3.167 \(\int \frac{A+B x^3}{\sqrt{x} (a+b x^3)^2} \, dx\)

Optimal. Leaf size=289 \[ -\frac{(a B+5 A b) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}-\frac{(a B+5 A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{11/6} b^{7/6}}+\frac{\sqrt{x} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

((A*b - a*B)*Sqrt[x])/(3*a*b*(a + b*x^3)) - ((5*A*b + a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*
a^(11/6)*b^(7/6)) + ((5*A*b + a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(11/6)*b^(7/6)) + ((5*
A*b + a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(11/6)*b^(7/6)) - ((5*A*b + a*B)*Log[a^(1/3) - Sqrt[3]*a^(1
/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(11/6)*b^(7/6)) + ((5*A*b + a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)
*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(11/6)*b^(7/6))

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Rubi [A]  time = 0.468368, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {457, 329, 209, 634, 618, 204, 628, 205} \[ -\frac{(a B+5 A b) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}-\frac{(a B+5 A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{11/6} b^{7/6}}+\frac{\sqrt{x} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^3)/(Sqrt[x]*(a + b*x^3)^2),x]

[Out]

((A*b - a*B)*Sqrt[x])/(3*a*b*(a + b*x^3)) - ((5*A*b + a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*
a^(11/6)*b^(7/6)) + ((5*A*b + a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(11/6)*b^(7/6)) + ((5*
A*b + a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(11/6)*b^(7/6)) - ((5*A*b + a*B)*Log[a^(1/3) - Sqrt[3]*a^(1
/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(11/6)*b^(7/6)) + ((5*A*b + a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)
*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(11/6)*b^(7/6))

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 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 209

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x] +
Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 +
s^2*x^2), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && PosQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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])

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{A+B x^3}{\sqrt{x} \left (a+b x^3\right )^2} \, dx &=\frac{(A b-a B) \sqrt{x}}{3 a b \left (a+b x^3\right )}+\frac{\left (\frac{5 A b}{2}+\frac{a B}{2}\right ) \int \frac{1}{\sqrt{x} \left (a+b x^3\right )} \, dx}{3 a b}\\ &=\frac{(A b-a B) \sqrt{x}}{3 a b \left (a+b x^3\right )}+\frac{\left (2 \left (\frac{5 A b}{2}+\frac{a B}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^6} \, dx,x,\sqrt{x}\right )}{3 a b}\\ &=\frac{(A b-a B) \sqrt{x}}{3 a b \left (a+b x^3\right )}+\frac{(5 A b+a B) \operatorname{Subst}\left (\int \frac{\sqrt [6]{a}-\frac{1}{2} \sqrt{3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{9 a^{11/6} b}+\frac{(5 A b+a B) \operatorname{Subst}\left (\int \frac{\sqrt [6]{a}+\frac{1}{2} \sqrt{3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{9 a^{11/6} b}+\frac{(5 A b+a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{9 a^{5/3} b}\\ &=\frac{(A b-a B) \sqrt{x}}{3 a b \left (a+b x^3\right )}+\frac{(5 A b+a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{11/6} b^{7/6}}-\frac{(5 A b+a B) \operatorname{Subst}\left (\int \frac{-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}+\frac{(5 A b+a B) \operatorname{Subst}\left (\int \frac{\sqrt{3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}+\frac{(5 A b+a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{36 a^{5/3} b}+\frac{(5 A b+a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{36 a^{5/3} b}\\ &=\frac{(A b-a B) \sqrt{x}}{3 a b \left (a+b x^3\right )}+\frac{(5 A b+a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{11/6} b^{7/6}}-\frac{(5 A b+a B) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}+\frac{(5 A b+a B) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}+\frac{(5 A b+a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt{3} \sqrt [6]{a}}\right )}{18 \sqrt{3} a^{11/6} b^{7/6}}-\frac{(5 A b+a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt{3} \sqrt [6]{a}}\right )}{18 \sqrt{3} a^{11/6} b^{7/6}}\\ &=\frac{(A b-a B) \sqrt{x}}{3 a b \left (a+b x^3\right )}-\frac{(5 A b+a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{11/6} b^{7/6}}+\frac{(5 A b+a B) \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{11/6} b^{7/6}}+\frac{(5 A b+a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{11/6} b^{7/6}}-\frac{(5 A b+a B) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}+\frac{(5 A b+a B) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}\\ \end{align*}

Mathematica [C]  time = 0.0295547, size = 68, normalized size = 0.24 \[ \frac{\sqrt{x} \left (\left (a+b x^3\right ) (a B+5 A b) \, _2F_1\left (\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right )+a (A b-a B)\right )}{3 a^2 b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^3)/(Sqrt[x]*(a + b*x^3)^2),x]

[Out]

(Sqrt[x]*(a*(A*b - a*B) + (5*A*b + a*B)*(a + b*x^3)*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/a)]))/(3*a^2*b*(a
+ b*x^3))

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Maple [A]  time = 0.037, size = 381, normalized size = 1.3 \begin{align*}{\frac{Ab-Ba}{3\,ab \left ( b{x}^{3}+a \right ) }\sqrt{x}}+{\frac{5\,A}{9\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{B}{9\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{5\,\sqrt{3}A}{36\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{\sqrt{3}B}{36\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{5\,A}{18\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ) }+{\frac{B}{18\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ) }+{\frac{5\,\sqrt{3}A}{36\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{\sqrt{3}B}{36\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{5\,A}{18\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }+{\frac{B}{18\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^3+A)/(b*x^3+a)^2/x^(1/2),x)

[Out]

1/3*(A*b-B*a)*x^(1/2)/a/b/(b*x^3+a)+5/9/a^2*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))*A+1/9/a/b*(a/b)^(1/6)*arct
an(x^(1/2)/(a/b)^(1/6))*B-5/36/a^2*3^(1/2)*(a/b)^(1/6)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))*A-1/36/a/
b*3^(1/2)*(a/b)^(1/6)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))*B+5/18/a^2*(a/b)^(1/6)*arctan(2*x^(1/2)/(a
/b)^(1/6)-3^(1/2))*A+1/18/a/b*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)-3^(1/2))*B+5/36/a^2*3^(1/2)*(a/b)^(1/6)
*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))*A+1/36/a/b*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)
+(a/b)^(1/3))*B+5/18/a^2*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))*A+1/18/a/b*(a/b)^(1/6)*arctan(2*x^(
1/2)/(a/b)^(1/6)+3^(1/2))*B

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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((B*x^3+A)/(b*x^3+a)^2/x^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.33607, size = 5873, normalized size = 20.32 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/(b*x^3+a)^2/x^(1/2),x, algorithm="fricas")

[Out]

1/36*(4*sqrt(3)*(a*b^2*x^3 + a^2*b)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 +
9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt(a^4*b^
2*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*
B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/3) + (B^2*a^2 + 10*A*B*a*b + 25*A^2*b^2)*x + (B*a^3*b + 5*A*a^2*b^2)*s
qrt(x)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750
*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6))*a^9*b^6*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2
500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(5/6) - 2*sqrt(3)*
(B*a^10*b^6 + 5*A*a^9*b^7)*sqrt(x)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 +
9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(5/6) + sqrt(3)*(B^6*a^6 + 30*A*B^5*a^5*
b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6))/(B
^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^
5 + 15625*A^6*b^6)) + 4*sqrt(3)*(a*b^2*x^3 + a^2*b)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A
^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)*arctan(1/3*(2*sqr
t(3)*sqrt(a^4*b^2*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*
b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/3) + (B^2*a^2 + 10*A*B*a*b + 25*A^2*b^2)*x - (B*a^3*b
+ 5*A*a^2*b^2)*sqrt(x)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2
*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6))*a^9*b^6*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2
*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(5
/6) - 2*sqrt(3)*(B*a^10*b^6 + 5*A*a^9*b^7)*sqrt(x)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^
3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(5/6) - sqrt(3)*(B^6*a^6
+ 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15
625*A^6*b^6))/(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 +
18750*A^5*B*a*b^5 + 15625*A^6*b^6)) + (a*b^2*x^3 + a^2*b)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 +
2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)*log(a^4*b^2
*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B
*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/3) + (B^2*a^2 + 10*A*B*a*b + 25*A^2*b^2)*x + (B*a^3*b + 5*A*a^2*b^2)*sq
rt(x)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*
A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)) - (a*b^2*x^3 + a^2*b)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^
4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)
*log(a^4*b^2*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 +
18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/3) + (B^2*a^2 + 10*A*B*a*b + 25*A^2*b^2)*x - (B*a^3*b + 5*A
*a^2*b^2)*sqrt(x)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*
b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)) + 2*(a*b^2*x^3 + a^2*b)*(-(B^6*a^6 + 30*A*B^5*a^5*
b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^
11*b^7))^(1/6)*log(a^2*b*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B
^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6) + (B*a + 5*A*b)*sqrt(x)) - 2*(a*b^2*x^3 + a^
2*b)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A
^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)*log(-a^2*b*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2
500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6) + (B*a + 5*A
*b)*sqrt(x)) - 12*(B*a - A*b)*sqrt(x))/(a*b^2*x^3 + a^2*b)

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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((B*x**3+A)/(b*x**3+a)**2/x**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.18149, size = 408, normalized size = 1.41 \begin{align*} \frac{\sqrt{3}{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \log \left (\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{36 \, a^{2} b^{2}} - \frac{\sqrt{3}{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \log \left (-\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{36 \, a^{2} b^{2}} - \frac{B a \sqrt{x} - A b \sqrt{x}}{3 \,{\left (b x^{3} + a\right )} a b} + \frac{{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} + 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a^{2} b^{2}} + \frac{{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (-\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} - 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a^{2} b^{2}} + \frac{{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{9 \, a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/(b*x^3+a)^2/x^(1/2),x, algorithm="giac")

[Out]

1/36*sqrt(3)*((a*b^5)^(1/6)*B*a + 5*(a*b^5)^(1/6)*A*b)*log(sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^2
*b^2) - 1/36*sqrt(3)*((a*b^5)^(1/6)*B*a + 5*(a*b^5)^(1/6)*A*b)*log(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1
/3))/(a^2*b^2) - 1/3*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^3 + a)*a*b) + 1/18*((a*b^5)^(1/6)*B*a + 5*(a*b^5)^(1/6)
*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6))/(a^2*b^2) + 1/18*((a*b^5)^(1/6)*B*a + 5*(a*b^5)^(1
/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2*sqrt(x))/(a/b)^(1/6))/(a^2*b^2) + 1/9*((a*b^5)^(1/6)*B*a + 5*(a*b^5)
^(1/6)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/(a^2*b^2)