\(\int \frac {x^2 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx\) [1473]

Optimal result

Integrand size = 25, antiderivative size = 435 \[ \int \frac {x^2 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx=\frac {2 \left (c \left (8 b c^2-7 a d^2\right )+d \left (2 b c^2-a d^2\right ) x\right ) \sqrt {a-b x^2}}{3 d^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {2 c^2 \left (a-b x^2\right )^{3/2}}{3 d \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} c \left (8 b c^2-7 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 d^4 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {4 \sqrt {a} \left (8 b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d^4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

2/3*(c*(-7*a*d^2+8*b*c^2)+d*(-a*d^2+2*b*c^2)*x)*(-b*x^2+a)^(1/2)/d^3/(-a*d 
^2+b*c^2)/(d*x+c)^(1/2)+2/3*c^2*(-b*x^2+a)^(3/2)/d/(-a*d^2+b*c^2)/(d*x+c)^ 
(3/2)-4/3*a^(1/2)*b^(1/2)*c*(-7*a*d^2+8*b*c^2)*(d*x+c)^(1/2)*(1-b*x^2/a)^( 
1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/ 
(b^(1/2)*c+a^(1/2)*d))^(1/2))/d^4/(-a*d^2+b*c^2)/(b^(1/2)*(d*x+c)/(b^(1/2) 
*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)+4/3*a^(1/2)*(-a*d^2+8*b*c^2)*(b^(1/2 
)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1/2)*EllipticF(1/2*(1- 
b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d)) 
^(1/2))/b^(1/2)/d^4/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.70 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.33 \[ \int \frac {x^2 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {2 a d^2 \left (6 c^2+8 c d x+d^2 x^2\right )-2 b c^2 \left (8 c^2+10 c d x+d^2 x^2\right )}{d^3 \left (-b c^2+a d^2\right ) (c+d x)}+\frac {4 \left (c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (8 b c^2-7 a d^2\right ) \left (-a+b x^2\right )-i \sqrt {b} c \left (8 b^{3/2} c^3-8 \sqrt {a} b c^2 d-7 a \sqrt {b} c d^2+7 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} d \left (8 b^{3/2} c^3-2 \sqrt {a} b c^2 d-7 a \sqrt {b} c d^2+a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{d^5 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-b c^2+a d^2\right ) \left (-a+b x^2\right )}\right )}{3 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*((2*a*d^2*(6*c^2 + 8*c*d*x + d^2*x^2) - 2*b*c^2*(8*c^2 + 
10*c*d*x + d^2*x^2))/(d^3*(-(b*c^2) + a*d^2)*(c + d*x)) + (4*(c*d^2*Sqrt[- 
c + (Sqrt[a]*d)/Sqrt[b]]*(8*b*c^2 - 7*a*d^2)*(-a + b*x^2) - I*Sqrt[b]*c*(8 
*b^(3/2)*c^3 - 8*Sqrt[a]*b*c^2*d - 7*a*Sqrt[b]*c*d^2 + 7*a^(3/2)*d^3)*Sqrt 
[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/( 
c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[ 
b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*S 
qrt[a]*d*(8*b^(3/2)*c^3 - 2*Sqrt[a]*b*c^2*d - 7*a*Sqrt[b]*c*d^2 + a^(3/2)* 
d^3)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] 
 - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a] 
*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]* 
d)]))/(d^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-(b*c^2) + a*d^2)*(-a + b*x^2)) 
))/(3*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {603, 27, 681, 25, 27, 600, 509, 508, 327, 512, 511, 321}

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.

\(\displaystyle \int \frac {x^2 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 681

\(\displaystyle \frac {\frac {2 \sqrt {a-b x^2} \left (x \left (2 b c^2-a d^2\right )+c \left (\frac {8 b c^2}{d}-7 a d\right )\right )}{3 d^2 \sqrt {c+d x}}-\frac {2 \int -\frac {a d \left (2 b c^2-a d^2\right )+b c \left (8 b c^2-7 a d^2\right ) x}{d \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 d^2}}{b c^2-a d^2}+\frac {2 c^2 \left (a-b x^2\right )^{3/2}}{3 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \int \frac {a d \left (2 b c^2-a d^2\right )+b c \left (8 b c^2-7 a d^2\right ) x}{d \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 d^2}+\frac {2 \sqrt {a-b x^2} \left (x \left (2 b c^2-a d^2\right )+c \left (\frac {8 b c^2}{d}-7 a d\right )\right )}{3 d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 c^2 \left (a-b x^2\right )^{3/2}}{3 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \int \frac {a d \left (2 b c^2-a d^2\right )+b c \left (8 b c^2-7 a d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 d^3}+\frac {2 \sqrt {a-b x^2} \left (x \left (2 b c^2-a d^2\right )+c \left (\frac {8 b c^2}{d}-7 a d\right )\right )}{3 d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 c^2 \left (a-b x^2\right )^{3/2}}{3 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {2 \left (\frac {b c \left (8 b c^2-7 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (8 b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{3 d^3}+\frac {2 \sqrt {a-b x^2} \left (x \left (2 b c^2-a d^2\right )+c \left (\frac {8 b c^2}{d}-7 a d\right )\right )}{3 d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 c^2 \left (a-b x^2\right )^{3/2}}{3 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {2 \left (\frac {b c \sqrt {1-\frac {b x^2}{a}} \left (8 b c^2-7 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (8 b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{3 d^3}+\frac {2 \sqrt {a-b x^2} \left (x \left (2 b c^2-a d^2\right )+c \left (\frac {8 b c^2}{d}-7 a d\right )\right )}{3 d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 c^2 \left (a-b x^2\right )^{3/2}}{3 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (8 b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-7 a d^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 d^3}+\frac {2 \sqrt {a-b x^2} \left (x \left (2 b c^2-a d^2\right )+c \left (\frac {8 b c^2}{d}-7 a d\right )\right )}{3 d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 c^2 \left (a-b x^2\right )^{3/2}}{3 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (8 b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-7 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 d^3}+\frac {2 \sqrt {a-b x^2} \left (x \left (2 b c^2-a d^2\right )+c \left (\frac {8 b c^2}{d}-7 a d\right )\right )}{3 d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 c^2 \left (a-b x^2\right )^{3/2}}{3 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {2 \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (8 b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-7 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 d^3}+\frac {2 \sqrt {a-b x^2} \left (x \left (2 b c^2-a d^2\right )+c \left (\frac {8 b c^2}{d}-7 a d\right )\right )}{3 d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 c^2 \left (a-b x^2\right )^{3/2}}{3 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (8 b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-7 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 d^3}+\frac {2 \sqrt {a-b x^2} \left (x \left (2 b c^2-a d^2\right )+c \left (\frac {8 b c^2}{d}-7 a d\right )\right )}{3 d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 c^2 \left (a-b x^2\right )^{3/2}}{3 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (8 b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-7 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 d^3}+\frac {2 \sqrt {a-b x^2} \left (x \left (2 b c^2-a d^2\right )+c \left (\frac {8 b c^2}{d}-7 a d\right )\right )}{3 d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 c^2 \left (a-b x^2\right )^{3/2}}{3 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

Input:

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

Output:

(2*c^2*(a - b*x^2)^(3/2))/(3*d*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) + ((2*(c*( 
(8*b*c^2)/d - 7*a*d) + (2*b*c^2 - a*d^2)*x)*Sqrt[a - b*x^2])/(3*d^2*Sqrt[c 
 + d*x]) + (2*((-2*Sqrt[a]*Sqrt[b]*c*(8*b*c^2 - 7*a*d^2)*Sqrt[c + d*x]*Sqr 
t[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], 
(2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + 
 Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(8*b*c^2 - a*d^ 
2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*E 
llipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c) 
/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(3*d^3))/(b*c^ 
2 - a*d^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

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 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

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

rule 511

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

rule 512

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

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 603

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde 
r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 
1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x) 
^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 
1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt 
Q[m, 1] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
 

rule 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(781\) vs. \(2(365)=730\).

Time = 4.91 (sec) , antiderivative size = 782, normalized size of antiderivative = 1.80

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 c^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 d^{5} \left (x +\frac {c}{d}\right )^{2}}+\frac {4 \left (-b d \,x^{2}+a d \right ) c \left (3 a \,d^{2}-4 b \,c^{2}\right )}{3 d^{4} \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}+\frac {2 \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 d^{3}}+\frac {2 \left (\frac {a \,d^{2}-3 b \,c^{2}}{d^{4}}+\frac {b \,c^{2}}{3 d^{4}}+\frac {2 b \,c^{2} \left (3 a \,d^{2}-4 b \,c^{2}\right )}{3 d^{4} \left (a \,d^{2}-b \,c^{2}\right )}-\frac {a}{3 d^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {8 b c}{3 d^{3}}+\frac {2 b c \left (3 a \,d^{2}-4 b \,c^{2}\right )}{3 d^{3} \left (a \,d^{2}-b \,c^{2}\right )}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(782\)
risch	\(\text {Expression too large to display}\)	\(1719\)
default	\(\text {Expression too large to display}\)	\(2498\)

Input:

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

Output:

((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-2/3/d^5*c^2*(- 
b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2+4/3*(-b*d*x^2+a*d)/d^4/(a*d^2-b 
*c^2)*c*(3*a*d^2-4*b*c^2)/((x+c/d)*(-b*d*x^2+a*d))^(1/2)+2/3/d^3*(-b*d*x^3 
-b*c*x^2+a*d*x+a*c)^(1/2)+2*(1/d^4*(a*d^2-3*b*c^2)+1/3*b/d^4*c^2+2/3*b/d^4 
*c^2*(3*a*d^2-4*b*c^2)/(a*d^2-b*c^2)-1/3/d^2*a)*(c/d-1/b*(a*b)^(1/2))*((x+ 
c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/ 
2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b* 
c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((- 
c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(8/3*b*c/d^3+2/3*b*c 
/d^3*(3*a*d^2-4*b*c^2)/(a*d^2-b*c^2))*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d- 
1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2) 
*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d* 
x+a*c)^(1/2)*((-c/d-1/b*(a*b)^(1/2))*EllipticE(((x+c/d)/(c/d-1/b*(a*b)^(1/ 
2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b 
)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^( 
1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.11 \[ \int \frac {x^2 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (2 \, {\left (8 \, b^{2} c^{6} - 13 \, a b c^{4} d^{2} + 3 \, a^{2} c^{2} d^{4} + {\left (8 \, b^{2} c^{4} d^{2} - 13 \, a b c^{2} d^{4} + 3 \, a^{2} d^{6}\right )} x^{2} + 2 \, {\left (8 \, b^{2} c^{5} d - 13 \, a b c^{3} d^{3} + 3 \, a^{2} c d^{5}\right )} x\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 6 \, {\left (8 \, b^{2} c^{5} d - 7 \, a b c^{3} d^{3} + {\left (8 \, b^{2} c^{3} d^{3} - 7 \, a b c d^{5}\right )} x^{2} + 2 \, {\left (8 \, b^{2} c^{4} d^{2} - 7 \, a b c^{2} d^{4}\right )} x\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (8 \, b^{2} c^{4} d^{2} - 6 \, a b c^{2} d^{4} + {\left (b^{2} c^{2} d^{4} - a b d^{6}\right )} x^{2} + 2 \, {\left (5 \, b^{2} c^{3} d^{3} - 4 \, a b c d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{9 \, {\left (b^{2} c^{4} d^{5} - a b c^{2} d^{7} + {\left (b^{2} c^{2} d^{7} - a b d^{9}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{6} - a b c d^{8}\right )} x\right )}} \] Input:

integrate(x^2*(-b*x^2+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

2/9*(2*(8*b^2*c^6 - 13*a*b*c^4*d^2 + 3*a^2*c^2*d^4 + (8*b^2*c^4*d^2 - 13*a 
*b*c^2*d^4 + 3*a^2*d^6)*x^2 + 2*(8*b^2*c^5*d - 13*a*b*c^3*d^3 + 3*a^2*c*d^ 
5)*x)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27* 
(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 6*(8*b^2*c^5*d - 7*a*b*c 
^3*d^3 + (8*b^2*c^3*d^3 - 7*a*b*c*d^5)*x^2 + 2*(8*b^2*c^4*d^2 - 7*a*b*c^2* 
d^4)*x)*sqrt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b 
*c^3 - 9*a*c*d^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^ 
2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(8*b^2*c^4* 
d^2 - 6*a*b*c^2*d^4 + (b^2*c^2*d^4 - a*b*d^6)*x^2 + 2*(5*b^2*c^3*d^3 - 4*a 
*b*c*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^2*c^4*d^5 - a*b*c^2*d^7 + 
(b^2*c^2*d^7 - a*b*d^9)*x^2 + 2*(b^2*c^3*d^6 - a*b*c*d^8)*x)
 

Sympy [F]

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

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

Output:

Integral(x**2*sqrt(a - b*x**2)/(c + d*x)**(5/2), x)
 

Maxima [F]

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

integrate(x^2*(-b*x^2+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(x^2*(-b*x^2+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(10*sqrt(c + d*x)*sqrt(a - b*x**2)*a*d - 12*sqrt(c + d*x)*sqrt(a - b*x**2) 
*b*c*x + 2*sqrt(c + d*x)*sqrt(a - b*x**2)*b*d*x**2 - 3*int((sqrt(c + d*x)* 
sqrt(a - b*x**2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x 
**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*a* 
b*c**2*d**2 - 6*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**3 + 3*a*c* 
*2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3 
*b*c*d**2*x**4 - b*d**3*x**5),x)*a*b*c*d**3*x - 3*int((sqrt(c + d*x)*sqrt( 
a - b*x**2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - 
 b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*a*b*d** 
4*x**2 - 24*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**3 + 3*a*c**2*d 
*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c 
*d**2*x**4 - b*d**3*x**5),x)*b**2*c**4 - 48*int((sqrt(c + d*x)*sqrt(a - b* 
x**2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c** 
3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*b**2*c**3*d*x 
 - 24*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 3 
*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2* 
x**4 - b*d**3*x**5),x)*b**2*c**2*d**2*x**2 + 15*int((sqrt(c + d*x)*sqrt(a 
- b*x**2))/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3 
*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*a**2*c**2*d**2 
 + 30*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**3 + 3*a*c**2*d*x + 3*a...