3.5.44 \(\int \sqrt {e x} (A+B x) (a+c x^2)^{3/2} \, dx\) [444]

Optimal. Leaf size=366 \[ \frac {8 a^2 A e x \sqrt {a+c x^2}}{15 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 a \sqrt {e x} (15 a B-77 A c x) \sqrt {a+c x^2}}{1155 c}-\frac {2 \sqrt {e x} (9 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{693 c}+\frac {2 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {8 a^{9/4} A e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {4 a^{9/4} \left (15 \sqrt {a} B-77 A \sqrt {c}\right ) e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 c^{5/4} \sqrt {e x} \sqrt {a+c x^2}} \]

[Out]

-2/693*(-77*A*c*x+9*B*a)*(c*x^2+a)^(3/2)*(e*x)^(1/2)/c+2/11*B*(c*x^2+a)^(5/2)*(e*x)^(1/2)/c+8/15*a^2*A*e*x*(c*
x^2+a)^(1/2)/c^(1/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-4/1155*a*(-77*A*c*x+15*B*a)*(e*x)^(1/2)*(c*x^2+a)^(1/2)/c
-8/15*a^(9/4)*A*e*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))*Elli
pticE(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x*c^(1/2))*x^(1/2)*((c*x^2+a)/(a^(1/2)+x*c^
(1/2))^2)^(1/2)/c^(3/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)-4/1155*a^(9/4)*e*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2
)^(1/2)/cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(
15*B*a^(1/2)-77*A*c^(1/2))*(a^(1/2)+x*c^(1/2))*x^(1/2)*((c*x^2+a)/(a^(1/2)+x*c^(1/2))^2)^(1/2)/c^(5/4)/(e*x)^(
1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.28, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {847, 829, 856, 854, 1212, 226, 1210} \begin {gather*} -\frac {4 a^{9/4} e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (15 \sqrt {a} B-77 A \sqrt {c}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 c^{5/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {8 a^{9/4} A e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^2 A e x \sqrt {a+c x^2}}{15 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (9 a B-77 A c x)}{693 c}-\frac {4 a \sqrt {e x} \sqrt {a+c x^2} (15 a B-77 A c x)}{1155 c}+\frac {2 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a^2*A*e*x*Sqrt[a + c*x^2])/(15*Sqrt[c]*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*a*Sqrt[e*x]*(15*a*B - 77*A*c*x
)*Sqrt[a + c*x^2])/(1155*c) - (2*Sqrt[e*x]*(9*a*B - 77*A*c*x)*(a + c*x^2)^(3/2))/(693*c) + (2*B*Sqrt[e*x]*(a +
c*x^2)^(5/2))/(11*c) - (8*a^(9/4)*A*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]
*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(15*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) - (4*a^(9/4)*(15*
Sqrt[a]*B - 77*A*Sqrt[c])*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[
2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(1155*c^(5/4)*Sqrt[e*x]*Sqrt[a + c*x^2])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 829

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m +
2*p + 2))), x] + Dist[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

Rule 847

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^
m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 854

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2, Subst[Int[(f + g*x^2)/Sqrt[
a + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, c, f, g}, x]

Rule 856

Int[((f_) + (g_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[x]/Sqrt[e*x], Int[
(f + g*x)/(Sqrt[x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, e, f, g}, x]

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1212

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(e + d*q)/q, Int
[1/Sqrt[a + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a
, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin {align*} \int \sqrt {e x} (A+B x) \left (a+c x^2\right )^{3/2} \, dx &=\frac {2 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {2 \int \frac {\left (-\frac {1}{2} a B e+\frac {11}{2} A c e x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt {e x}} \, dx}{11 c}\\ &=-\frac {2 \sqrt {e x} (9 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{693 c}+\frac {2 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {8 \int \frac {\left (-\frac {9}{4} a^2 B c e^3+\frac {77}{4} a A c^2 e^3 x\right ) \sqrt {a+c x^2}}{\sqrt {e x}} \, dx}{231 c^2 e^2}\\ &=-\frac {4 a \sqrt {e x} (15 a B-77 A c x) \sqrt {a+c x^2}}{1155 c}-\frac {2 \sqrt {e x} (9 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{693 c}+\frac {2 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {32 \int \frac {-\frac {45}{8} a^3 B c^2 e^5+\frac {231}{8} a^2 A c^3 e^5 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{3465 c^3 e^4}\\ &=-\frac {4 a \sqrt {e x} (15 a B-77 A c x) \sqrt {a+c x^2}}{1155 c}-\frac {2 \sqrt {e x} (9 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{693 c}+\frac {2 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {\left (32 \sqrt {x}\right ) \int \frac {-\frac {45}{8} a^3 B c^2 e^5+\frac {231}{8} a^2 A c^3 e^5 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{3465 c^3 e^4 \sqrt {e x}}\\ &=-\frac {4 a \sqrt {e x} (15 a B-77 A c x) \sqrt {a+c x^2}}{1155 c}-\frac {2 \sqrt {e x} (9 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{693 c}+\frac {2 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {\left (64 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-\frac {45}{8} a^3 B c^2 e^5+\frac {231}{8} a^2 A c^3 e^5 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3465 c^3 e^4 \sqrt {e x}}\\ &=-\frac {4 a \sqrt {e x} (15 a B-77 A c x) \sqrt {a+c x^2}}{1155 c}-\frac {2 \sqrt {e x} (9 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{693 c}+\frac {2 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {\left (8 a^{5/2} \left (15 \sqrt {a} B-77 A \sqrt {c}\right ) e \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{1155 c \sqrt {e x}}-\frac {\left (8 a^{5/2} A e \sqrt {x}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15 \sqrt {c} \sqrt {e x}}\\ &=\frac {8 a^2 A e x \sqrt {a+c x^2}}{15 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 a \sqrt {e x} (15 a B-77 A c x) \sqrt {a+c x^2}}{1155 c}-\frac {2 \sqrt {e x} (9 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{693 c}+\frac {2 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {8 a^{9/4} A e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {4 a^{9/4} \left (15 \sqrt {a} B-77 A \sqrt {c}\right ) e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 c^{5/4} \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.06, size = 116, normalized size = 0.32 \begin {gather*} \frac {2 \sqrt {e x} \sqrt {a+c x^2} \left (3 B \left (a+c x^2\right )^2 \sqrt {1+\frac {c x^2}{a}}-3 a^2 B \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{a}\right )+11 a A c x \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )\right )}{33 c \sqrt {1+\frac {c x^2}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e*x]*Sqrt[a + c*x^2]*(3*B*(a + c*x^2)^2*Sqrt[1 + (c*x^2)/a] - 3*a^2*B*Hypergeometric2F1[-3/2, 1/4, 5/4
, -((c*x^2)/a)] + 11*a*A*c*x*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2)/a)]))/(33*c*Sqrt[1 + (c*x^2)/a])

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Maple [A]
time = 0.59, size = 357, normalized size = 0.98

method	result	size
default	\(-\frac {2 \sqrt {e x}\, \left (-315 B \,c^{4} x^{7}-385 A \,c^{4} x^{6}+462 A \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3} c -924 A \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3} c +90 B \sqrt {-a c}\, \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-900 a B \,c^{3} x^{5}-1232 a A \,c^{3} x^{4}-765 a^{2} B \,c^{2} x^{3}-847 a^{2} A \,c^{2} x^{2}-180 a^{3} B c x \right )}{3465 \sqrt {c \,x^{2}+a}\, c^{2} x}\)	\(357\)
risch	\(\frac {2 \left (315 B \,c^{2} x^{4}+385 A \,c^{2} x^{3}+585 a B c \,x^{2}+847 a A c x +180 a^{2} B \right ) x \sqrt {c \,x^{2}+a}\, e}{3465 c \sqrt {e x}}+\frac {4 a^{2} \left (\frac {77 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}-\frac {15 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) e \sqrt {\left (c \,x^{2}+a \right ) e x}}{1155 c \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(368\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B c \,x^{4} \sqrt {c e \,x^{3}+a e x}}{11}+\frac {2 A c \,x^{3} \sqrt {c e \,x^{3}+a e x}}{9}+\frac {26 B a \,x^{2} \sqrt {c e \,x^{3}+a e x}}{77}+\frac {22 a A x \sqrt {c e \,x^{3}+a e x}}{45}+\frac {8 B \,a^{2} \sqrt {c e \,x^{3}+a e x}}{77 c}-\frac {4 B \,a^{3} e \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{77 c^{2} \sqrt {c e \,x^{3}+a e x}}+\frac {4 a^{2} A e \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{15 c \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\)	\(415\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(1/2)*(B*x+A)*(c*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/3465*(e*x)^(1/2)/(c*x^2+a)^(1/2)*(-315*B*c^4*x^7-385*A*c^4*x^6+462*A*2^(1/2)*((c*x+(-a*c)^(1/2))/(-a*c)^(1/
2))^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a
*c)^(1/2))^(1/2),1/2*2^(1/2))*a^3*c-924*A*2^(1/2)*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*((-c*x+(-a*c)^(1/2))
/(-a*c)^(1/2))^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)*EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*
a^3*c+90*B*(-a*c)^(1/2)*2^(1/2)*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/
2)*(-x/(-a*c)^(1/2)*c)^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a^3-900*a*B*c^3*x^
5-1232*a*A*c^3*x^4-765*a^2*B*c^2*x^3-847*a^2*A*c^2*x^2-180*a^3*B*c*x)/c^2/x

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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)^(1/2)*(B*x+A)*(c*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

e^(1/2)*integrate((c*x^2 + a)^(3/2)*(B*x + A)*sqrt(x), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.49, size = 114, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (180 \, B a^{3} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 924 \, A a^{2} c^{\frac {3}{2}} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (315 \, B c^{3} x^{4} + 385 \, A c^{3} x^{3} + 585 \, B a c^{2} x^{2} + 847 \, A a c^{2} x + 180 \, B a^{2} c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {1}{2}}\right )}}{3465 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(180*B*a^3*sqrt(c)*e^(1/2)*weierstrassPInverse(-4*a/c, 0, x) + 924*A*a^2*c^(3/2)*e^(1/2)*weierstrassZe
ta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (315*B*c^3*x^4 + 385*A*c^3*x^3 + 585*B*a*c^2*x^2 + 847*A*a*
c^2*x + 180*B*a^2*c)*sqrt(c*x^2 + a)*sqrt(x)*e^(1/2))/c^2

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Sympy [C] Result contains complex when optimal does not.
time = 5.91, size = 197, normalized size = 0.54 \begin {gather*} \frac {A a^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e \Gamma \left (\frac {7}{4}\right )} + \frac {A \sqrt {a} c \left (e x\right )^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{3} \Gamma \left (\frac {11}{4}\right )} + \frac {B a^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{2} \Gamma \left (\frac {9}{4}\right )} + \frac {B \sqrt {a} c \left (e x\right )^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{4} \Gamma \left (\frac {13}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**(3/2)*(e*x)**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), c*x**2*exp_polar(I*pi)/a)/(2*e*gamma(7/4)) + A*s
qrt(a)*c*(e*x)**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**3*gamma(11/4)) +
B*a**(3/2)*(e*x)**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**2*gamma(9/4)) +
B*sqrt(a)*c*(e*x)**(9/2)*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**4*gamma(13/4))

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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)^(1/2)*(B*x+A)*(c*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(3/2)*(B*x + A)*sqrt(x)*e^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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