3.9.54 \(\int \frac {(a+b x^2)^2}{(e x)^{3/2} (c+d x^2)^{3/2}} \, dx\) [854]

Optimal. Leaf size=393 \[ -\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt {c+d x^2}}+\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{c^2 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}} \]

[Out]

-(3*a^2*d^2-2*a*b*c*d+b^2*c^2)*(e*x)^(3/2)/c^2/d/e^3/(d*x^2+c)^(1/2)-2*a^2/c/e/(e*x)^(1/2)/(d*x^2+c)^(1/2)+(3*
a^2*d^2-2*a*b*c*d+3*b^2*c^2)*(e*x)^(1/2)*(d*x^2+c)^(1/2)/c^2/d^(3/2)/e^2/(c^(1/2)+x*d^(1/2))-(3*a^2*d^2-2*a*b*
c*d+3*b^2*c^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)))*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)/c^(7/4)/d^(7/4)/e^(3/2)/(d*x^2+c)^(1/2)+1/2*(3*a^2*d^2-2*a*b*c*d+3*b^
2*c^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^(7/4)/d^(7/4)/e^(3/2)/(d*x^2+c)^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 393, 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 = {473, 468, 335, 311, 226, 1210} \begin {gather*} \frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 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 )}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 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 )}{c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c^2 d e^3 \sqrt {c+d x^2}}+\frac {\sqrt {e x} \sqrt {c+d x^2} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{c^2 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2)/(c*e*Sqrt[e*x]*Sqrt[c + d*x^2]) - ((b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*(e*x)^(3/2))/(c^2*d*e^3*Sqrt[c +
d*x^2]) + ((3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*Sqrt[e*x]*Sqrt[c + d*x^2])/(c^2*d^(3/2)*e^2*(Sqrt[c] + Sqrt[d]
*x)) - ((3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*El
lipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(c^(7/4)*d^(7/4)*e^(3/2)*Sqrt[c + d*x^2]) + ((3
*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^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)*Sqrt[e])], 1/2])/(2*c^(7/4)*d^(7/4)*e^(3/2)*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 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 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(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}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx &=-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}+\frac {2 \int \frac {\sqrt {e x} \left (\frac {1}{2} a (2 b c-3 a d)+\frac {1}{2} b^2 c x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{c e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt {c+d x^2}}-\frac {\left (2 a b-\frac {3 b^2 c}{d}-\frac {3 a^2 d}{c}\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{2 c e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt {c+d x^2}}-\frac {\left (2 a b-\frac {3 b^2 c}{d}-\frac {3 a^2 d}{c}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{c e^3}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt {c+d x^2}}+\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{c^{3/2} d^{3/2} e^2}-\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\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 )}{c^{3/2} d^{3/2} e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt {c+d x^2}}+\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{c^2 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\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 )}{c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\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 )}{2 c^{7/4} d^{7/4} e^{3/2} \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 = 10.10, size = 126, normalized size = 0.32 \begin {gather*} \frac {x \left (-3 b^2 c^2 x^2+6 a b c d x^2-3 a^2 d \left (2 c+3 d x^2\right )+\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) x^2 \sqrt {1+\frac {d x^2}{c}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {d x^2}{c}\right )\right )}{3 c^2 d (e x)^{3/2} \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-3*b^2*c^2*x^2 + 6*a*b*c*d*x^2 - 3*a^2*d*(2*c + 3*d*x^2) + (3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*x^2*Sqrt[1
+ (d*x^2)/c]*Hypergeometric2F1[1/2, 3/4, 7/4, -((d*x^2)/c)]))/(3*c^2*d*(e*x)^(3/2)*Sqrt[c + d*x^2])

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Maple [A]
time = 0.14, size = 594, normalized size = 1.51

method	result	size
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {x^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d e \,c^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}-\frac {2 \left (d e \,x^{2}+c e \right ) a^{2}}{c^{2} e^{2} \sqrt {x \left (d e \,x^{2}+c e \right )}}+\frac {\left (\frac {b^{2}}{e d}+\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 d \,c^{2} e}+\frac {d \,a^{2}}{c^{2} e}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(327\)
risch	\(-\frac {2 a^{2} \sqrt {d \,x^{2}+c}}{c^{2} e \sqrt {e x}}+\frac {\left (\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d^{2} \sqrt {d e \,x^{3}+c e x}}-\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {x^{2}}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}-\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )}{d}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{c^{2} e \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(448\)
default	\(-\frac {3 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-2 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +3 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-6 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}+4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d -6 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}+6 a^{2} d^{3} x^{2}-4 a b c \,d^{2} x^{2}+2 b^{2} c^{2} d \,x^{2}+4 a^{2} c \,d^{2}}{2 \sqrt {d \,x^{2}+c}\, d^{2} e \sqrt {e x}\, c^{2}}\)	\(594\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*((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^2-2*((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^2*d+3*((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^3-6*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c*d^2+4*((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-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)*EllipticE(((d*x+(-c*d)^(1/2))
/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^3+6*a^2*d^3*x^2-4*a*b*c*d^2*x^2+2*b^2*c^2*d*x^2+4*a^2*c*d^2)/(d*x^2+c)
^(1/2)/d^2/e/(e*x)^(1/2)/c^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/(e*x)^(3/2)/(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.26, size = 158, normalized size = 0.40 \begin {gather*} -\frac {{\left ({\left ({\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{3} + {\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {d} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (2 \, a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {3}{2}\right )}}{c^{2} d^{3} x^{3} + c^{3} d^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(((3*b^2*c^2*d - 2*a*b*c*d^2 + 3*a^2*d^3)*x^3 + (3*b^2*c^3 - 2*a*b*c^2*d + 3*a^2*c*d^2)*x)*sqrt(d)*weierstras
sZeta(-4*c/d, 0, weierstrassPInverse(-4*c/d, 0, x)) + (2*a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + 3*a^2*d^3)*x^2
)*sqrt(d*x^2 + c)*sqrt(x))*e^(-3/2)/(c^2*d^3*x^3 + c^3*d^2*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{2}}{\left (e x\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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