\(\int \frac {1}{(e x)^{5/2} (c+d x) (b c^2-b d^2 x^2)^{3/2}} \, dx\) [550]

Optimal result

Integrand size = 34, antiderivative size = 339 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\frac {9 c-7 d x}{6 b c^4 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}+\frac {1}{3 b c^2 e (e x)^{3/2} (c+d x) \sqrt {b c^2-b d^2 x^2}}-\frac {5 \sqrt {b c^2-b d^2 x^2}}{2 b^2 c^5 e (e x)^{3/2}}+\frac {7 d \sqrt {b c^2-b d^2 x^2}}{2 b^2 c^6 e^2 \sqrt {e x}}+\frac {7 d^{3/2} \sqrt {1-\frac {d^2 x^2}{c^2}} E\left (\left .\arcsin \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )\right |-1\right )}{2 b c^{9/2} e^{5/2} \sqrt {b c^2-b d^2 x^2}}-\frac {d^{3/2} \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ),-1\right )}{b c^{9/2} e^{5/2} \sqrt {b c^2-b d^2 x^2}} \] Output:

1/6*(-7*d*x+9*c)/b/c^4/e/(e*x)^(3/2)/(-b*d^2*x^2+b*c^2)^(1/2)+1/3/b/c^2/e/ 
(e*x)^(3/2)/(d*x+c)/(-b*d^2*x^2+b*c^2)^(1/2)-5/2*(-b*d^2*x^2+b*c^2)^(1/2)/ 
b^2/c^5/e/(e*x)^(3/2)+7/2*d*(-b*d^2*x^2+b*c^2)^(1/2)/b^2/c^6/e^2/(e*x)^(1/ 
2)+7/2*d^(3/2)*(1-d^2*x^2/c^2)^(1/2)*EllipticE(d^(1/2)*(e*x)^(1/2)/c^(1/2) 
/e^(1/2),I)/b/c^(9/2)/e^(5/2)/(-b*d^2*x^2+b*c^2)^(1/2)-d^(3/2)*(1-d^2*x^2/ 
c^2)^(1/2)*EllipticF(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2),I)/b/c^(9/2)/e^(5 
/2)/(-b*d^2*x^2+b*c^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.29 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=-\frac {2 x \sqrt {1-\frac {d^2 x^2}{c^2}} \left (c \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {5}{2},\frac {1}{4},\frac {d^2 x^2}{c^2}\right )-3 d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {5}{2},\frac {3}{4},\frac {d^2 x^2}{c^2}\right )\right )}{3 b c^4 (e x)^{5/2} \sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:

Integrate[1/((e*x)^(5/2)*(c + d*x)*(b*c^2 - b*d^2*x^2)^(3/2)),x]
 

Output:

(-2*x*Sqrt[1 - (d^2*x^2)/c^2]*(c*Hypergeometric2F1[-3/4, 5/2, 1/4, (d^2*x^ 
2)/c^2] - 3*d*x*Hypergeometric2F1[-1/4, 5/2, 3/4, (d^2*x^2)/c^2]))/(3*b*c^ 
4*(e*x)^(5/2)*Sqrt[b*(c^2 - d^2*x^2)])
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.04, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {583, 551, 27, 551, 27, 553, 27, 553, 27, 556, 555, 1513, 27, 765, 762, 1390, 1389, 327}

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.

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

\(\Big \downarrow \) 583

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

\(\Big \downarrow \) 551

\(\displaystyle b \left (\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}-\frac {\int -\frac {9 c-7 d x}{2 (e x)^{5/2} \left (b c^2-b d^2 x^2\right )^{3/2}}dx}{3 b c^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle b \left (\frac {\int \frac {9 c-7 d x}{(e x)^{5/2} \left (b c^2-b d^2 x^2\right )^{3/2}}dx}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 551

\(\displaystyle b \left (\frac {\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}-\frac {\int -\frac {3 (15 c-7 d x)}{2 (e x)^{5/2} \sqrt {b c^2-b d^2 x^2}}dx}{b c^2}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle b \left (\frac {\frac {3 \int \frac {15 c-7 d x}{(e x)^{5/2} \sqrt {b c^2-b d^2 x^2}}dx}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 553

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {2 \int \frac {3 b c d (7 c-5 d x)}{2 (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}dx}{3 b c^2 e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \int \frac {7 c-5 d x}{(e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}dx}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 553

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \left (-\frac {2 \int \frac {b c d (5 c+7 d x)}{2 \sqrt {e x} \sqrt {b c^2-b d^2 x^2}}dx}{b c^2 e}-\frac {14 \sqrt {b c^2-b d^2 x^2}}{b c e \sqrt {e x}}\right )}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \left (-\frac {d \int \frac {5 c+7 d x}{\sqrt {e x} \sqrt {b c^2-b d^2 x^2}}dx}{c e}-\frac {14 \sqrt {b c^2-b d^2 x^2}}{b c e \sqrt {e x}}\right )}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 556

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \left (-\frac {d \sqrt {x} \int \frac {5 c+7 d x}{\sqrt {x} \sqrt {b c^2-b d^2 x^2}}dx}{c e \sqrt {e x}}-\frac {14 \sqrt {b c^2-b d^2 x^2}}{b c e \sqrt {e x}}\right )}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 555

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \left (-\frac {2 d \sqrt {x} \int \frac {5 c+7 d x}{\sqrt {b c^2-b d^2 x^2}}d\sqrt {x}}{c e \sqrt {e x}}-\frac {14 \sqrt {b c^2-b d^2 x^2}}{b c e \sqrt {e x}}\right )}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 1513

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \left (-\frac {2 d \sqrt {x} \left (7 c \int \frac {c+d x}{c \sqrt {b c^2-b d^2 x^2}}d\sqrt {x}-2 c \int \frac {1}{\sqrt {b c^2-b d^2 x^2}}d\sqrt {x}\right )}{c e \sqrt {e x}}-\frac {14 \sqrt {b c^2-b d^2 x^2}}{b c e \sqrt {e x}}\right )}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \left (-\frac {2 d \sqrt {x} \left (7 \int \frac {c+d x}{\sqrt {b c^2-b d^2 x^2}}d\sqrt {x}-2 c \int \frac {1}{\sqrt {b c^2-b d^2 x^2}}d\sqrt {x}\right )}{c e \sqrt {e x}}-\frac {14 \sqrt {b c^2-b d^2 x^2}}{b c e \sqrt {e x}}\right )}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 765

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \left (-\frac {2 d \sqrt {x} \left (7 \int \frac {c+d x}{\sqrt {b c^2-b d^2 x^2}}d\sqrt {x}-\frac {2 c \sqrt {1-\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {1-\frac {d^2 x^2}{c^2}}}d\sqrt {x}}{\sqrt {b c^2-b d^2 x^2}}\right )}{c e \sqrt {e x}}-\frac {14 \sqrt {b c^2-b d^2 x^2}}{b c e \sqrt {e x}}\right )}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 762

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \left (-\frac {2 d \sqrt {x} \left (7 \int \frac {c+d x}{\sqrt {b c^2-b d^2 x^2}}d\sqrt {x}-\frac {2 c^{3/2} \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right ),-1\right )}{\sqrt {d} \sqrt {b c^2-b d^2 x^2}}\right )}{c e \sqrt {e x}}-\frac {14 \sqrt {b c^2-b d^2 x^2}}{b c e \sqrt {e x}}\right )}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 1390

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \left (-\frac {2 d \sqrt {x} \left (\frac {7 \sqrt {1-\frac {d^2 x^2}{c^2}} \int \frac {c+d x}{\sqrt {1-\frac {d^2 x^2}{c^2}}}d\sqrt {x}}{\sqrt {b c^2-b d^2 x^2}}-\frac {2 c^{3/2} \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right ),-1\right )}{\sqrt {d} \sqrt {b c^2-b d^2 x^2}}\right )}{c e \sqrt {e x}}-\frac {14 \sqrt {b c^2-b d^2 x^2}}{b c e \sqrt {e x}}\right )}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 1389

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \left (-\frac {2 d \sqrt {x} \left (\frac {7 c \sqrt {1-\frac {d^2 x^2}{c^2}} \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {1-\frac {d x}{c}}}d\sqrt {x}}{\sqrt {b c^2-b d^2 x^2}}-\frac {2 c^{3/2} \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right ),-1\right )}{\sqrt {d} \sqrt {b c^2-b d^2 x^2}}\right )}{c e \sqrt {e x}}-\frac {14 \sqrt {b c^2-b d^2 x^2}}{b c e \sqrt {e x}}\right )}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 327

\(\displaystyle b \left (\frac {\frac {3 \left (-\frac {d \left (-\frac {2 d \sqrt {x} \left (\frac {7 c^{3/2} \sqrt {1-\frac {d^2 x^2}{c^2}} E\left (\left .\arcsin \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )\right |-1\right )}{\sqrt {d} \sqrt {b c^2-b d^2 x^2}}-\frac {2 c^{3/2} \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right ),-1\right )}{\sqrt {d} \sqrt {b c^2-b d^2 x^2}}\right )}{c e \sqrt {e x}}-\frac {14 \sqrt {b c^2-b d^2 x^2}}{b c e \sqrt {e x}}\right )}{c e}-\frac {10 \sqrt {b c^2-b d^2 x^2}}{b c e (e x)^{3/2}}\right )}{2 b c^2}+\frac {9 c-7 d x}{b c^2 e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}}{6 b c^2}+\frac {c-d x}{3 b c^2 e (e x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}\right )\)

Input:

Int[1/((e*x)^(5/2)*(c + d*x)*(b*c^2 - b*d^2*x^2)^(3/2)),x]
 

Output:

b*((c - d*x)/(3*b*c^2*e*(e*x)^(3/2)*(b*c^2 - b*d^2*x^2)^(3/2)) + ((9*c - 7 
*d*x)/(b*c^2*e*(e*x)^(3/2)*Sqrt[b*c^2 - b*d^2*x^2]) + (3*((-10*Sqrt[b*c^2 
- b*d^2*x^2])/(b*c*e*(e*x)^(3/2)) - (d*((-14*Sqrt[b*c^2 - b*d^2*x^2])/(b*c 
*e*Sqrt[e*x]) - (2*d*Sqrt[x]*((7*c^(3/2)*Sqrt[1 - (d^2*x^2)/c^2]*EllipticE 
[ArcSin[(Sqrt[d]*Sqrt[x])/Sqrt[c]], -1])/(Sqrt[d]*Sqrt[b*c^2 - b*d^2*x^2]) 
 - (2*c^(3/2)*Sqrt[1 - (d^2*x^2)/c^2]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[x])/S 
qrt[c]], -1])/(Sqrt[d]*Sqrt[b*c^2 - b*d^2*x^2])))/(c*e*Sqrt[e*x])))/(c*e)) 
)/(2*b*c^2))/(6*b*c^2))
 

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 327

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

rule 551

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[(-(e*x)^(m + 1))*(c + d*x)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1) 
)), x] + Simp[1/(2*a*(p + 1))   Int[(e*x)^m*(c*(m + 2*p + 3) + d*(m + 2*p + 
 4)*x)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p 
, -1] && LtQ[m, 0]
 

rule 553

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp 
[1/(a*e*(m + 1))   Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 
2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[m, -1]
 

rule 555

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> 
Simp[2   Subst[Int[(f + g*x^2)/Sqrt[a + c*x^4], x], x, Sqrt[x]], x] /; Free 
Q[{a, c, f, g}, x]
 

rule 556

Int[((c_) + (d_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symb 
ol] :> Simp[Sqrt[x]/Sqrt[e*x]   Int[(c + d*x)/(Sqrt[x]*Sqrt[a + b*x^2]), x] 
, x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 583

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

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 1389

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

rule 1390

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

rule 1513

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

Maple [A] (verified)

Time = 3.95 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {\left (18 \sqrt {\frac {d x +c}{c}}\, \sqrt {2}\, \sqrt {\frac {-d x +c}{c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \frac {\sqrt {2}}{2}\right ) c^{2} d^{2} x^{2}-21 \sqrt {\frac {d x +c}{c}}\, \sqrt {2}\, \sqrt {\frac {-d x +c}{c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \frac {\sqrt {2}}{2}\right ) c^{2} d^{2} x^{2}+18 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \frac {\sqrt {2}}{2}\right ) c^{3} d x \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {-d x +c}{c}}\, \sqrt {-\frac {x d}{c}}-21 \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \frac {\sqrt {2}}{2}\right ) c^{3} d x \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {-d x +c}{c}}\, \sqrt {-\frac {x d}{c}}-21 d^{4} x^{4}-6 c \,d^{3} x^{3}+29 d^{2} c^{2} x^{2}+8 c^{3} d x -4 c^{4}\right ) \sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}}{6 x \,e^{2} b^{2} c^{6} \sqrt {e x}\, \left (-d x +c \right ) \left (d x +c \right )^{2}}\)	\(336\)
elliptic	\(\frac {\sqrt {b e x \left (-d^{2} x^{2}+c^{2}\right )}\, \left (-\frac {\left (-b \,d^{2} e \,x^{2}-b c d e x \right ) d}{4 c^{6} b^{2} e^{3} \sqrt {\left (x -\frac {c}{d}\right ) \left (-b \,d^{2} e \,x^{2}-b c d e x \right )}}+\frac {\sqrt {-b \,d^{2} e \,x^{3}+b \,c^{2} e x}}{6 c^{5} b^{2} e^{3} \left (x +\frac {c}{d}\right )^{2}}+\frac {7 \left (-b \,d^{2} e \,x^{2}+b c d e x \right ) d}{4 c^{6} b^{2} e^{3} \sqrt {\left (x +\frac {c}{d}\right ) \left (-b \,d^{2} e \,x^{2}+b c d e x \right )}}-\frac {2 \sqrt {-b \,d^{2} e \,x^{3}+b \,c^{2} e x}}{3 c^{5} b^{2} e^{3} x^{2}}+\frac {2 \left (-b \,d^{2} e \,x^{2}+b \,c^{2} e \right ) d}{c^{6} b^{2} e^{3} \sqrt {x \left (-b \,d^{2} e \,x^{2}+b \,c^{2} e \right )}}+\frac {5 d \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {-\frac {2 \left (x -\frac {c}{d}\right ) d}{c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \frac {\sqrt {2}}{2}\right )}{4 c^{4} e^{2} b \sqrt {-b \,d^{2} e \,x^{3}+b \,c^{2} e x}}+\frac {7 d^{2} \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {-\frac {2 \left (x -\frac {c}{d}\right ) d}{c}}\, \sqrt {-\frac {x d}{c}}\, \left (-\frac {2 c \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {c \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{4 c^{5} e^{2} b \sqrt {-b \,d^{2} e \,x^{3}+b \,c^{2} e x}}\right )}{\sqrt {e x}\, \sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}}\)	\(497\)
risch	\(\text {Expression too large to display}\)	\(1153\)

Input:

int(1/(e*x)^(5/2)/(d*x+c)/(-b*d^2*x^2+b*c^2)^(3/2),x,method=_RETURNVERBOSE 
)
 

Output:

1/6/x/e^2/b^2*(18*((d*x+c)/c)^(1/2)*2^(1/2)*((-d*x+c)/c)^(1/2)*(-1/c*x*d)^ 
(1/2)*EllipticF(((d*x+c)/c)^(1/2),1/2*2^(1/2))*c^2*d^2*x^2-21*((d*x+c)/c)^ 
(1/2)*2^(1/2)*((-d*x+c)/c)^(1/2)*(-1/c*x*d)^(1/2)*EllipticE(((d*x+c)/c)^(1 
/2),1/2*2^(1/2))*c^2*d^2*x^2+18*2^(1/2)*EllipticF(((d*x+c)/c)^(1/2),1/2*2^ 
(1/2))*c^3*d*x*((d*x+c)/c)^(1/2)*((-d*x+c)/c)^(1/2)*(-1/c*x*d)^(1/2)-21*2^ 
(1/2)*EllipticE(((d*x+c)/c)^(1/2),1/2*2^(1/2))*c^3*d*x*((d*x+c)/c)^(1/2)*( 
(-d*x+c)/c)^(1/2)*(-1/c*x*d)^(1/2)-21*d^4*x^4-6*c*d^3*x^3+29*d^2*c^2*x^2+8 
*c^3*d*x-4*c^4)*(b*(-d^2*x^2+c^2))^(1/2)/c^6/(e*x)^(1/2)/(-d*x+c)/(d*x+c)^ 
2
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=-\frac {15 \, {\left (c d^{3} x^{5} + c^{2} d^{2} x^{4} - c^{3} d x^{3} - c^{4} x^{2}\right )} \sqrt {-b d^{2} e} {\rm weierstrassPInverse}\left (\frac {4 \, c^{2}}{d^{2}}, 0, x\right ) - 21 \, {\left (d^{4} x^{5} + c d^{3} x^{4} - c^{2} d^{2} x^{3} - c^{3} d x^{2}\right )} \sqrt {-b d^{2} e} {\rm weierstrassZeta}\left (\frac {4 \, c^{2}}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, c^{2}}{d^{2}}, 0, x\right )\right ) - {\left (21 \, d^{4} x^{4} + 6 \, c d^{3} x^{3} - 29 \, c^{2} d^{2} x^{2} - 8 \, c^{3} d x + 4 \, c^{4}\right )} \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {e x}}{6 \, {\left (b^{2} c^{6} d^{3} e^{3} x^{5} + b^{2} c^{7} d^{2} e^{3} x^{4} - b^{2} c^{8} d e^{3} x^{3} - b^{2} c^{9} e^{3} x^{2}\right )}} \] Input:

integrate(1/(e*x)^(5/2)/(d*x+c)/(-b*d^2*x^2+b*c^2)^(3/2),x, algorithm="fri 
cas")
 

Output:

-1/6*(15*(c*d^3*x^5 + c^2*d^2*x^4 - c^3*d*x^3 - c^4*x^2)*sqrt(-b*d^2*e)*we 
ierstrassPInverse(4*c^2/d^2, 0, x) - 21*(d^4*x^5 + c*d^3*x^4 - c^2*d^2*x^3 
 - c^3*d*x^2)*sqrt(-b*d^2*e)*weierstrassZeta(4*c^2/d^2, 0, weierstrassPInv 
erse(4*c^2/d^2, 0, x)) - (21*d^4*x^4 + 6*c*d^3*x^3 - 29*c^2*d^2*x^2 - 8*c^ 
3*d*x + 4*c^4)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(e*x))/(b^2*c^6*d^3*e^3*x^5 + 
b^2*c^7*d^2*e^3*x^4 - b^2*c^8*d*e^3*x^3 - b^2*c^9*e^3*x^2)
 

Sympy [F]

\[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {5}{2}} \left (- b \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:

integrate(1/(e*x)**(5/2)/(d*x+c)/(-b*d**2*x**2+b*c**2)**(3/2),x)
 

Output:

Integral(1/((e*x)**(5/2)*(-b*(-c + d*x)*(c + d*x))**(3/2)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b d^{2} x^{2} + b c^{2}\right )}^{\frac {3}{2}} {\left (d x + c\right )} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(5/2)/(d*x+c)/(-b*d^2*x^2+b*c^2)^(3/2),x, algorithm="max 
ima")
 

Output:

integrate(1/((-b*d^2*x^2 + b*c^2)^(3/2)*(d*x + c)*(e*x)^(5/2)), x)
 

Giac [F]

\[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b d^{2} x^{2} + b c^{2}\right )}^{\frac {3}{2}} {\left (d x + c\right )} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(5/2)/(d*x+c)/(-b*d^2*x^2+b*c^2)^(3/2),x, algorithm="gia 
c")
 

Output:

integrate(1/((-b*d^2*x^2 + b*c^2)^(3/2)*(d*x + c)*(e*x)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{5/2}\,{\left (b\,c^2-b\,d^2\,x^2\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:

int(1/((e*x)^(5/2)*(b*c^2 - b*d^2*x^2)^(3/2)*(c + d*x)),x)
 

Output:

int(1/((e*x)^(5/2)*(b*c^2 - b*d^2*x^2)^(3/2)*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (b c^2-b d^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(1/(e*x)^(5/2)/(d*x+c)/(-b*d^2*x^2+b*c^2)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

(sqrt(e)*sqrt(b)*( - 2*sqrt(c**2 - d**2*x**2)*c + 4*sqrt(c**2 - d**2*x**2) 
*d*x + 11*sqrt(x)*int(sqrt(c**2 - d**2*x**2)/(sqrt(x)*c**5 + sqrt(x)*c**4* 
d*x - 2*sqrt(x)*c**3*d**2*x**2 - 2*sqrt(x)*c**2*d**3*x**3 + sqrt(x)*c*d**4 
*x**4 + sqrt(x)*d**5*x**5),x)*c**4*d**2*x + 11*sqrt(x)*int(sqrt(c**2 - d** 
2*x**2)/(sqrt(x)*c**5 + sqrt(x)*c**4*d*x - 2*sqrt(x)*c**3*d**2*x**2 - 2*sq 
rt(x)*c**2*d**3*x**3 + sqrt(x)*c*d**4*x**4 + sqrt(x)*d**5*x**5),x)*c**3*d* 
*3*x**2 - 11*sqrt(x)*int(sqrt(c**2 - d**2*x**2)/(sqrt(x)*c**5 + sqrt(x)*c* 
*4*d*x - 2*sqrt(x)*c**3*d**2*x**2 - 2*sqrt(x)*c**2*d**3*x**3 + sqrt(x)*c*d 
**4*x**4 + sqrt(x)*d**5*x**5),x)*c**2*d**4*x**3 - 11*sqrt(x)*int(sqrt(c**2 
 - d**2*x**2)/(sqrt(x)*c**5 + sqrt(x)*c**4*d*x - 2*sqrt(x)*c**3*d**2*x**2 
- 2*sqrt(x)*c**2*d**3*x**3 + sqrt(x)*c*d**4*x**4 + sqrt(x)*d**5*x**5),x)*c 
*d**5*x**4 - 10*sqrt(x)*int((sqrt(c**2 - d**2*x**2)*x)/(sqrt(x)*c**5 + sqr 
t(x)*c**4*d*x - 2*sqrt(x)*c**3*d**2*x**2 - 2*sqrt(x)*c**2*d**3*x**3 + sqrt 
(x)*c*d**4*x**4 + sqrt(x)*d**5*x**5),x)*c**3*d**3*x - 10*sqrt(x)*int((sqrt 
(c**2 - d**2*x**2)*x)/(sqrt(x)*c**5 + sqrt(x)*c**4*d*x - 2*sqrt(x)*c**3*d* 
*2*x**2 - 2*sqrt(x)*c**2*d**3*x**3 + sqrt(x)*c*d**4*x**4 + sqrt(x)*d**5*x* 
*5),x)*c**2*d**4*x**2 + 10*sqrt(x)*int((sqrt(c**2 - d**2*x**2)*x)/(sqrt(x) 
*c**5 + sqrt(x)*c**4*d*x - 2*sqrt(x)*c**3*d**2*x**2 - 2*sqrt(x)*c**2*d**3* 
x**3 + sqrt(x)*c*d**4*x**4 + sqrt(x)*d**5*x**5),x)*c*d**5*x**3 + 10*sqrt(x 
)*int((sqrt(c**2 - d**2*x**2)*x)/(sqrt(x)*c**5 + sqrt(x)*c**4*d*x - 2*s...