3.9.1 \(\int \frac {(e x)^{3/2} (A+B x^2)}{\sqrt {a+b x^2}} \, dx\) [801]

Optimal. Leaf size=174 \[ \frac {2 (7 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {a^{3/4} (7 A b-5 a B) e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 b^{9/4} \sqrt {a+b x^2}} \]

[Out]

2/7*B*(e*x)^(5/2)*(b*x^2+a)^(1/2)/b/e+2/21*(7*A*b-5*B*a)*e*(e*x)^(1/2)*(b*x^2+a)^(1/2)/b^2-1/21*a^(3/4)*(7*A*b
-5*B*a)*e^(3/2)*(cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(e*x)^(1/2)/
a^(1/4)/e^(1/2)))*EllipticF(sin(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2)
)*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(9/4)/(b*x^2+a)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {470, 327, 335, 226} \begin {gather*} -\frac {a^{3/4} e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-5 a B) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 b^{9/4} \sqrt {a+b x^2}}+\frac {2 e \sqrt {e x} \sqrt {a+b x^2} (7 A b-5 a B)}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(7*A*b - 5*a*B)*e*Sqrt[e*x]*Sqrt[a + b*x^2])/(21*b^2) + (2*B*(e*x)^(5/2)*Sqrt[a + b*x^2])/(7*b*e) - (a^(3/4
)*(7*A*b - 5*a*B)*e^(3/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(
b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(21*b^(9/4)*Sqrt[a + b*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 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 \frac {(e x)^{3/2} \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx &=\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {\left (2 \left (-\frac {7 A b}{2}+\frac {5 a B}{2}\right )\right ) \int \frac {(e x)^{3/2}}{\sqrt {a+b x^2}} \, dx}{7 b}\\ &=\frac {2 (7 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {\left (a (7 A b-5 a B) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{21 b^2}\\ &=\frac {2 (7 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {(2 a (7 A b-5 a B) e) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b^2}\\ &=\frac {2 (7 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {a^{3/4} (7 A b-5 a B) e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 b^{9/4} \sqrt {a+b 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 = 10.09, size = 96, normalized size = 0.55 \begin {gather*} \frac {2 e \sqrt {e x} \left (-\left (\left (a+b x^2\right ) \left (-7 A b+5 a B-3 b B x^2\right )\right )+a (-7 A b+5 a B) \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{21 b^2 \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*Sqrt[e*x]*(-((a + b*x^2)*(-7*A*b + 5*a*B - 3*b*B*x^2)) + a*(-7*A*b + 5*a*B)*Sqrt[1 + (b*x^2)/a]*Hypergeom
etric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(21*b^2*Sqrt[a + b*x^2])

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Maple [A]
time = 0.10, size = 250, normalized size = 1.44

method	result	size
risch	\(\frac {2 \left (3 b B \,x^{2}+7 A b -5 B a \right ) x \sqrt {b \,x^{2}+a}\, e^{2}}{21 b^{2} \sqrt {e x}}-\frac {a \left (7 A b -5 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) e^{2} \sqrt {\left (b \,x^{2}+a \right ) e x}}{21 b^{3} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(190\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B e \,x^{2} \sqrt {b e \,x^{3}+a e x}}{7 b}+\frac {2 \left (A \,e^{2}-\frac {5 B \,e^{2} a}{7 b}\right ) \sqrt {b e \,x^{3}+a e x}}{3 b e}-\frac {\left (A \,e^{2}-\frac {5 B \,e^{2} a}{7 b}\right ) a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 b^{2} \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\)	\(222\)
default	\(-\frac {e \sqrt {e x}\, \left (7 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b -5 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2}-6 B \,b^{3} x^{5}-14 A \,b^{3} x^{3}+4 B a \,b^{2} x^{3}-14 A a \,b^{2} x +10 B \,a^{2} b x \right )}{21 x \sqrt {b \,x^{2}+a}\, b^{3}}\)	\(250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*e/x*(e*x)^(1/2)/(b*x^2+a)^(1/2)*(7*A*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2)
)/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))
*(-a*b)^(1/2)*a*b-5*B*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)
*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*a^2-6*B
*b^3*x^5-14*A*b^3*x^3+4*B*a*b^2*x^3-14*A*a*b^2*x+10*B*a^2*b*x)/b^3

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.23, size = 69, normalized size = 0.40 \begin {gather*} \frac {2 \, {\left ({\left (5 \, B a^{2} - 7 \, A a b\right )} \sqrt {b} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (3 \, B b^{2} x^{2} - 5 \, B a b + 7 \, A b^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x} e^{\frac {3}{2}}\right )}}{21 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*((5*B*a^2 - 7*A*a*b)*sqrt(b)*e^(3/2)*weierstrassPInverse(-4*a/b, 0, x) + (3*B*b^2*x^2 - 5*B*a*b + 7*A*b^2
)*sqrt(b*x^2 + a)*sqrt(x)*e^(3/2))/b^3

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Sympy [C] Result contains complex when optimal does not.
time = 4.15, size = 94, normalized size = 0.54 \begin {gather*} \frac {A e^{\frac {3}{2}} x^{\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 {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {B e^{\frac {3}{2}} x^{\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 {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), b*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*gamma(9/4)) + B*
e**(3/2)*x**(9/2)*gamma(9/4)*hyper((1/2, 9/4), (13/4,), b*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*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)^(3/2)*(B*x^2+A)/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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