3.807 \(\int \frac {(e x)^{7/2} (A+B x^2)}{(a+b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=211 \[ -\frac {5 a^{3/4} e^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-9 a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{42 b^{13/4} \sqrt {a+b x^2}}+\frac {5 e^3 \sqrt {e x} \sqrt {a+b x^2} (7 A b-9 a B)}{21 b^3}-\frac {e (e x)^{5/2} (7 A b-9 a B)}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}} \]

[Out]

-1/7*(7*A*b-9*B*a)*e*(e*x)^(5/2)/b^2/(b*x^2+a)^(1/2)+2/7*B*(e*x)^(9/2)/b/e/(b*x^2+a)^(1/2)+5/21*(7*A*b-9*B*a)*
e^3*(e*x)^(1/2)*(b*x^2+a)^(1/2)/b^3-5/42*a^(3/4)*(7*A*b-9*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)/b^(13/4)/(b*x
^2+a)^(1/2)

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Rubi [A]  time = 0.14, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {459, 288, 321, 329, 220} \[ -\frac {5 a^{3/4} e^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-9 a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{42 b^{13/4} \sqrt {a+b x^2}}+\frac {5 e^3 \sqrt {e x} \sqrt {a+b x^2} (7 A b-9 a B)}{21 b^3}-\frac {e (e x)^{5/2} (7 A b-9 a B)}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]

[Out]

-((7*A*b - 9*a*B)*e*(e*x)^(5/2))/(7*b^2*Sqrt[a + b*x^2]) + (2*B*(e*x)^(9/2))/(7*b*e*Sqrt[a + b*x^2]) + (5*(7*A
*b - 9*a*B)*e^3*Sqrt[e*x]*Sqrt[a + b*x^2])/(21*b^3) - (5*a^(3/4)*(7*A*b - 9*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])/(4
2*b^(13/4)*Sqrt[a + b*x^2])

Rule 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 288

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 321

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 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 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]

Rubi steps

\begin {align*} \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}-\frac {\left (2 \left (-\frac {7 A b}{2}+\frac {9 a B}{2}\right )\right ) \int \frac {(e x)^{7/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{7 b}\\ &=-\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {\left (5 (7 A b-9 a B) e^2\right ) \int \frac {(e x)^{3/2}}{\sqrt {a+b x^2}} \, dx}{14 b^2}\\ &=-\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {5 (7 A b-9 a B) e^3 \sqrt {e x} \sqrt {a+b x^2}}{21 b^3}-\frac {\left (5 a (7 A b-9 a B) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{42 b^3}\\ &=-\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {5 (7 A b-9 a B) e^3 \sqrt {e x} \sqrt {a+b x^2}}{21 b^3}-\frac {\left (5 a (7 A b-9 a B) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b^3}\\ &=-\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {5 (7 A b-9 a B) e^3 \sqrt {e x} \sqrt {a+b x^2}}{21 b^3}-\frac {5 a^{3/4} (7 A b-9 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 )}{42 b^{13/4} \sqrt {a+b x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.15, size = 111, normalized size = 0.53 \[ \frac {e^3 \sqrt {e x} \left (-45 a^2 B+5 a \sqrt {\frac {b x^2}{a}+1} (9 a B-7 A b) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {b x^2}{a}\right )+a b \left (35 A-18 B x^2\right )+2 b^2 x^2 \left (7 A+3 B x^2\right )\right )}{21 b^3 \sqrt {a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]

[Out]

(e^3*Sqrt[e*x]*(-45*a^2*B + a*b*(35*A - 18*B*x^2) + 2*b^2*x^2*(7*A + 3*B*x^2) + 5*a*(-7*A*b + 9*a*B)*Sqrt[1 +
(b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(21*b^3*Sqrt[a + b*x^2])

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fricas [F]  time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e^{3} x^{5} + A e^{3} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(7/2)*(B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

integral((B*e^3*x^5 + A*e^3*x^3)*sqrt(b*x^2 + a)*sqrt(e*x)/(b^2*x^4 + 2*a*b*x^2 + a^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(7/2)*(B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*(e*x)^(7/2)/(b*x^2 + a)^(3/2), x)

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maple [A]  time = 0.05, size = 252, normalized size = 1.19 \[ -\frac {\sqrt {e x}\, \left (-12 B \,b^{3} x^{5}-28 A \,b^{3} x^{3}+36 B a \,b^{2} x^{3}-70 A a \,b^{2} x +90 B \,a^{2} b x +35 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, A a b \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-45 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, B \,a^{2} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )\right ) e^{3}}{42 \sqrt {b \,x^{2}+a}\, b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(7/2)*(B*x^2+A)/(b*x^2+a)^(3/2),x)

[Out]

-1/42*e^3/x*(e*x)^(1/2)*(35*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)*(-1/(-a*b)^(1/2)*b*x)^(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-45*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)*(-1/(-a*b
)^(1/2)*b*x)^(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^2-12*B*b^3*x^
5-28*A*b^3*x^3+36*B*a*b^2*x^3-70*A*a*b^2*x+90*B*a^2*b*x)/(b*x^2+a)^(1/2)/b^4

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(7/2)*(B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*(e*x)^(7/2)/(b*x^2 + a)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{7/2}}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(e*x)^(7/2))/(a + b*x^2)^(3/2),x)

[Out]

int(((A + B*x^2)*(e*x)^(7/2))/(a + b*x^2)^(3/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(7/2)*(B*x**2+A)/(b*x**2+a)**(3/2),x)

[Out]

Timed out

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