3.9.38 \(\int \frac {(a+b x^2)^2 (c+d x^2)^{3/2}}{(e x)^{7/2}} \, dx\) [838]

Optimal. Leaf size=468 \[ \frac {4 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c e^5}+\frac {8 \left (b^2 c^2+9 a d (2 b c+a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{15 \sqrt {d} e^4 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 c^2 e^5}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac {2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt {e x}}-\frac {8 \sqrt [4]{c} \left (b^2 c^2+9 a d (2 b c+a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt {c+d x^2}}+\frac {4 \sqrt [4]{c} \left (b^2 c^2+9 a d (2 b c+a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt {c+d x^2}} \]

[Out]

2/9*(b^2*c^2+9*a*d*(a*d+2*b*c))*(e*x)^(3/2)*(d*x^2+c)^(3/2)/c^2/e^5-2/5*a^2*(d*x^2+c)^(5/2)/c/e/(e*x)^(5/2)-2*
a*(a*d+2*b*c)*(d*x^2+c)^(5/2)/c^2/e^3/(e*x)^(1/2)+4/15*(b^2*c^2+9*a*d*(a*d+2*b*c))*(e*x)^(3/2)*(d*x^2+c)^(1/2)
/c/e^5+8/15*(b^2*c^2+9*a*d*(a*d+2*b*c))*(e*x)^(1/2)*(d*x^2+c)^(1/2)/e^4/d^(1/2)/(c^(1/2)+x*d^(1/2))-8/15*c^(1/
4)*(b^2*c^2+9*a*d*(a*d+2*b*c))*(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)))*EllipticE(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)/d^(3/4)/e^(7/2)/(d*x^2+c)^(1/2)+4/15*c^(1/4)*(b^2*c^2
+9*a*d*(a*d+2*b*c))*(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)/d^(3/4)/e^(7/2)/(d*x^2+c)^(1/2)

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Rubi [A]
time = 0.31, antiderivative size = 468, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {473, 464, 285, 335, 311, 226, 1210} \begin {gather*} -\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}+\frac {4 \sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (9 a d (a d+2 b c)+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 )}{15 d^{3/4} e^{7/2} \sqrt {c+d x^2}}-\frac {8 \sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt {c+d x^2}}+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{9 c^2 e^5}+\frac {4 (e x)^{3/2} \sqrt {c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 c e^5}+\frac {8 \sqrt {e x} \sqrt {c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 \sqrt {d} e^4 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 a \left (c+d x^2\right )^{5/2} (a d+2 b c)}{c^2 e^3 \sqrt {e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(15*c*e^5) + (8*(b^2*c^2 + 9*a*d*(2*b*c + a*d)
)*Sqrt[e*x]*Sqrt[c + d*x^2])/(15*Sqrt[d]*e^4*(Sqrt[c] + Sqrt[d]*x)) + (2*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(e*x)
^(3/2)*(c + d*x^2)^(3/2))/(9*c^2*e^5) - (2*a^2*(c + d*x^2)^(5/2))/(5*c*e*(e*x)^(5/2)) - (2*a*(2*b*c + a*d)*(c
+ d*x^2)^(5/2))/(c^2*e^3*Sqrt[e*x]) - (8*c^(1/4)*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c
+ d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(15*d^(3/4
)*e^(7/2)*Sqrt[c + d*x^2]) + (4*c^(1/4)*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(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)*Sqrt[e])], 1/2])/(15*d^(3/4)*e^(7/2)
*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 285

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

Rule 311

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

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 464

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

Rule 473

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

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{7/2}} \, dx &=-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}+\frac {2 \int \frac {\left (\frac {5}{2} a (2 b c+a d)+\frac {5}{2} b^2 c x^2\right ) \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx}{5 c e^2}\\ &=-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac {2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt {e x}}+\frac {\left (b^2 c^2+9 a d (2 b c+a d)\right ) \int \sqrt {e x} \left (c+d x^2\right )^{3/2} \, dx}{c^2 e^4}\\ &=\frac {2 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 c^2 e^5}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac {2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt {e x}}+\frac {\left (2 \left (b^2 c^2+9 a d (2 b c+a d)\right )\right ) \int \sqrt {e x} \sqrt {c+d x^2} \, dx}{3 c e^4}\\ &=\frac {4 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c e^5}+\frac {2 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 c^2 e^5}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac {2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt {e x}}+\frac {\left (4 \left (b^2 c^2+9 a d (2 b c+a d)\right )\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{15 e^4}\\ &=\frac {4 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c e^5}+\frac {2 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 c^2 e^5}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac {2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt {e x}}+\frac {\left (8 \left (b^2 c^2+9 a d (2 b c+a d)\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 e^5}\\ &=\frac {4 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c e^5}+\frac {2 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 c^2 e^5}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac {2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt {e x}}+\frac {\left (8 \sqrt {c} \left (b^2 c^2+9 a d (2 b c+a d)\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 \sqrt {d} e^4}-\frac {\left (8 \sqrt {c} \left (b^2 c^2+9 a d (2 b c+a d)\right )\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 \sqrt {d} e^4}\\ &=\frac {4 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c e^5}+\frac {8 \left (b^2 c^2+9 a d (2 b c+a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{15 \sqrt {d} e^4 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 c^2 e^5}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac {2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt {e x}}-\frac {8 \sqrt [4]{c} \left (b^2 c^2+9 a d (2 b c+a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt {c+d x^2}}+\frac {4 \sqrt [4]{c} \left (b^2 c^2+9 a d (2 b c+a d)\right ) \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 )}{15 d^{3/4} e^{7/2} \sqrt {c+d 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 = 20.13, size = 141, normalized size = 0.30 \begin {gather*} \frac {x \left (-2 \left (c+d x^2\right ) \left (-18 a b x^2 \left (-5 c+d x^2\right )-b^2 x^4 \left (11 c+5 d x^2\right )+9 a^2 \left (c+7 d x^2\right )\right )+24 \left (b^2 c^2+18 a b c d+9 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x^4 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )\right )}{45 (e x)^{7/2} \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-2*(c + d*x^2)*(-18*a*b*x^2*(-5*c + d*x^2) - b^2*x^4*(11*c + 5*d*x^2) + 9*a^2*(c + 7*d*x^2)) + 24*(b^2*c^2
+ 18*a*b*c*d + 9*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^4*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(45*(e*x)
^(7/2)*Sqrt[c + d*x^2])

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Maple [A]
time = 0.13, size = 668, normalized size = 1.43

method	result	size
risch	\(-\frac {2 \sqrt {d \,x^{2}+c}\, \left (-5 b^{2} d \,x^{6}-18 a b d \,x^{4}-11 b^{2} c \,x^{4}+63 a^{2} d \,x^{2}+90 a b c \,x^{2}+9 a^{2} c \right )}{45 x^{2} e^{3} \sqrt {e x}}+\frac {\left (\frac {12}{5} a^{2} d^{2}+\frac {24}{5} a b c d +\frac {4}{15} b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{d \sqrt {d e \,x^{3}+c e x}\, e^{3} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(285\)
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 c \,a^{2} \sqrt {d e \,x^{3}+c e x}}{5 e^{4} x^{3}}-\frac {2 \left (d e \,x^{2}+c e \right ) a \left (7 a d +10 b c \right )}{5 e^{4} \sqrt {x \left (d e \,x^{2}+c e \right )}}+\frac {2 b^{2} d \,x^{3} \sqrt {d e \,x^{3}+c e x}}{9 e^{4}}+\frac {2 \left (\frac {2 b d \left (a d +b c \right )}{e^{3}}-\frac {7 b^{2} d c}{9 e^{3}}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (\frac {a^{2} d^{2}+4 a b c d +b^{2} c^{2}}{e^{3}}+\frac {d a \left (7 a d +10 b c \right )}{5 e^{3}}-\frac {3 \left (\frac {2 b d \left (a d +b c \right )}{e^{3}}-\frac {7 b^{2} d c}{9 e^{3}}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(397\)
default	\(\frac {\frac {2 b^{2} d^{3} x^{8}}{9}+\frac {24 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2} x^{2}}{5}+\frac {48 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d \,x^{2}}{5}+\frac {8 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3} x^{2}}{15}-\frac {12 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2} x^{2}}{5}-\frac {24 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d \,x^{2}}{5}-\frac {4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3} x^{2}}{15}+\frac {4 a b \,d^{3} x^{6}}{5}+\frac {32 b^{2} c \,d^{2} x^{6}}{45}-\frac {14 a^{2} d^{3} x^{4}}{5}-\frac {16 a b c \,d^{2} x^{4}}{5}+\frac {22 b^{2} c^{2} d \,x^{4}}{45}-\frac {16 a^{2} c \,d^{2} x^{2}}{5}-4 a b \,c^{2} d \,x^{2}-\frac {2 a^{2} c^{2} d}{5}}{x^{2} \sqrt {d \,x^{2}+c}\, d \,e^{3} \sqrt {e x}}\)	\(668\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^(3/2)/(e*x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/45/x^2*(5*b^2*d^3*x^8+108*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c*d^2*x^2+
216*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c^2*d*x^2+12*((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)*EllipticE(((d*x+
(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^3*x^2-54*((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*c*d^2*x^2-108*((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^2*d*x^2-6*((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^3*x^2+18*a*b*d^3*x^6+16*
b^2*c*d^2*x^6-63*a^2*d^3*x^4-72*a*b*c*d^2*x^4+11*b^2*c^2*d*x^4-72*a^2*c*d^2*x^2-90*a*b*c^2*d*x^2-9*a^2*c^2*d)/
(d*x^2+c)^(1/2)/d/e^3/(e*x)^(1/2)

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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((b*x^2+a)^2*(d*x^2+c)^(3/2)/(e*x)^(7/2),x, algorithm="maxima")

[Out]

e^(-7/2)*integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)/x^(7/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.31, size = 130, normalized size = 0.28 \begin {gather*} -\frac {2 \, {\left (12 \, {\left (b^{2} c^{2} + 18 \, a b c d + 9 \, a^{2} d^{2}\right )} \sqrt {d} x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (5 \, b^{2} d^{2} x^{6} + {\left (11 \, b^{2} c d + 18 \, a b d^{2}\right )} x^{4} - 9 \, a^{2} c d - 9 \, {\left (10 \, a b c d + 7 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {7}{2}\right )}}{45 \, d x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45*(12*(b^2*c^2 + 18*a*b*c*d + 9*a^2*d^2)*sqrt(d)*x^3*weierstrassZeta(-4*c/d, 0, weierstrassPInverse(-4*c/d
, 0, x)) - (5*b^2*d^2*x^6 + (11*b^2*c*d + 18*a*b*d^2)*x^4 - 9*a^2*c*d - 9*(10*a*b*c*d + 7*a^2*d^2)*x^2)*sqrt(d
*x^2 + c)*sqrt(x))*e^(-7/2)/(d*x^3)

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Sympy [C] Result contains complex when optimal does not.
time = 32.51, size = 320, normalized size = 0.68 \begin {gather*} \frac {a^{2} c^{\frac {3}{2}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {a^{2} \sqrt {c} d \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {a b c^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {a b \sqrt {c} d x^{\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 {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {b^{2} c^{\frac {3}{2}} x^{\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 {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {b^{2} \sqrt {c} d x^{\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 {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {7}{2}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/(e*x)**(7/2),x)

[Out]

a**2*c**(3/2)*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(7/2)*x**(5/2)*gamma(-1
/4)) + a**2*sqrt(c)*d*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(7/2)*sqrt(x)*ga
mma(3/4)) + a*b*c**(3/2)*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), d*x**2*exp_polar(I*pi)/c)/(e**(7/2)*sqrt(x)*g
amma(3/4)) + a*b*sqrt(c)*d*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), d*x**2*exp_polar(I*pi)/c)/(e**(7/2)*
gamma(7/4)) + b**2*c**(3/2)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(7/
2)*gamma(7/4)) + b**2*sqrt(c)*d*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(2*e
**(7/2)*gamma(11/4))

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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((b*x^2+a)^2*(d*x^2+c)^(3/2)/(e*x)^(7/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)*e^(-7/2)/x^(7/2), 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 )}^2\,{\left (d\,x^2+c\right )}^{3/2}}{{\left (e\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(7/2),x)

[Out]

int(((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(7/2), x)

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