\(\int \frac {(e x)^{7/2}}{(a-b x^2)^2 (c-d x^2)^{5/2}} \, dx\) [1157]

Optimal result

Integrand size = 30, antiderivative size = 454 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {(2 b c+3 a d) e^3 \sqrt {e x}}{6 b (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 \sqrt {e x}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {5 (b c+2 a d) e^3 \sqrt {e x}}{6 (b c-a d)^3 \sqrt {c-d x^2}}+\frac {5 \sqrt [4]{c} (b c+2 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {5 \sqrt [4]{c} (b c+a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {5 \sqrt [4]{c} (b c+a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \] Output:

1/6*(3*a*d+2*b*c)*e^3*(e*x)^(1/2)/b/(-a*d+b*c)^2/(-d*x^2+c)^(3/2)+1/2*a*e^ 
3*(e*x)^(1/2)/b/(-a*d+b*c)/(-b*x^2+a)/(-d*x^2+c)^(3/2)+5/6*(2*a*d+b*c)*e^3 
*(e*x)^(1/2)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)+5/6*c^(1/4)*(2*a*d+b*c)*e^(7/2) 
*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/d^(1/4 
)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)-5/4*c^(1/4)*(a*d+b*c)*e^(7/2)*(1-d*x^2/c)^ 
(1/2)*EllipticPi(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),-b^(1/2)*c^(1/2)/a^(1 
/2)/d^(1/2),I)/d^(1/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)-5/4*c^(1/4)*(a*d+b*c) 
*e^(7/2)*(1-d*x^2/c)^(1/2)*EllipticPi(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2), 
b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)/d^(1/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.30 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.56 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {e^3 \sqrt {e x} \left (a \left (b^2 c x^2 \left (7 c-5 d x^2\right )+a^2 d \left (-5 c+7 d x^2\right )-2 a b \left (5 c^2-8 c d x^2+5 d^2 x^4\right )\right )+5 a (2 b c+a d) \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+b (b c+2 a d) x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{6 a (b c-a d)^3 \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \] Input:

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

Output:

(e^3*Sqrt[e*x]*(a*(b^2*c*x^2*(7*c - 5*d*x^2) + a^2*d*(-5*c + 7*d*x^2) - 2* 
a*b*(5*c^2 - 8*c*d*x^2 + 5*d^2*x^4)) + 5*a*(2*b*c + a*d)*(a - b*x^2)*(c - 
d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a 
] + b*(b*c + 2*a*d)*x^2*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*Appell 
F1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(6*a*(b*c - a*d)^3*(-a + b*x^ 
2)*(c - d*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {368, 27, 970, 1024, 27, 1024, 27, 1021, 765, 762, 925, 27, 1543, 1542}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle \frac {2 \int \frac {e^8 x^4}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \int \frac {e^4 x^4}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\)

\(\Big \downarrow \) 970

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {(4 b c+5 a d) x^2 e^2+a c e^2}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1024

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {-\frac {\int -\frac {10 b c \left ((2 b c+3 a d) x^2 e^2+a c e^2\right )}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{6 c (b c-a d)}-\frac {\sqrt {e x} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {5 b \int \frac {(2 b c+3 a d) x^2 e^2+a c e^2}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 (b c-a d)}-\frac {\sqrt {e x} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1024

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {5 b \left (-\frac {\int -\frac {2 c \left (b (b c+2 a d) x^2 e^2+a (2 b c+a d) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}-\frac {\sqrt {e x} (2 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {\sqrt {e x} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {5 b \left (\frac {\int \frac {b (b c+2 a d) x^2 e^2+a (2 b c+a d) e^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b c-a d}-\frac {\sqrt {e x} (2 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {\sqrt {e x} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1021

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {5 b \left (\frac {3 a e^2 (a d+b c) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}-(2 a d+b c) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{b c-a d}-\frac {\sqrt {e x} (2 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {\sqrt {e x} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 765

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {5 b \left (\frac {3 a e^2 (a d+b c) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}-\frac {\sqrt {1-\frac {d x^2}{c}} (2 a d+b c) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{\sqrt {c-d x^2}}}{b c-a d}-\frac {\sqrt {e x} (2 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {\sqrt {e x} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 762

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {5 b \left (\frac {3 a e^2 (a d+b c) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (2 a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{d} \sqrt {c-d x^2}}}{b c-a d}-\frac {\sqrt {e x} (2 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {\sqrt {e x} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 925

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {5 b \left (\frac {3 a e^2 (a d+b c) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (2 a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{d} \sqrt {c-d x^2}}}{b c-a d}-\frac {\sqrt {e x} (2 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {\sqrt {e x} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {5 b \left (\frac {3 a e^2 (a d+b c) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (2 a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{d} \sqrt {c-d x^2}}}{b c-a d}-\frac {\sqrt {e x} (2 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {\sqrt {e x} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1543

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {5 b \left (\frac {3 a e^2 (a d+b c) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (2 a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{d} \sqrt {c-d x^2}}}{b c-a d}-\frac {\sqrt {e x} (2 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {\sqrt {e x} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1542

\(\displaystyle 2 e^3 \left (\frac {a e^2 \sqrt {e x}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {5 b \left (\frac {3 a e^2 (a d+b c) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (2 a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{d} \sqrt {c-d x^2}}}{b c-a d}-\frac {\sqrt {e x} (2 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {\sqrt {e x} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\)

Input:

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

Output:

2*e^3*((a*e^2*Sqrt[e*x])/(4*b*(b*c - a*d)*(c - d*x^2)^(3/2)*(a*e^2 - b*e^2 
*x^2)) - (-1/3*((2*b*c + 3*a*d)*Sqrt[e*x])/((b*c - a*d)*(c - d*x^2)^(3/2)) 
 + (5*b*(-(((b*c + 2*a*d)*Sqrt[e*x])/((b*c - a*d)*Sqrt[c - d*x^2])) + (-(( 
c^(1/4)*(b*c + 2*a*d)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4 
)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(d^(1/4)*Sqrt[c - d*x^2])) + 3*a*(b* 
c + a*d)*e^2*((c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/ 
(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2 
*a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2]) + (c^(1/4)*Sqrt[1 - (d*x^2)/c]*Ellipti 
cPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/ 
4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2])))/(b*c - a*d)))/( 
3*(b*c - a*d)))/(4*b*(b*c - a*d)))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 368

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[x^(k*(m + 1) 
 - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], 
 x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m 
] && IntegerQ[p]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 925

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 
1/(2*c)   Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 
*c)   Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, 
 c, d}, x] && NeQ[b*c - a*d, 0]
 

rule 970

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

rule 1021

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x 
_)^(n_)]), x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* 
e - a*f)/b   Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, 
 d, e, f, n}, x]
 

rule 1024

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f 
_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c 
+ d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( 
p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b 
*c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
 

rule 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

rule 1543

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 
Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) 
]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1192\) vs. \(2(360)=720\).

Time = 3.52 (sec) , antiderivative size = 1193, normalized size of antiderivative = 2.63

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1193\)
default	\(\text {Expression too large to display}\)	\(4391\)

Input:

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

Output:

1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2*d*e^3*a*b/( 
a*d-b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(-d*e*x^3+c*e*x)^(1/2)/(b*d*x^2-a*d)+ 
1/3*e^3*c/d^2/(a*d-b*c)^2*(-d*e*x^3+c*e*x)^(1/2)/(x^2-c/d)^2-1/6*e^4*x*(7* 
a*d+5*b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/(-(x^2-c/d)*d*e*x)^(1/2)- 
5/6*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x* 
d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2)) 
*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*e^4/(a*d-b*c)/(a^2*d^2-2*a*b*c*d+b^2* 
c^2)-5/12/d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1 
/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d 
)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*e^4/(a^2*d^2-2*a*b*c*d+b^2*c^2) 
/(a*d-b*c)*b*c-5/8*a^2*e^4/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/(a*b)^(1/ 
2)*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d 
/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/ 
2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/ 
(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2))-5/8*a*e^4/(a^2*d^2-2*a*b*c 
*d+b^2*c^2)/(a*d-b*c)/(a*b)^(1/2)/d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)* 
(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2 
)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c* 
d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1/2*2^ 
(1/2))*b*c+5/8*a^2*e^4/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/(a*b)^(1/2...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:

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

Output:

(sqrt(e)*e**3*(sqrt(c - d*x**2)*int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c**3 - 
3*sqrt(x)*a**2*c**2*d*x**2 + 3*sqrt(x)*a**2*c*d**2*x**4 - sqrt(x)*a**2*d** 
3*x**6 - 2*sqrt(x)*a*b*c**3*x**2 + 6*sqrt(x)*a*b*c**2*d*x**4 - 6*sqrt(x)*a 
*b*c*d**2*x**6 + 2*sqrt(x)*a*b*d**3*x**8 + sqrt(x)*b**2*c**3*x**4 - 3*sqrt 
(x)*b**2*c**2*d*x**6 + 3*sqrt(x)*b**2*c*d**2*x**8 - sqrt(x)*b**2*d**3*x**1 
0),x)*a**2*c**2 - sqrt(c - d*x**2)*int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c**3 
 - 3*sqrt(x)*a**2*c**2*d*x**2 + 3*sqrt(x)*a**2*c*d**2*x**4 - sqrt(x)*a**2* 
d**3*x**6 - 2*sqrt(x)*a*b*c**3*x**2 + 6*sqrt(x)*a*b*c**2*d*x**4 - 6*sqrt(x 
)*a*b*c*d**2*x**6 + 2*sqrt(x)*a*b*d**3*x**8 + sqrt(x)*b**2*c**3*x**4 - 3*s 
qrt(x)*b**2*c**2*d*x**6 + 3*sqrt(x)*b**2*c*d**2*x**8 - sqrt(x)*b**2*d**3*x 
**10),x)*a**2*c*d*x**2 - sqrt(c - d*x**2)*int(sqrt(c - d*x**2)/(sqrt(x)*a* 
*2*c**3 - 3*sqrt(x)*a**2*c**2*d*x**2 + 3*sqrt(x)*a**2*c*d**2*x**4 - sqrt(x 
)*a**2*d**3*x**6 - 2*sqrt(x)*a*b*c**3*x**2 + 6*sqrt(x)*a*b*c**2*d*x**4 - 6 
*sqrt(x)*a*b*c*d**2*x**6 + 2*sqrt(x)*a*b*d**3*x**8 + sqrt(x)*b**2*c**3*x** 
4 - 3*sqrt(x)*b**2*c**2*d*x**6 + 3*sqrt(x)*b**2*c*d**2*x**8 - sqrt(x)*b**2 
*d**3*x**10),x)*a*b*c**2*x**2 + sqrt(c - d*x**2)*int(sqrt(c - d*x**2)/(sqr 
t(x)*a**2*c**3 - 3*sqrt(x)*a**2*c**2*d*x**2 + 3*sqrt(x)*a**2*c*d**2*x**4 - 
 sqrt(x)*a**2*d**3*x**6 - 2*sqrt(x)*a*b*c**3*x**2 + 6*sqrt(x)*a*b*c**2*d*x 
**4 - 6*sqrt(x)*a*b*c*d**2*x**6 + 2*sqrt(x)*a*b*d**3*x**8 + sqrt(x)*b**2*c 
**3*x**4 - 3*sqrt(x)*b**2*c**2*d*x**6 + 3*sqrt(x)*b**2*c*d**2*x**8 - sq...