3.6.33 \(\int \frac {(a+b x^3)^{3/2} (A+B x^3)}{(e x)^{5/2}} \, dx\) [533]

Optimal. Leaf size=152 \[ \frac {(4 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{4 e^4}+\frac {(4 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 a e^4}-\frac {2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {a (4 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{4 \sqrt {b} e^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {464, 285, 335, 281, 223, 212} \begin {gather*} \frac {a (a B+4 A b) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{4 \sqrt {b} e^{5/2}}+\frac {(e x)^{3/2} \left (a+b x^3\right )^{3/2} (a B+4 A b)}{6 a e^4}+\frac {(e x)^{3/2} \sqrt {a+b x^3} (a B+4 A b)}{4 e^4}-\frac {2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^(3/2)*(A + B*x^3))/(e*x)^(5/2),x]

[Out]

((4*A*b + a*B)*(e*x)^(3/2)*Sqrt[a + b*x^3])/(4*e^4) + ((4*A*b + a*B)*(e*x)^(3/2)*(a + b*x^3)^(3/2))/(6*a*e^4)
- (2*A*(a + b*x^3)^(5/2))/(3*a*e*(e*x)^(3/2)) + (a*(4*A*b + a*B)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a
+ b*x^3])])/(4*Sqrt[b]*e^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 285

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

Rule 335

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 464

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{(e x)^{5/2}} \, dx &=-\frac {2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {(4 A b+a B) \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \, dx}{a e^3}\\ &=\frac {(4 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 a e^4}-\frac {2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {(3 (4 A b+a B)) \int \sqrt {e x} \sqrt {a+b x^3} \, dx}{4 e^3}\\ &=\frac {(4 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{4 e^4}+\frac {(4 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 a e^4}-\frac {2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {(3 a (4 A b+a B)) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{8 e^3}\\ &=\frac {(4 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{4 e^4}+\frac {(4 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 a e^4}-\frac {2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {(3 a (4 A b+a B)) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{4 e^4}\\ &=\frac {(4 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{4 e^4}+\frac {(4 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 a e^4}-\frac {2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {(a (4 A b+a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^2}{e^3}}} \, dx,x,(e x)^{3/2}\right )}{4 e^4}\\ &=\frac {(4 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{4 e^4}+\frac {(4 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 a e^4}-\frac {2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {(a (4 A b+a B)) \text {Subst}\left (\int \frac {1}{1-\frac {b x^2}{e^3}} \, dx,x,\frac {(e x)^{3/2}}{\sqrt {a+b x^3}}\right )}{4 e^4}\\ &=\frac {(4 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{4 e^4}+\frac {(4 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 a e^4}-\frac {2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {a (4 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{4 \sqrt {b} e^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 100, normalized size = 0.66 \begin {gather*} \frac {x \left (\sqrt {b} \sqrt {a+b x^3} \left (-8 a A+4 A b x^3+5 a B x^3+2 b B x^6\right )+3 a (4 A b+a B) x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {b} x^{3/2}}\right )\right )}{12 \sqrt {b} (e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/(e*x)^(5/2),x]

[Out]

(x*(Sqrt[b]*Sqrt[a + b*x^3]*(-8*a*A + 4*A*b*x^3 + 5*a*B*x^3 + 2*b*B*x^6) + 3*a*(4*A*b + a*B)*x^(3/2)*ArcTanh[S
qrt[a + b*x^3]/(Sqrt[b]*x^(3/2))]))/(12*Sqrt[b]*(e*x)^(5/2))

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.40, size = 7108, normalized size = 46.76

method	result	size
risch	\(\text {Expression too large to display}\)	\(1067\)
elliptic	\(\text {Expression too large to display}\)	\(1140\)
default	\(\text {Expression too large to display}\)	\(7108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(3/2)*(B*x^3+A)/(e*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (105) = 210\).
time = 0.54, size = 223, normalized size = 1.47 \begin {gather*} -\frac {1}{24} \, {\left (4 \, {\left (3 \, a \sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right ) + \frac {4 \, \sqrt {b x^{3} + a} a}{x^{\frac {3}{2}}} + \frac {2 \, \sqrt {b x^{3} + a} a b}{{\left (b - \frac {b x^{3} + a}{x^{3}}\right )} x^{\frac {3}{2}}}\right )} A + {\left (\frac {3 \, a^{2} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{\sqrt {b}} + \frac {2 \, {\left (\frac {3 \, \sqrt {b x^{3} + a} a^{2} b}{x^{\frac {3}{2}}} - \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {9}{2}}}\right )}}{b^{2} - \frac {2 \, {\left (b x^{3} + a\right )} b}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2}}{x^{6}}}\right )} B\right )} e^{\left (-\frac {5}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(3/2)*(B*x^3+A)/(e*x)^(5/2),x, algorithm="maxima")

[Out]

-1/24*(4*(3*a*sqrt(b)*log(-(sqrt(b) - sqrt(b*x^3 + a)/x^(3/2))/(sqrt(b) + sqrt(b*x^3 + a)/x^(3/2))) + 4*sqrt(b
*x^3 + a)*a/x^(3/2) + 2*sqrt(b*x^3 + a)*a*b/((b - (b*x^3 + a)/x^3)*x^(3/2)))*A + (3*a^2*log(-(sqrt(b) - sqrt(b
*x^3 + a)/x^(3/2))/(sqrt(b) + sqrt(b*x^3 + a)/x^(3/2)))/sqrt(b) + 2*(3*sqrt(b*x^3 + a)*a^2*b/x^(3/2) - 5*(b*x^
3 + a)^(3/2)*a^2/x^(9/2))/(b^2 - 2*(b*x^3 + a)*b/x^3 + (b*x^3 + a)^2/x^6))*B)*e^(-5/2)

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Fricas [A]
time = 3.88, size = 232, normalized size = 1.53 \begin {gather*} \left [\frac {{\left (3 \, {\left (B a^{2} + 4 \, A a b\right )} \sqrt {b} x^{2} \log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - 4 \, {\left (2 \, b x^{4} + a x\right )} \sqrt {b x^{3} + a} \sqrt {b} \sqrt {x} - a^{2}\right ) + 4 \, {\left (2 \, B b^{2} x^{6} + {\left (5 \, B a b + 4 \, A b^{2}\right )} x^{3} - 8 \, A a b\right )} \sqrt {b x^{3} + a} \sqrt {x}\right )} e^{\left (-\frac {5}{2}\right )}}{48 \, b x^{2}}, -\frac {{\left (3 \, {\left (B a^{2} + 4 \, A a b\right )} \sqrt {-b} x^{2} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-b} x^{\frac {3}{2}}}{2 \, b x^{3} + a}\right ) - 2 \, {\left (2 \, B b^{2} x^{6} + {\left (5 \, B a b + 4 \, A b^{2}\right )} x^{3} - 8 \, A a b\right )} \sqrt {b x^{3} + a} \sqrt {x}\right )} e^{\left (-\frac {5}{2}\right )}}{24 \, b x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(3/2)*(B*x^3+A)/(e*x)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(3*(B*a^2 + 4*A*a*b)*sqrt(b)*x^2*log(-8*b^2*x^6 - 8*a*b*x^3 - 4*(2*b*x^4 + a*x)*sqrt(b*x^3 + a)*sqrt(b)*
sqrt(x) - a^2) + 4*(2*B*b^2*x^6 + (5*B*a*b + 4*A*b^2)*x^3 - 8*A*a*b)*sqrt(b*x^3 + a)*sqrt(x))*e^(-5/2)/(b*x^2)
, -1/24*(3*(B*a^2 + 4*A*a*b)*sqrt(-b)*x^2*arctan(2*sqrt(b*x^3 + a)*sqrt(-b)*x^(3/2)/(2*b*x^3 + a)) - 2*(2*B*b^
2*x^6 + (5*B*a*b + 4*A*b^2)*x^3 - 8*A*a*b)*sqrt(b*x^3 + a)*sqrt(x))*e^(-5/2)/(b*x^2)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (138) = 276\).
time = 12.90, size = 289, normalized size = 1.90 \begin {gather*} - \frac {2 A a^{\frac {3}{2}}}{3 e^{\frac {5}{2}} x^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {A \sqrt {a} b x^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}}{3 e^{\frac {5}{2}}} - \frac {2 A \sqrt {a} b x^{\frac {3}{2}}}{3 e^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {A a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {3}{2}}}{\sqrt {a}} \right )}}{e^{\frac {5}{2}}} + \frac {B a^{\frac {3}{2}} x^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}}{3 e^{\frac {5}{2}}} + \frac {B a^{\frac {3}{2}} x^{\frac {3}{2}}}{12 e^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {B \sqrt {a} b x^{\frac {9}{2}}}{4 e^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {B a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {3}{2}}}{\sqrt {a}} \right )}}{4 \sqrt {b} e^{\frac {5}{2}}} + \frac {B b^{2} x^{\frac {15}{2}}}{6 \sqrt {a} e^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(3/2)*(B*x**3+A)/(e*x)**(5/2),x)

[Out]

-2*A*a**(3/2)/(3*e**(5/2)*x**(3/2)*sqrt(1 + b*x**3/a)) + A*sqrt(a)*b*x**(3/2)*sqrt(1 + b*x**3/a)/(3*e**(5/2))
- 2*A*sqrt(a)*b*x**(3/2)/(3*e**(5/2)*sqrt(1 + b*x**3/a)) + A*a*sqrt(b)*asinh(sqrt(b)*x**(3/2)/sqrt(a))/e**(5/2
) + B*a**(3/2)*x**(3/2)*sqrt(1 + b*x**3/a)/(3*e**(5/2)) + B*a**(3/2)*x**(3/2)/(12*e**(5/2)*sqrt(1 + b*x**3/a))
+ B*sqrt(a)*b*x**(9/2)/(4*e**(5/2)*sqrt(1 + b*x**3/a)) + B*a**2*asinh(sqrt(b)*x**(3/2)/sqrt(a))/(4*sqrt(b)*e*
*(5/2)) + B*b**2*x**(15/2)/(6*sqrt(a)*e**(5/2)*sqrt(1 + b*x**3/a))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(3/2)*(B*x^3+A)/(e*x)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*e^(-5/2)/x^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (B\,x^3+A\right )\,{\left (b\,x^3+a\right )}^{3/2}}{{\left (e\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3)^(3/2))/(e*x)^(5/2),x)

[Out]

int(((A + B*x^3)*(a + b*x^3)^(3/2))/(e*x)^(5/2), x)

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