3*a*x^(3/2)) + ((A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4)*b^(1/4)) -
((A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4)*b^(1/4)) + ((A*b - a*B)*Log[Sqrt
[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*b^(1/4)) - ((A*b - a*B)*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*b^(1/4))

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])

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 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 453

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

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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 168, normalized size = 0.71 \[ \frac {(A b-a B) \left (\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )\right )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {2 A}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^(5/2)*(a + b*x^2)),x]

[Out]

(-2*A)/(3*a*x^(3/2)) + ((A*b - a*B)*(2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)] + 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]))/(2*Sqrt[2]*a^(7/4)*b^(1/4))

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fricas [B]  time = 0.78, size = 653, normalized size = 2.76 \[ -\frac {12 \, a x^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{4} \sqrt {-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}} + {\left (B^{2} a^{2} - 2 \, A B a b + A^{2} b^{2}\right )} x} a^{5} b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {3}{4}} + {\left (B a^{6} b - A a^{5} b^{2}\right )} \sqrt {x} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {3}{4}}}{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}\right ) + 3 \, a x^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} \log \left (a^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) - 3 \, a x^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} \log \left (-a^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) + 4 \, A \sqrt {x}}{6 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(12*a*x^2*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^7*b))^(1/4)*arctan
((sqrt(a^4*sqrt(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^7*b)) + (B^2*a^2 -
2*A*B*a*b + A^2*b^2)*x)*a^5*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^7*
b))^(3/4) + (B*a^6*b - A*a^5*b^2)*sqrt(x)*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4
*b^4)/(a^7*b))^(3/4))/(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)) + 3*a*x^2*(-(B^
4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^7*b))^(1/4)*log(a^2*(-(B^4*a^4 - 4*A*B
^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^7*b))^(1/4) - (B*a - A*b)*sqrt(x)) - 3*a*x^2*(-(B^4
*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^7*b))^(1/4)*log(-a^2*(-(B^4*a^4 - 4*A*B
^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^7*b))^(1/4) - (B*a - A*b)*sqrt(x)) + 4*A*sqrt(x))/(
a*x^2)

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giac [A]  time = 0.36, size = 251, normalized size = 1.06 \[ \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{2} b} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{2} b} - \frac {2 \, A}{3 \, a x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*((a*b^3)^(1/4)*B*a - (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))/(a^2*b) + 1/2*sqrt(2)*((a*b^3)^(1/4)*B*a - (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))/(a^2*b) + 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*log(sqrt(2)*sqrt(x)*(
a/b)^(1/4) + x + sqrt(a/b))/(a^2*b) - 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*log(-sqrt(2)*sqrt(x)
*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b) - 2/3*A/(a*x^(3/2))

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maple [A]  time = 0.01, size = 280, normalized size = 1.18 \[ -\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 a^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 a^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 a^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 a}-\frac {2 A}{3 a \,x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^(5/2)/(b*x^2+a),x)

[Out]

-1/4/a^2*(a/b)^(1/4)*2^(1/2)*A*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(
a/b)^(1/2)))*b-1/2/a^2*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*b-1/2/a^2*(a/b)^(1/4)*2^(1/
2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*b+1/4/a*(a/b)^(1/4)*2^(1/2)*B*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/
b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))+1/2/a*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*
x^(1/2)+1)+1/2/a*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-2/3*A/a/x^(3/2)

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maxima [A]  time = 2.37, size = 218, normalized size = 0.92 \[ \frac {\frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{4 \, a} - \frac {2 \, A}{3 \, a x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*sqrt(2)*(B*a - A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(
b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(B*a - A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*
sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(B*a - A*b)*log(sqrt(2)*a^(1
/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(B*a - A*b)*log(-sqrt(2)*a^(1/4)*b^(1/4
)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/a - 2/3*A/(a*x^(3/2))

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mupad [B]  time = 0.38, size = 811, normalized size = 3.42 \[ -\frac {2\,A}{3\,a\,x^{3/2}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )-\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}+\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )+\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}}{\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )-\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}-\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )+\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}}\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{7/4}\,b^{1/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )-\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}+\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )+\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}}{\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )-\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}-\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )+\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}}\right )\,\left (A\,b-B\,a\right )}{{\left (-a\right )}^{7/4}\,b^{1/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^(5/2)*(a + b*x^2)),x)

[Out]

- (2*A)/(3*a*x^(3/2)) - (atan((((A*b - B*a)*(x^(1/2)*(16*A^2*a^3*b^5 + 16*B^2*a^5*b^3 - 32*A*B*a^4*b^4) - ((A*
b - B*a)*(32*A*a^5*b^4 - 32*B*a^6*b^3))/(2*(-a)^(7/4)*b^(1/4)))*1i)/(2*(-a)^(7/4)*b^(1/4)) + ((A*b - B*a)*(x^(
1/2)*(16*A^2*a^3*b^5 + 16*B^2*a^5*b^3 - 32*A*B*a^4*b^4) + ((A*b - B*a)*(32*A*a^5*b^4 - 32*B*a^6*b^3))/(2*(-a)^
(7/4)*b^(1/4)))*1i)/(2*(-a)^(7/4)*b^(1/4)))/(((A*b - B*a)*(x^(1/2)*(16*A^2*a^3*b^5 + 16*B^2*a^5*b^3 - 32*A*B*a
^4*b^4) - ((A*b - B*a)*(32*A*a^5*b^4 - 32*B*a^6*b^3))/(2*(-a)^(7/4)*b^(1/4))))/(2*(-a)^(7/4)*b^(1/4)) - ((A*b
- B*a)*(x^(1/2)*(16*A^2*a^3*b^5 + 16*B^2*a^5*b^3 - 32*A*B*a^4*b^4) + ((A*b - B*a)*(32*A*a^5*b^4 - 32*B*a^6*b^3
))/(2*(-a)^(7/4)*b^(1/4))))/(2*(-a)^(7/4)*b^(1/4))))*(A*b - B*a)*1i)/((-a)^(7/4)*b^(1/4)) - (atan((((A*b - B*a
)*(x^(1/2)*(16*A^2*a^3*b^5 + 16*B^2*a^5*b^3 - 32*A*B*a^4*b^4) - ((A*b - B*a)*(32*A*a^5*b^4 - 32*B*a^6*b^3)*1i)
/(2*(-a)^(7/4)*b^(1/4))))/(2*(-a)^(7/4)*b^(1/4)) + ((A*b - B*a)*(x^(1/2)*(16*A^2*a^3*b^5 + 16*B^2*a^5*b^3 - 32
*A*B*a^4*b^4) + ((A*b - B*a)*(32*A*a^5*b^4 - 32*B*a^6*b^3)*1i)/(2*(-a)^(7/4)*b^(1/4))))/(2*(-a)^(7/4)*b^(1/4))
)/(((A*b - B*a)*(x^(1/2)*(16*A^2*a^3*b^5 + 16*B^2*a^5*b^3 - 32*A*B*a^4*b^4) - ((A*b - B*a)*(32*A*a^5*b^4 - 32*
B*a^6*b^3)*1i)/(2*(-a)^(7/4)*b^(1/4)))*1i)/(2*(-a)^(7/4)*b^(1/4)) - ((A*b - B*a)*(x^(1/2)*(16*A^2*a^3*b^5 + 16
*B^2*a^5*b^3 - 32*A*B*a^4*b^4) + ((A*b - B*a)*(32*A*a^5*b^4 - 32*B*a^6*b^3)*1i)/(2*(-a)^(7/4)*b^(1/4)))*1i)/(2
*(-a)^(7/4)*b^(1/4))))*(A*b - B*a))/((-a)^(7/4)*b^(1/4))

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sympy [A]  time = 31.26, size = 364, normalized size = 1.54 \[ \begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} + 2 B \sqrt {x}}{a} & \text {for}\: b = 0 \\- \frac {2 A}{3 a x^{\frac {3}{2}}} + \frac {\sqrt [4]{-1} A b \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {7}{4}}} - \frac {\sqrt [4]{-1} A b \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {7}{4}}} + \frac {\sqrt [4]{-1} A b \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{a^{\frac {7}{4}}} - \frac {\sqrt [4]{-1} B \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {3}{4}}} + \frac {\sqrt [4]{-1} B \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {3}{4}}} - \frac {\sqrt [4]{-1} B \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{a^{\frac {3}{4}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**(5/2)/(b*x**2+a),x)

[Out]

Piecewise((zoo*(-2*A/(7*x**(7/2)) - 2*B/(3*x**(3/2))), Eq(a, 0) & Eq(b, 0)), ((-2*A/(7*x**(7/2)) - 2*B/(3*x**(
3/2)))/b, Eq(a, 0)), ((-2*A/(3*x**(3/2)) + 2*B*sqrt(x))/a, Eq(b, 0)), (-2*A/(3*a*x**(3/2)) + (-1)**(1/4)*A*b*(
1/b)**(1/4)*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*a**(7/4)) - (-1)**(1/4)*A*b*(1/b)**(1/4)*log(
(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*a**(7/4)) + (-1)**(1/4)*A*b*(1/b)**(1/4)*atan((-1)**(3/4)*sqrt
(x)/(a**(1/4)*(1/b)**(1/4)))/a**(7/4) - (-1)**(1/4)*B*(1/b)**(1/4)*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sq
rt(x))/(2*a**(3/4)) + (-1)**(1/4)*B*(1/b)**(1/4)*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*a**(3/4))
- (-1)**(1/4)*B*(1/b)**(1/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b)**(1/4)))/a**(3/4), True))

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