3.2.97 \(\int \frac {A+B x^3}{x^4 (a+b x^3)^{3/2}} \, dx\)

Optimal. Leaf size=86 \[ \frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}}+\frac {2 a B-3 A b}{3 a^2 \sqrt {a+b x^3}}-\frac {A}{3 a x^3 \sqrt {a+b x^3}} \]

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Rubi [A]  time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 51, 63, 208} \begin {gather*} -\frac {3 A b-2 a B}{3 a^2 \sqrt {a+b x^3}}+\frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}}-\frac {A}{3 a x^3 \sqrt {a+b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^3)/(x^4*(a + b*x^3)^(3/2)),x]

[Out]

-(3*A*b - 2*a*B)/(3*a^2*Sqrt[a + b*x^3]) - A/(3*a*x^3*Sqrt[a + b*x^3]) + ((3*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x
^3]/Sqrt[a]])/(3*a^(5/2))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 208

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

Rule 446

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

Rubi steps

\begin {align*} \int \frac {A+B x^3}{x^4 \left (a+b x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {A}{3 a x^3 \sqrt {a+b x^3}}+\frac {\left (-\frac {3 A b}{2}+a B\right ) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^3\right )}{3 a}\\ &=-\frac {3 A b-2 a B}{3 a^2 \sqrt {a+b x^3}}-\frac {A}{3 a x^3 \sqrt {a+b x^3}}-\frac {(3 A b-2 a B) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{6 a^2}\\ &=-\frac {3 A b-2 a B}{3 a^2 \sqrt {a+b x^3}}-\frac {A}{3 a x^3 \sqrt {a+b x^3}}-\frac {(3 A b-2 a B) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 a^2 b}\\ &=-\frac {3 A b-2 a B}{3 a^2 \sqrt {a+b x^3}}-\frac {A}{3 a x^3 \sqrt {a+b x^3}}+\frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 57, normalized size = 0.66 \begin {gather*} \frac {x^3 (2 a B-3 A b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x^3}{a}+1\right )-a A}{3 a^2 x^3 \sqrt {a+b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^3)/(x^4*(a + b*x^3)^(3/2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.13, size = 77, normalized size = 0.90 \begin {gather*} \frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}}+\frac {-a A+2 a B x^3-3 A b x^3}{3 a^2 x^3 \sqrt {a+b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x^3)/(x^4*(a + b*x^3)^(3/2)),x]

[Out]

(-(a*A) - 3*A*b*x^3 + 2*a*B*x^3)/(3*a^2*x^3*Sqrt[a + b*x^3]) + ((3*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a
]])/(3*a^(5/2))

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fricas [A]  time = 0.96, size = 233, normalized size = 2.71 \begin {gather*} \left [-\frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{6} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3}\right )} \sqrt {a} \log \left (\frac {b x^{3} + 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) - 2 \, {\left ({\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3} - A a^{2}\right )} \sqrt {b x^{3} + a}}{6 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}, \frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{6} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left ({\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3} - A a^{2}\right )} \sqrt {b x^{3} + a}}{3 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^4/(b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/6*(((2*B*a*b - 3*A*b^2)*x^6 + (2*B*a^2 - 3*A*a*b)*x^3)*sqrt(a)*log((b*x^3 + 2*sqrt(b*x^3 + a)*sqrt(a) + 2*
a)/x^3) - 2*((2*B*a^2 - 3*A*a*b)*x^3 - A*a^2)*sqrt(b*x^3 + a))/(a^3*b*x^6 + a^4*x^3), 1/3*(((2*B*a*b - 3*A*b^2
)*x^6 + (2*B*a^2 - 3*A*a*b)*x^3)*sqrt(-a)*arctan(sqrt(b*x^3 + a)*sqrt(-a)/a) + ((2*B*a^2 - 3*A*a*b)*x^3 - A*a^
2)*sqrt(b*x^3 + a))/(a^3*b*x^6 + a^4*x^3)]

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giac [A]  time = 0.17, size = 99, normalized size = 1.15 \begin {gather*} \frac {{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a} a^{2}} + \frac {2 \, {\left (b x^{3} + a\right )} B a - 2 \, B a^{2} - 3 \, {\left (b x^{3} + a\right )} A b + 2 \, A a b}{3 \, {\left ({\left (b x^{3} + a\right )}^{\frac {3}{2}} - \sqrt {b x^{3} + a} a\right )} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^4/(b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 100, normalized size = 1.16 \begin {gather*} \left (\frac {b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}-\frac {2 b}{3 \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}\, a^{2}}-\frac {\sqrt {b \,x^{3}+a}}{3 a^{2} x^{3}}\right ) A +\left (-\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {3}{2}}}+\frac {2}{3 \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}\, a}\right ) B \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^3+A)/x^4/(b*x^3+a)^(3/2),x)

[Out]

A*(-1/3*(b*x^3+a)^(1/2)/a^2/x^3-2/3/((x^3+a/b)*b)^(1/2)/a^2*b+b*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(5/2))+B*(2
/3/((x^3+a/b)*b)^(1/2)/a-2/3*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(3/2))

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maxima [B]  time = 1.39, size = 144, normalized size = 1.67 \begin {gather*} -\frac {1}{6} \, A {\left (\frac {2 \, {\left (3 \, {\left (b x^{3} + a\right )} b - 2 \, a b\right )}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2} - \sqrt {b x^{3} + a} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )} + \frac {1}{3} \, B {\left (\frac {\log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {b x^{3} + a} a}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^4/(b*x^3+a)^(3/2),x, algorithm="maxima")

[Out]

-1/6*A*(2*(3*(b*x^3 + a)*b - 2*a*b)/((b*x^3 + a)^(3/2)*a^2 - sqrt(b*x^3 + a)*a^3) + 3*b*log((sqrt(b*x^3 + a) -
sqrt(a))/(sqrt(b*x^3 + a) + sqrt(a)))/a^(5/2)) + 1/3*B*(log((sqrt(b*x^3 + a) - sqrt(a))/(sqrt(b*x^3 + a) + sq
rt(a)))/a^(3/2) + 2/(sqrt(b*x^3 + a)*a))

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mupad [B]  time = 2.93, size = 131, normalized size = 1.52 \begin {gather*} \frac {\ln \left (\frac {\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )\,{\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}^3}{x^6}\right )\,\left (3\,A\,b-2\,B\,a\right )}{6\,a^{5/2}}-\frac {\frac {2\,B\,a^2-3\,A\,a\,b}{2\,a^3}-\frac {a\,\left (\frac {A\,b^2}{3\,a^3}+\frac {5\,b\,\left (2\,B\,a^2-3\,A\,a\,b\right )}{6\,a^4}\right )}{b}}{\sqrt {b\,x^3+a}}-\frac {A\,\sqrt {b\,x^3+a}}{3\,a^2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^3)/(x^4*(a + b*x^3)^(3/2)),x)

[Out]

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

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sympy [B]  time = 79.79, size = 264, normalized size = 3.07 \begin {gather*} A \left (- \frac {1}{3 a \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b}}{a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{a^{\frac {5}{2}}}\right ) + B \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{3}}{a}}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} + \frac {a^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} + \frac {a^{2} b x^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} - \frac {2 a^{2} b x^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**3+A)/x**4/(b*x**3+a)**(3/2),x)

[Out]

A*(-1/(3*a*sqrt(b)*x**(9/2)*sqrt(a/(b*x**3) + 1)) - sqrt(b)/(a**2*x**(3/2)*sqrt(a/(b*x**3) + 1)) + b*asinh(sqr
t(a)/(sqrt(b)*x**(3/2)))/a**(5/2)) + B*(2*a**3*sqrt(1 + b*x**3/a)/(3*a**(9/2) + 3*a**(7/2)*b*x**3) + a**3*log(
b*x**3/a)/(3*a**(9/2) + 3*a**(7/2)*b*x**3) - 2*a**3*log(sqrt(1 + b*x**3/a) + 1)/(3*a**(9/2) + 3*a**(7/2)*b*x**
3) + a**2*b*x**3*log(b*x**3/a)/(3*a**(9/2) + 3*a**(7/2)*b*x**3) - 2*a**2*b*x**3*log(sqrt(1 + b*x**3/a) + 1)/(3
*a**(9/2) + 3*a**(7/2)*b*x**3))

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