3.6.36 \(\int (e x)^{5/2} (a+b x^3)^{5/2} (A+B x^3) \, dx\) [536]

Optimal. Leaf size=404 \[ \frac {81 a^3 (4 A b-a B) e^2 \sqrt {e x} \sqrt {a+b x^3}}{5632 b^2}+\frac {27 a^2 (4 A b-a B) (e x)^{7/2} \sqrt {a+b x^3}}{1408 b e}+\frac {15 a (4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{704 b e}+\frac {(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}-\frac {27\ 3^{3/4} a^{11/3} (4 A b-a B) e^2 \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{11264 b^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

15/704*a*(4*A*b-B*a)*(e*x)^(7/2)*(b*x^3+a)^(3/2)/b/e+1/44*(4*A*b-B*a)*(e*x)^(7/2)*(b*x^3+a)^(5/2)/b/e+1/14*B*(
e*x)^(7/2)*(b*x^3+a)^(7/2)/b/e+27/1408*a^2*(4*A*b-B*a)*(e*x)^(7/2)*(b*x^3+a)^(1/2)/b/e+81/5632*a^3*(4*A*b-B*a)
*e^2*(e*x)^(1/2)*(b*x^3+a)^(1/2)/b^2-27/11264*3^(3/4)*a^(11/3)*(4*A*b-B*a)*e^2*(a^(1/3)+b^(1/3)*x)*((a^(1/3)+b
^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)/(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))*(a^(1/3)+b^(1
/3)*x*(1+3^(1/2)))*EllipticF((1-(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2),1/4
*6^(1/2)+1/4*2^(1/2))*(e*x)^(1/2)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^
(1/2)/b^2/(b*x^3+a)^(1/2)/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)

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Rubi [A]
time = 0.27, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {470, 285, 327, 335, 231} \begin {gather*} -\frac {27\ 3^{3/4} a^{11/3} e^2 \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) F\left (\text {ArcCos}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{11264 b^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {81 a^3 e^2 \sqrt {e x} \sqrt {a+b x^3} (4 A b-a B)}{5632 b^2}+\frac {27 a^2 (e x)^{7/2} \sqrt {a+b x^3} (4 A b-a B)}{1408 b e}+\frac {15 a (e x)^{7/2} \left (a+b x^3\right )^{3/2} (4 A b-a B)}{704 b e}+\frac {(e x)^{7/2} \left (a+b x^3\right )^{5/2} (4 A b-a B)}{44 b e}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(81*a^3*(4*A*b - a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^3])/(5632*b^2) + (27*a^2*(4*A*b - a*B)*(e*x)^(7/2)*Sqrt[a + b
*x^3])/(1408*b*e) + (15*a*(4*A*b - a*B)*(e*x)^(7/2)*(a + b*x^3)^(3/2))/(704*b*e) + ((4*A*b - a*B)*(e*x)^(7/2)*
(a + b*x^3)^(5/2))/(44*b*e) + (B*(e*x)^(7/2)*(a + b*x^3)^(7/2))/(14*b*e) - (27*3^(3/4)*a^(11/3)*(4*A*b - a*B)*
e^2*Sqrt[e*x]*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*
b^(1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 +
Sqrt[3])/4])/(11264*b^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a +
b*x^3])

Rule 231

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s +
r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*(
(s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^
2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x]

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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && 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 470

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]

Rubi steps

\begin {align*} \int (e x)^{5/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx &=\frac {B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}-\frac {\left (-14 A b+\frac {7 a B}{2}\right ) \int (e x)^{5/2} \left (a+b x^3\right )^{5/2} \, dx}{14 b}\\ &=\frac {(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}+\frac {(15 a (4 A b-a B)) \int (e x)^{5/2} \left (a+b x^3\right )^{3/2} \, dx}{88 b}\\ &=\frac {15 a (4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{704 b e}+\frac {(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}+\frac {\left (135 a^2 (4 A b-a B)\right ) \int (e x)^{5/2} \sqrt {a+b x^3} \, dx}{1408 b}\\ &=\frac {27 a^2 (4 A b-a B) (e x)^{7/2} \sqrt {a+b x^3}}{1408 b e}+\frac {15 a (4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{704 b e}+\frac {(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}+\frac {\left (81 a^3 (4 A b-a B)\right ) \int \frac {(e x)^{5/2}}{\sqrt {a+b x^3}} \, dx}{2816 b}\\ &=\frac {81 a^3 (4 A b-a B) e^2 \sqrt {e x} \sqrt {a+b x^3}}{5632 b^2}+\frac {27 a^2 (4 A b-a B) (e x)^{7/2} \sqrt {a+b x^3}}{1408 b e}+\frac {15 a (4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{704 b e}+\frac {(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}-\frac {\left (81 a^4 (4 A b-a B) e^3\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^3}} \, dx}{11264 b^2}\\ &=\frac {81 a^3 (4 A b-a B) e^2 \sqrt {e x} \sqrt {a+b x^3}}{5632 b^2}+\frac {27 a^2 (4 A b-a B) (e x)^{7/2} \sqrt {a+b x^3}}{1408 b e}+\frac {15 a (4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{704 b e}+\frac {(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}-\frac {\left (81 a^4 (4 A b-a B) e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{5632 b^2}\\ &=\frac {81 a^3 (4 A b-a B) e^2 \sqrt {e x} \sqrt {a+b x^3}}{5632 b^2}+\frac {27 a^2 (4 A b-a B) (e x)^{7/2} \sqrt {a+b x^3}}{1408 b e}+\frac {15 a (4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{704 b e}+\frac {(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}-\frac {27\ 3^{3/4} a^{11/3} (4 A b-a B) e^2 \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{11264 b^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.14, size = 116, normalized size = 0.29 \begin {gather*} \frac {e^2 \sqrt {e x} \sqrt {a+b x^3} \left (-\left (a+b x^3\right )^3 \sqrt {1+\frac {b x^3}{a}} \left (-28 A b+7 a B-22 b B x^3\right )+7 a^3 (-4 A b+a B) \, _2F_1\left (-\frac {5}{2},\frac {1}{6};\frac {7}{6};-\frac {b x^3}{a}\right )\right )}{308 b^2 \sqrt {1+\frac {b x^3}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^2*Sqrt[e*x]*Sqrt[a + b*x^3]*(-((a + b*x^3)^3*Sqrt[1 + (b*x^3)/a]*(-28*A*b + 7*a*B - 22*b*B*x^3)) + 7*a^3*(-
4*A*b + a*B)*Hypergeometric2F1[-5/2, 1/6, 7/6, -((b*x^3)/a)]))/(308*b^2*Sqrt[1 + (b*x^3)/a])

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Maple [C] Result contains complex when optimal does not.
time = 0.33, size = 5063, normalized size = 12.53

method	result	size
risch	\(\frac {\left (2816 B \,b^{4} x^{12}+3584 A \,b^{4} x^{9}+7552 B a \,b^{3} x^{9}+10528 a \,b^{3} A \,x^{6}+5816 B \,a^{2} b^{2} x^{6}+9968 A \,a^{2} b^{2} x^{3}+324 B \,a^{3} b \,x^{3}+2268 A \,a^{3} b -567 B \,a^{4}\right ) x \sqrt {b \,x^{3}+a}\, e^{3}}{39424 b^{2} \sqrt {e x}}-\frac {81 a^{4} \left (4 A b -B a \right ) \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )^{2} \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}, \sqrt {\frac {\left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{\left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right ) e^{3} \sqrt {\left (b \,x^{3}+a \right ) e x}}{5632 b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {b e x \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}\, \sqrt {e x}\, \sqrt {b \,x^{3}+a}}\)	\(825\)
elliptic	\(\text {Expression too large to display}\)	\(1134\)
default	\(\text {Expression too large to display}\)	\(5063\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integral((B*b^2*x^11 + (2*B*a*b + A*b^2)*x^8 + (B*a^2 + 2*A*a*b)*x^5 + A*a^2*x^2)*sqrt(b*x^3 + a)*sqrt(x)*e^(5
/2), x)

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Sympy [C] Result contains complex when optimal does not.
time = 175.61, size = 308, normalized size = 0.76 \begin {gather*} \frac {A a^{\frac {5}{2}} e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {13}{6}\right )} + \frac {2 A a^{\frac {3}{2}} b e^{\frac {5}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{6} \\ \frac {19}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {19}{6}\right )} + \frac {A \sqrt {a} b^{2} e^{\frac {5}{2}} x^{\frac {19}{2}} \Gamma \left (\frac {19}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {19}{6} \\ \frac {25}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {25}{6}\right )} + \frac {B a^{\frac {5}{2}} e^{\frac {5}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{6} \\ \frac {19}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {19}{6}\right )} + \frac {2 B a^{\frac {3}{2}} b e^{\frac {5}{2}} x^{\frac {19}{2}} \Gamma \left (\frac {19}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {19}{6} \\ \frac {25}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {25}{6}\right )} + \frac {B \sqrt {a} b^{2} e^{\frac {5}{2}} x^{\frac {25}{2}} \Gamma \left (\frac {25}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {25}{6} \\ \frac {31}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {31}{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**(5/2)*e**(5/2)*x**(7/2)*gamma(7/6)*hyper((-1/2, 7/6), (13/6,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(13/6))
+ 2*A*a**(3/2)*b*e**(5/2)*x**(13/2)*gamma(13/6)*hyper((-1/2, 13/6), (19/6,), b*x**3*exp_polar(I*pi)/a)/(3*gamm
a(19/6)) + A*sqrt(a)*b**2*e**(5/2)*x**(19/2)*gamma(19/6)*hyper((-1/2, 19/6), (25/6,), b*x**3*exp_polar(I*pi)/a
)/(3*gamma(25/6)) + B*a**(5/2)*e**(5/2)*x**(13/2)*gamma(13/6)*hyper((-1/2, 13/6), (19/6,), b*x**3*exp_polar(I*
pi)/a)/(3*gamma(19/6)) + 2*B*a**(3/2)*b*e**(5/2)*x**(19/2)*gamma(19/6)*hyper((-1/2, 19/6), (25/6,), b*x**3*exp
_polar(I*pi)/a)/(3*gamma(25/6)) + B*sqrt(a)*b**2*e**(5/2)*x**(25/2)*gamma(25/6)*hyper((-1/2, 25/6), (31/6,), b
*x**3*exp_polar(I*pi)/a)/(3*gamma(31/6))

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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