3.9.15 \(\int \frac {(e x)^{7/2} (A+B x^2)}{(a+b x^2)^{5/2}} \, dx\) [815]

Optimal. Leaf size=208 \[ -\frac {(A b-3 a B) e (e x)^{5/2}}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}-\frac {5 (A b-3 a B) e^3 \sqrt {e x}}{6 b^3 \sqrt {a+b x^2}}+\frac {5 (A b-3 a B) e^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{12 \sqrt [4]{a} b^{13/4} \sqrt {a+b x^2}} \]

[Out]

-1/3*(A*b-3*B*a)*e*(e*x)^(5/2)/b^2/(b*x^2+a)^(3/2)+2/3*B*(e*x)^(9/2)/b/e/(b*x^2+a)^(3/2)-5/6*(A*b-3*B*a)*e^3*(
e*x)^(1/2)/b^3/(b*x^2+a)^(1/2)+5/12*(A*b-3*B*a)*e^(7/2)*(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)/a^(1/4)/b^(13/4)/(b*x^2+a)^(
1/2)

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Rubi [A]
time = 0.09, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {470, 294, 335, 226} \begin {gather*} \frac {5 e^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (A b-3 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 )}{12 \sqrt [4]{a} b^{13/4} \sqrt {a+b x^2}}-\frac {5 e^3 \sqrt {e x} (A b-3 a B)}{6 b^3 \sqrt {a+b x^2}}-\frac {e (e x)^{5/2} (A b-3 a B)}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

-1/3*((A*b - 3*a*B)*e*(e*x)^(5/2))/(b^2*(a + b*x^2)^(3/2)) + (2*B*(e*x)^(9/2))/(3*b*e*(a + b*x^2)^(3/2)) - (5*
(A*b - 3*a*B)*e^3*Sqrt[e*x])/(6*b^3*Sqrt[a + b*x^2]) + (5*(A*b - 3*a*B)*e^(7/2)*(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])/(12*a^(1/4)
*b^(13/4)*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 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 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)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}-\frac {\left (2 \left (-\frac {3 A b}{2}+\frac {9 a B}{2}\right )\right ) \int \frac {(e x)^{7/2}}{\left (a+b x^2\right )^{5/2}} \, dx}{3 b}\\ &=-\frac {(A b-3 a B) e (e x)^{5/2}}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}+\frac {\left (5 (A b-3 a B) e^2\right ) \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{6 b^2}\\ &=-\frac {(A b-3 a B) e (e x)^{5/2}}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}-\frac {5 (A b-3 a B) e^3 \sqrt {e x}}{6 b^3 \sqrt {a+b x^2}}+\frac {\left (5 (A b-3 a B) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{12 b^3}\\ &=-\frac {(A b-3 a B) e (e x)^{5/2}}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}-\frac {5 (A b-3 a B) e^3 \sqrt {e x}}{6 b^3 \sqrt {a+b x^2}}+\frac {\left (5 (A b-3 a B) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 b^3}\\ &=-\frac {(A b-3 a B) e (e x)^{5/2}}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}-\frac {5 (A b-3 a B) e^3 \sqrt {e x}}{6 b^3 \sqrt {a+b x^2}}+\frac {5 (A b-3 a B) e^{7/2} \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 )}{12 \sqrt [4]{a} b^{13/4} \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 = 10.14, size = 116, normalized size = 0.56 \begin {gather*} \frac {e^3 \sqrt {e x} \left (15 a^2 B+b^2 x^2 \left (-7 A+4 B x^2\right )+a \left (-5 A b+21 b B x^2\right )+5 (A b-3 a B) \left (a+b x^2\right ) \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 )}{6 b^3 \left (a+b x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

(e^3*Sqrt[e*x]*(15*a^2*B + b^2*x^2*(-7*A + 4*B*x^2) + a*(-5*A*b + 21*b*B*x^2) + 5*(A*b - 3*a*B)*(a + b*x^2)*Sq
rt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(6*b^3*(a + b*x^2)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(438\) vs. \(2(209)=418\).
time = 0.16, size = 439, normalized size = 2.11

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {a \,e^{3} \left (A b -B a \right ) \sqrt {b e \,x^{3}+a e x}}{3 b^{5} \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {e^{4} x \left (7 A b -13 B a \right )}{6 b^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {2 B \,e^{3} \sqrt {b e \,x^{3}+a e x}}{3 b^{3}}+\frac {\left (\frac {\left (A b -2 B a \right ) e^{4}}{b^{3}}-\frac {e^{4} \left (7 A b -13 B a \right )}{12 b^{3}}-\frac {B \,e^{4} a}{3 b^{3}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\)	\(283\)
default	\(\frac {\left (5 A \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, b^{2} x^{2}-15 B \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, a b \,x^{2}+5 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b -15 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2}+8 B \,b^{3} x^{5}-14 A \,b^{3} x^{3}+42 B a \,b^{2} x^{3}-10 A a \,b^{2} x +30 B \,a^{2} b x \right ) e^{3} \sqrt {e x}}{12 x \,b^{4} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(439\)
risch	\(\frac {2 B x \sqrt {b \,x^{2}+a}\, e^{4}}{3 b^{3} \sqrt {e x}}+\frac {\left (\frac {3 A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b e \,x^{3}+a e x}}-\frac {7 B a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}+3 a^{2} \left (A b -B a \right ) \left (\frac {\sqrt {b e \,x^{3}+a e x}}{3 a e \,b^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {5 x}{6 a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {5 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{12 a^{2} b \sqrt {b e \,x^{3}+a e x}}\right )-3 a \left (2 A b -3 B a \right ) \left (\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )\right ) e^{4} \sqrt {\left (b \,x^{2}+a \right ) e x}}{3 b^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(610\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(7/2)*(B*x^2+A)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/12*(5*A*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*((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)*b^2*x^2-15*B*Ellipt
icF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*((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)*a*b*x^2+5*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-15*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+8*B*b^3*x^5-14*A*b^3*x^3+42*B*a*b^2*x^3-10*A*a*b^2*x+30*B*a^2*b*x)*e
^3/x*(e*x)^(1/2)/b^4/(b*x^2+a)^(3/2)

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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)^(7/2)*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

e^(7/2)*integrate((B*x^2 + A)*x^(7/2)/(b*x^2 + a)^(5/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.33, size = 156, normalized size = 0.75 \begin {gather*} -\frac {5 \, {\left ({\left (3 \, B a b^{2} - A b^{3}\right )} x^{4} + 3 \, B a^{3} - A a^{2} b + 2 \, {\left (3 \, B a^{2} b - A a b^{2}\right )} x^{2}\right )} \sqrt {b} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (4 \, B b^{3} x^{4} + 15 \, B a^{2} b - 5 \, A a b^{2} + 7 \, {\left (3 \, B a b^{2} - A b^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x} e^{\frac {7}{2}}}{6 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(7/2)*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

-1/6*(5*((3*B*a*b^2 - A*b^3)*x^4 + 3*B*a^3 - A*a^2*b + 2*(3*B*a^2*b - A*a*b^2)*x^2)*sqrt(b)*e^(7/2)*weierstras
sPInverse(-4*a/b, 0, x) - (4*B*b^3*x^4 + 15*B*a^2*b - 5*A*a*b^2 + 7*(3*B*a*b^2 - A*b^3)*x^2)*sqrt(b*x^2 + a)*s
qrt(x)*e^(7/2))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(7/2)*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

Timed out

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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)^(7/2)*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*x^(7/2)*e^(7/2)/(b*x^2 + a)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{7/2}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(e*x)^(7/2))/(a + b*x^2)^(5/2),x)

[Out]

int(((A + B*x^2)*(e*x)^(7/2))/(a + b*x^2)^(5/2), x)

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