3.9.57 \(\int \frac {(e x)^{7/2} (a+b x^2)^2}{(c+d x^2)^{5/2}} \, dx\) [857]

Optimal. Leaf size=302 \[ \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}} \]

[Out]

1/3*(-a*d+b*c)^2*(e*x)^(9/2)/c/d^2/e/(d*x^2+c)^(3/2)+1/14*(7*a^2*d^2-42*a*b*c*d+39*b^2*c^2)*e*(e*x)^(5/2)/c/d^
3/(d*x^2+c)^(1/2)+2/7*b^2*(e*x)^(9/2)/d^2/e/(d*x^2+c)^(1/2)-5/42*(7*a^2*d^2-42*a*b*c*d+39*b^2*c^2)*e^3*(e*x)^(
1/2)*(d*x^2+c)^(1/2)/c/d^4+5/84*(7*a^2*d^2-42*a*b*c*d+39*b^2*c^2)*e^(7/2)*(cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^
(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))*EllipticF(sin(2*arctan(d^(1/4)*(e*
x)^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2))*(c^(1/2)+x*d^(1/2))*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/c^(1/4)/d^
(17/4)/(d*x^2+c)^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {474, 470, 294, 327, 335, 226} \begin {gather*} \frac {5 e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}}-\frac {5 e^3 \sqrt {e x} \sqrt {c+d x^2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{42 c d^4}+\frac {e (e x)^{5/2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{14 c d^3 \sqrt {c+d x^2}}+\frac {(e x)^{9/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^2*(e*x)^(9/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) + ((39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^2)*e*(e*x)^(5/
2))/(14*c*d^3*Sqrt[c + d*x^2]) + (2*b^2*(e*x)^(9/2))/(7*d^2*e*Sqrt[c + d*x^2]) - (5*(39*b^2*c^2 - 42*a*b*c*d +
7*a^2*d^2)*e^3*Sqrt[e*x]*Sqrt[c + d*x^2])/(42*c*d^4) + (5*(39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^2)*e^(7/2)*(Sqrt
[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqr
t[e])], 1/2])/(84*c^(1/4)*d^(17/4)*Sqrt[c + d*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 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]

Rule 474

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[(-(b*c - a*
d)^2)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b^2*e*n*(p + 1))), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a +
b*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a
, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {(e x)^{7/2} \left (-\frac {3}{2} \left (2 a^2 d^2-3 (b c-a d)^2\right )-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2}\\ &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) \int \frac {(e x)^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{14 c d^2}\\ &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^2\right ) \int \frac {(e x)^{3/2}}{\sqrt {c+d x^2}} \, dx}{28 c d^3}\\ &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{84 d^4}\\ &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{42 d^4}\\ &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.22, size = 222, normalized size = 0.74 \begin {gather*} \frac {(e x)^{7/2} \left (\frac {\sqrt {x} \left (-7 a^2 d^2 \left (5 c+7 d x^2\right )+14 a b d \left (15 c^2+21 c d x^2+4 d^2 x^4\right )-b^2 \left (195 c^3+273 c^2 d x^2+52 c d^2 x^4-12 d^3 x^6\right )\right )}{d^4 \left (c+d x^2\right )}+\frac {5 i \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right )\right |-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^4}\right )}{42 x^{7/2} \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*x)^(7/2)*((Sqrt[x]*(-7*a^2*d^2*(5*c + 7*d*x^2) + 14*a*b*d*(15*c^2 + 21*c*d*x^2 + 4*d^2*x^4) - b^2*(195*c^3
+ 273*c^2*d*x^2 + 52*c*d^2*x^4 - 12*d^3*x^6)))/(d^4*(c + d*x^2)) + ((5*I)*(39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^
2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d
]]*d^4)))/(42*x^(7/2)*Sqrt[c + d*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(695\) vs. \(2(297)=594\).
time = 0.19, size = 696, normalized size = 2.30 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/84*(35*(-c*d)^(1/2)*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)
*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*d^3*x^2-210*(-c*
d)^(1/2)*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(
1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c*d^2*x^2+195*(-c*d)^(1/2)*((
d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/
2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^2*d*x^2+24*b^2*d^4*x^7+35*((d*x+(-c*d)
^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*Ellipti
cF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*a^2*c*d^2-210*((d*x+(-c*d)^(1/2))/(-c*d)^
(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)
^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*a*b*c^2*d+195*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^
(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^
(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*b^2*c^3+112*a*b*d^4*x^5-104*b^2*c*d^3*x^5-98*a^2*d^4*x^3+588*a*b*c*d^3*
x^3-546*b^2*c^2*d^2*x^3-70*a^2*c*d^3*x+420*a*b*c^2*d^2*x-390*b^2*c^3*d*x)*e^3/x*(e*x)^(1/2)/d^5/(d*x^2+c)^(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)^2/(d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

e^(7/2)*integrate((b*x^2 + a)^2*x^(7/2)/(d*x^2 + c)^(5/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.27, size = 249, normalized size = 0.82 \begin {gather*} \frac {5 \, {\left (39 \, b^{2} c^{4} - 42 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2} + {\left (39 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (39 \, b^{2} c^{3} d - 42 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (12 \, b^{2} d^{4} x^{6} - 195 \, b^{2} c^{3} d + 210 \, a b c^{2} d^{2} - 35 \, a^{2} c d^{3} - 4 \, {\left (13 \, b^{2} c d^{3} - 14 \, a b d^{4}\right )} x^{4} - 7 \, {\left (39 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {7}{2}}}{42 \, {\left (d^{7} x^{4} + 2 \, c d^{6} x^{2} + c^{2} d^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/42*(5*(39*b^2*c^4 - 42*a*b*c^3*d + 7*a^2*c^2*d^2 + (39*b^2*c^2*d^2 - 42*a*b*c*d^3 + 7*a^2*d^4)*x^4 + 2*(39*b
^2*c^3*d - 42*a*b*c^2*d^2 + 7*a^2*c*d^3)*x^2)*sqrt(d)*e^(7/2)*weierstrassPInverse(-4*c/d, 0, x) + (12*b^2*d^4*
x^6 - 195*b^2*c^3*d + 210*a*b*c^2*d^2 - 35*a^2*c*d^3 - 4*(13*b^2*c*d^3 - 14*a*b*d^4)*x^4 - 7*(39*b^2*c^2*d^2 -
42*a*b*c*d^3 + 7*a^2*d^4)*x^2)*sqrt(d*x^2 + c)*sqrt(x)*e^(7/2))/(d^7*x^4 + 2*c*d^6*x^2 + c^2*d^5)

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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)**2/(d*x**2+c)**(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)^2/(d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*x^(7/2)*e^(7/2)/(d*x^2 + c)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x)

[Out]

int(((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x)

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