3.538 \(\int \sqrt{e x} (a+b x^3)^{5/2} (A+B x^3) \, dx\)

Optimal. Leaf size=201 \[ \frac{5 a^3 \sqrt{e} (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{192 b^{3/2}}+\frac{5 a^2 (e x)^{3/2} \sqrt{a+b x^3} (8 A b-a B)}{192 b e}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{5/2} (8 A b-a B)}{72 b e}+\frac{5 a (e x)^{3/2} \left (a+b x^3\right )^{3/2} (8 A b-a B)}{288 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e} \]

[Out]

(5*a^2*(8*A*b - a*B)*(e*x)^(3/2)*Sqrt[a + b*x^3])/(192*b*e) + (5*a*(8*A*b - a*B)*(e*x)^(3/2)*(a + b*x^3)^(3/2)
)/(288*b*e) + ((8*A*b - a*B)*(e*x)^(3/2)*(a + b*x^3)^(5/2))/(72*b*e) + (B*(e*x)^(3/2)*(a + b*x^3)^(7/2))/(12*b
*e) + (5*a^3*(8*A*b - a*B)*Sqrt[e]*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(192*b^(3/2))

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Rubi [A]  time = 0.127124, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {459, 279, 329, 275, 217, 206} \[ \frac{5 a^3 \sqrt{e} (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{192 b^{3/2}}+\frac{5 a^2 (e x)^{3/2} \sqrt{a+b x^3} (8 A b-a B)}{192 b e}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{5/2} (8 A b-a B)}{72 b e}+\frac{5 a (e x)^{3/2} \left (a+b x^3\right )^{3/2} (8 A b-a B)}{288 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a^2*(8*A*b - a*B)*(e*x)^(3/2)*Sqrt[a + b*x^3])/(192*b*e) + (5*a*(8*A*b - a*B)*(e*x)^(3/2)*(a + b*x^3)^(3/2)
)/(288*b*e) + ((8*A*b - a*B)*(e*x)^(3/2)*(a + b*x^3)^(5/2))/(72*b*e) + (B*(e*x)^(3/2)*(a + b*x^3)^(7/2))/(12*b
*e) + (5*a^3*(8*A*b - a*B)*Sqrt[e]*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(192*b^(3/2))

Rule 459

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

Rule 279

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 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 275

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 217

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 206

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

Rubi steps

\begin{align*} \int \sqrt{e x} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx &=\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}-\frac{\left (-12 A b+\frac{3 a B}{2}\right ) \int \sqrt{e x} \left (a+b x^3\right )^{5/2} \, dx}{12 b}\\ &=\frac{(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac{(5 a (8 A b-a B)) \int \sqrt{e x} \left (a+b x^3\right )^{3/2} \, dx}{48 b}\\ &=\frac{5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac{(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac{\left (5 a^2 (8 A b-a B)\right ) \int \sqrt{e x} \sqrt{a+b x^3} \, dx}{64 b}\\ &=\frac{5 a^2 (8 A b-a B) (e x)^{3/2} \sqrt{a+b x^3}}{192 b e}+\frac{5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac{(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac{\left (5 a^3 (8 A b-a B)\right ) \int \frac{\sqrt{e x}}{\sqrt{a+b x^3}} \, dx}{128 b}\\ &=\frac{5 a^2 (8 A b-a B) (e x)^{3/2} \sqrt{a+b x^3}}{192 b e}+\frac{5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac{(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac{\left (5 a^3 (8 A b-a B)\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{64 b e}\\ &=\frac{5 a^2 (8 A b-a B) (e x)^{3/2} \sqrt{a+b x^3}}{192 b e}+\frac{5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac{(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac{\left (5 a^3 (8 A b-a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^2}{e^3}}} \, dx,x,(e x)^{3/2}\right )}{192 b e}\\ &=\frac{5 a^2 (8 A b-a B) (e x)^{3/2} \sqrt{a+b x^3}}{192 b e}+\frac{5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac{(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac{\left (5 a^3 (8 A b-a B)\right ) \operatorname{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 )}{192 b e}\\ &=\frac{5 a^2 (8 A b-a B) (e x)^{3/2} \sqrt{a+b x^3}}{192 b e}+\frac{5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac{(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac{5 a^3 (8 A b-a B) \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{192 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.376751, size = 146, normalized size = 0.73 \[ \frac{x \sqrt{e x} \sqrt{a+b x^3} \left (\frac{(8 A b-a B) \left (\sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1} \left (33 a^2+26 a b x^3+8 b^2 x^6\right )+15 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )\right )}{48 \sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1}}+B \left (a+b x^3\right )^3\right )}{12 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[e*x]*Sqrt[a + b*x^3]*(B*(a + b*x^3)^3 + ((8*A*b - a*B)*(Sqrt[b]*x^(3/2)*Sqrt[1 + (b*x^3)/a]*(33*a^2 +
26*a*b*x^3 + 8*b^2*x^6) + 15*a^(5/2)*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(48*Sqrt[b]*x^(3/2)*Sqrt[1 + (b*x^3)
/a])))/(12*b)

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Maple [C]  time = 0.049, size = 7702, normalized size = 38.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(5/2)*(B*x^3+A)*(e*x)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \sqrt{e x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(5/2)*(B*x^3+A)*(e*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 4.31232, size = 738, normalized size = 3.67 \begin{align*} \left [-\frac{15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (48 \, B b^{3} x^{10} + 8 \,{\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{7} + 2 \,{\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x^{4} + 3 \,{\left (5 \, B a^{3} + 88 \, A a^{2} b\right )} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{2304 \, b}, \frac{15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} b x \sqrt{-\frac{e}{b}}}{2 \, b e x^{3} + a e}\right ) + 2 \,{\left (48 \, B b^{3} x^{10} + 8 \,{\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{7} + 2 \,{\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x^{4} + 3 \,{\left (5 \, B a^{3} + 88 \, A a^{2} b\right )} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{1152 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(5/2)*(B*x^3+A)*(e*x)^(1/2),x, algorithm="fricas")

[Out]

[-1/2304*(15*(B*a^4 - 8*A*a^3*b)*sqrt(e/b)*log(-8*b^2*e*x^6 - 8*a*b*e*x^3 - a^2*e - 4*(2*b^2*x^4 + a*b*x)*sqrt
(b*x^3 + a)*sqrt(e*x)*sqrt(e/b)) - 4*(48*B*b^3*x^10 + 8*(17*B*a*b^2 + 8*A*b^3)*x^7 + 2*(59*B*a^2*b + 104*A*a*b
^2)*x^4 + 3*(5*B*a^3 + 88*A*a^2*b)*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b, 1/1152*(15*(B*a^4 - 8*A*a^3*b)*sqrt(-e/b)*
arctan(2*sqrt(b*x^3 + a)*sqrt(e*x)*b*x*sqrt(-e/b)/(2*b*e*x^3 + a*e)) + 2*(48*B*b^3*x^10 + 8*(17*B*a*b^2 + 8*A*
b^3)*x^7 + 2*(59*B*a^2*b + 104*A*a*b^2)*x^4 + 3*(5*B*a^3 + 88*A*a^2*b)*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b]

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Sympy [B]  time = 111.188, size = 413, normalized size = 2.05 \begin{align*} \frac{A a^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 e} + \frac{A a^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}}}{8 e \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{35 A a^{\frac{3}{2}} b \left (e x\right )^{\frac{9}{2}}}{72 e^{4} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{17 A \sqrt{a} b^{2} \left (e x\right )^{\frac{15}{2}}}{36 e^{7} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{5 A a^{3} \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{24 \sqrt{b}} + \frac{A b^{3} \left (e x\right )^{\frac{21}{2}}}{9 \sqrt{a} e^{10} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{5 B a^{\frac{7}{2}} \left (e x\right )^{\frac{3}{2}}}{192 b e \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{133 B a^{\frac{5}{2}} \left (e x\right )^{\frac{9}{2}}}{576 e^{4} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{127 B a^{\frac{3}{2}} b \left (e x\right )^{\frac{15}{2}}}{288 e^{7} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{23 B \sqrt{a} b^{2} \left (e x\right )^{\frac{21}{2}}}{72 e^{10} \sqrt{1 + \frac{b x^{3}}{a}}} - \frac{5 B a^{4} \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{192 b^{\frac{3}{2}}} + \frac{B b^{3} \left (e x\right )^{\frac{27}{2}}}{12 \sqrt{a} e^{13} \sqrt{1 + \frac{b x^{3}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(5/2)*(B*x**3+A)*(e*x)**(1/2),x)

[Out]

A*a**(5/2)*(e*x)**(3/2)*sqrt(1 + b*x**3/a)/(3*e) + A*a**(5/2)*(e*x)**(3/2)/(8*e*sqrt(1 + b*x**3/a)) + 35*A*a**
(3/2)*b*(e*x)**(9/2)/(72*e**4*sqrt(1 + b*x**3/a)) + 17*A*sqrt(a)*b**2*(e*x)**(15/2)/(36*e**7*sqrt(1 + b*x**3/a
)) + 5*A*a**3*sqrt(e)*asinh(sqrt(b)*(e*x)**(3/2)/(sqrt(a)*e**(3/2)))/(24*sqrt(b)) + A*b**3*(e*x)**(21/2)/(9*sq
rt(a)*e**10*sqrt(1 + b*x**3/a)) + 5*B*a**(7/2)*(e*x)**(3/2)/(192*b*e*sqrt(1 + b*x**3/a)) + 133*B*a**(5/2)*(e*x
)**(9/2)/(576*e**4*sqrt(1 + b*x**3/a)) + 127*B*a**(3/2)*b*(e*x)**(15/2)/(288*e**7*sqrt(1 + b*x**3/a)) + 23*B*s
qrt(a)*b**2*(e*x)**(21/2)/(72*e**10*sqrt(1 + b*x**3/a)) - 5*B*a**4*sqrt(e)*asinh(sqrt(b)*(e*x)**(3/2)/(sqrt(a)
*e**(3/2)))/(192*b**(3/2)) + B*b**3*(e*x)**(27/2)/(12*sqrt(a)*e**13*sqrt(1 + b*x**3/a))

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Giac [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)^(5/2)*(B*x^3+A)*(e*x)^(1/2),x, algorithm="giac")

[Out]

Timed out