3.178 \(\int \frac{A+B x^3}{x^{7/2} (a+b x^3)^3} \, dx\)

Optimal. Leaf size=351 \[ \frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac{11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac{11 (17 A b-5 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}+\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}+\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2} \]

[Out]

(-11*(17*A*b - 5*a*B))/(180*a^3*b*x^(5/2)) + (A*b - a*B)/(6*a*b*x^(5/2)*(a + b*x^3)^2) + (17*A*b - 5*a*B)/(36*
a^2*b*x^(5/2)*(a + b*x^3)) + (11*(17*A*b - 5*a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(23/6)
*b^(1/6)) - (11*(17*A*b - 5*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(23/6)*b^(1/6)) - (11*(
17*A*b - 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(108*a^(23/6)*b^(1/6)) + (11*(17*A*b - 5*a*B)*Log[a^(1/3) -
Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqrt[3]*a^(23/6)*b^(1/6)) - (11*(17*A*b - 5*a*B)*Log[a^(1/
3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqrt[3]*a^(23/6)*b^(1/6))

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Rubi [A]  time = 0.541375, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {457, 290, 325, 329, 209, 634, 618, 204, 628, 205} \[ \frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac{11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac{11 (17 A b-5 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}+\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}+\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^3),x]

[Out]

(-11*(17*A*b - 5*a*B))/(180*a^3*b*x^(5/2)) + (A*b - a*B)/(6*a*b*x^(5/2)*(a + b*x^3)^2) + (17*A*b - 5*a*B)/(36*
a^2*b*x^(5/2)*(a + b*x^3)) + (11*(17*A*b - 5*a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(23/6)
*b^(1/6)) - (11*(17*A*b - 5*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(23/6)*b^(1/6)) - (11*(
17*A*b - 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(108*a^(23/6)*b^(1/6)) + (11*(17*A*b - 5*a*B)*Log[a^(1/3) -
Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqrt[3]*a^(23/6)*b^(1/6)) - (11*(17*A*b - 5*a*B)*Log[a^(1/
3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqrt[3]*a^(23/6)*b^(1/6))

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 290

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 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}{x^{7/2} \left (a+b x^3\right )^3} \, dx &=\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac{\left (\frac{17 A b}{2}-\frac{5 a B}{2}\right ) \int \frac{1}{x^{7/2} \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}+\frac{(11 (17 A b-5 a B)) \int \frac{1}{x^{7/2} \left (a+b x^3\right )} \, dx}{72 a^2 b}\\ &=-\frac{11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac{(11 (17 A b-5 a B)) \int \frac{1}{\sqrt{x} \left (a+b x^3\right )} \, dx}{72 a^3}\\ &=-\frac{11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac{(11 (17 A b-5 a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^6} \, dx,x,\sqrt{x}\right )}{36 a^3}\\ &=-\frac{11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac{(11 (17 A b-5 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 )}{108 a^{23/6}}-\frac{(11 (17 A b-5 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 )}{108 a^{23/6}}-\frac{(11 (17 A b-5 a B)) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{108 a^{11/3}}\\ &=-\frac{11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}-\frac{(11 (17 A b-5 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 )}{432 a^{11/3}}-\frac{(11 (17 A b-5 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 )}{432 a^{11/3}}+\frac{(11 (17 A b-5 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 )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}-\frac{(11 (17 A b-5 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 )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}\\ &=-\frac{11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}+\frac{11 (17 A b-5 a B) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}-\frac{(11 (17 A b-5 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 )}{216 \sqrt{3} a^{23/6} \sqrt [6]{b}}+\frac{(11 (17 A b-5 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 )}{216 \sqrt{3} a^{23/6} \sqrt [6]{b}}\\ &=-\frac{11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}+\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}+\frac{11 (17 A b-5 a B) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}\\ \end{align*}

Mathematica [C]  time = 0.110109, size = 96, normalized size = 0.27 \[ \frac{\frac{a \left (a^2 \left (85 B x^3-72 A\right )+a \left (55 b B x^6-289 A b x^3\right )-187 A b^2 x^6\right )}{\left (a+b x^3\right )^2}+55 x^3 (5 a B-17 A b) \, _2F_1\left (\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right )}{180 a^4 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^3),x]

[Out]

((a*(-187*A*b^2*x^6 + a^2*(-72*A + 85*B*x^3) + a*(-289*A*b*x^3 + 55*b*B*x^6)))/(a + b*x^3)^2 + 55*(-17*A*b + 5
*a*B)*x^3*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/a)])/(180*a^4*x^(5/2))

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Maple [A]  time = 0.043, size = 429, normalized size = 1.2 \begin{align*} -{\frac{23\,A{b}^{2}}{36\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{11\,Bb}{36\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{29\,Ab}{36\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}\sqrt{x}}+{\frac{17\,B}{36\,a \left ( b{x}^{3}+a \right ) ^{2}}\sqrt{x}}-{\frac{187\,Ab}{108\,{a}^{4}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{187\,Ab\sqrt{3}}{432\,{a}^{4}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{187\,Ab}{216\,{a}^{4}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ) }-{\frac{187\,Ab\sqrt{3}}{432\,{a}^{4}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{187\,Ab}{216\,{a}^{4}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }+{\frac{55\,B}{108\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{55\,B\sqrt{3}}{432\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{55\,B}{216\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ) }+{\frac{55\,B\sqrt{3}}{432\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{55\,B}{216\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }-{\frac{2\,A}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x)

[Out]

-23/36/a^3/(b*x^3+a)^2*x^(7/2)*A*b^2+11/36/a^2/(b*x^3+a)^2*x^(7/2)*b*B-29/36/a^2/(b*x^3+a)^2*A*x^(1/2)*b+17/36
/a/(b*x^3+a)^2*B*x^(1/2)-187/108/a^4*A*b*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))+187/432/a^4*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))-187/216/a^4*A*b*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)-
3^(1/2))-187/432/a^4*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))-187/216/a^4*A*b*(a/
b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))+55/108/a^3*B*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))-55/432/a^3
*B*3^(1/2)*(a/b)^(1/6)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+55/216/a^3*B*(a/b)^(1/6)*arctan(2*x^(1/2)
/(a/b)^(1/6)-3^(1/2))+55/432/a^3*B*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+55/216/a^
3*B*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))-2/5*A/a^3/x^(5/2)

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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((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.40146, size = 7052, normalized size = 20.09 \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)/x^(7/2)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

-1/2160*(220*sqrt(3)*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2
*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/
(a^23*b))^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt(a^8*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2
- 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1
/3) + (25*B^2*a^2 - 170*A*B*a*b + 289*A^2*b^2)*x + (5*B*a^5 - 17*A*a^4*b)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*
B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b
^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6))*a^19*b*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2
- 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(5
/6) + 2*sqrt(3)*(5*B*a^20*b - 17*A*a^19*b^2)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a
^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*
b))^(5/6) - sqrt(3)*(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 +
31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6))/(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 27
09375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*
A^6*b^6)) + 220*sqrt(3)*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*
A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^
6)/(a^23*b))^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt(a^8*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*
b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))
^(1/3) + (25*B^2*a^2 - 170*A*B*a*b + 289*A^2*b^2)*x - (5*B*a^5 - 17*A*a^4*b)*sqrt(x)*(-(15625*B^6*a^6 - 318750
*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*
a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6))*a^19*b*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*
b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))
^(5/6) + 2*sqrt(3)*(5*B*a^20*b - 17*A*a^19*b^2)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^
4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^
23*b))^(5/6) + sqrt(3)*(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^
3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6))/(15625*B^6*a^6 - 318750*A*B^5*a^5*b +
2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 241375
69*A^6*b^6)) - 55*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^
4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^
23*b))^(1/6)*log(121*a^8*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^
3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/3) + 121*(25*B^2*a^2
- 170*A*B*a*b + 289*A^2*b^2)*x + 121*(5*B*a^5 - 17*A*a^4*b)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 27
09375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*
A^6*b^6)/(a^23*b))^(1/6)) + 55*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2
709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569
*A^6*b^6)/(a^23*b))^(1/6)*log(121*a^8*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 122825
00*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/3) + 121
*(25*B^2*a^2 - 170*A*B*a*b + 289*A^2*b^2)*x - 121*(5*B*a^5 - 17*A*a^4*b)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B
^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^
5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)) + 110*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A
*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*
b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)*log(11*a^4*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4
*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b)
)^(1/6) - 11*(5*B*a - 17*A*b)*sqrt(x)) - 110*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A
*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*
b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)*log(-11*a^4*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^
4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b
))^(1/6) - 11*(5*B*a - 17*A*b)*sqrt(x)) - 12*(11*(5*B*a*b - 17*A*b^2)*x^6 + 17*(5*B*a^2 - 17*A*a*b)*x^3 - 72*A
*a^2)*sqrt(x))/(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^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((B*x**3+A)/x**(7/2)/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.17512, size = 451, normalized size = 1.28 \begin{align*} \frac{11 \, \sqrt{3}{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 17 \, \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 )}{432 \, a^{4} b} - \frac{11 \, \sqrt{3}{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 17 \, \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 )}{432 \, a^{4} b} + \frac{11 \,{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 17 \, \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 )}{216 \, a^{4} b} + \frac{11 \,{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 17 \, \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 )}{216 \, a^{4} b} + \frac{11 \,{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 17 \, \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 )}{108 \, a^{4} b} + \frac{11 \, B a b x^{\frac{7}{2}} - 23 \, A b^{2} x^{\frac{7}{2}} + 17 \, B a^{2} \sqrt{x} - 29 \, A a b \sqrt{x}}{36 \,{\left (b x^{3} + a\right )}^{2} a^{3}} - \frac{2 \, A}{5 \, a^{3} x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

11/432*sqrt(3)*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*log(sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))
/(a^4*b) - 11/432*sqrt(3)*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*log(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x +
(a/b)^(1/3))/(a^4*b) + 11/216*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqr
t(x))/(a/b)^(1/6))/(a^4*b) + 11/216*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6)
- 2*sqrt(x))/(a/b)^(1/6))/(a^4*b) + 11/108*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*arctan(sqrt(x)/(a/b)^(
1/6))/(a^4*b) + 1/36*(11*B*a*b*x^(7/2) - 23*A*b^2*x^(7/2) + 17*B*a^2*sqrt(x) - 29*A*a*b*sqrt(x))/((b*x^3 + a)^
2*a^3) - 2/5*A/(a^3*x^(5/2))