\(\int \frac {x^2}{(c+d x)^2 \sqrt [4]{a+b x^2}} \, dx\) [1652]

Optimal result

Integrand size = 22, antiderivative size = 510 \[ \int \frac {x^2}{(c+d x)^2 \sqrt [4]{a+b x^2}} \, dx=\frac {\left (2 a+\frac {3 b c^2}{d^2}\right ) x}{\left (b c^2+a d^2\right ) \sqrt [4]{a+b x^2}}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d \left (b c^2+a d^2\right ) (c+d x)}-\frac {c \left (3 b c^2+4 a d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{b c^2+a d^2}}\right )}{2 d^{5/2} \left (b c^2+a d^2\right )^{5/4}}+\frac {c \left (3 b c^2+4 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{b c^2+a d^2}}\right )}{2 d^{5/2} \left (b c^2+a d^2\right )^{5/4}}-\frac {\sqrt {a} \left (3 b c^2+2 a d^2\right ) \sqrt [4]{\frac {a+b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} d^2 \left (b c^2+a d^2\right ) \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} c^2 \left (3 b c^2+4 a d^2\right ) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b c^2+a d^2}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{2 d^3 \left (b c^2+a d^2\right )^{3/2} x}-\frac {\sqrt [4]{a} c^2 \left (3 b c^2+4 a d^2\right ) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} d}{\sqrt {b c^2+a d^2}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{2 d^3 \left (b c^2+a d^2\right )^{3/2} x} \] Output:

(2*a+3*b*c^2/d^2)*x/(a*d^2+b*c^2)/(b*x^2+a)^(1/4)-c^2*(b*x^2+a)^(3/4)/d/(a 
*d^2+b*c^2)/(d*x+c)-1/2*c*(4*a*d^2+3*b*c^2)*arctan(d^(1/2)*(b*x^2+a)^(1/4) 
/(a*d^2+b*c^2)^(1/4))/d^(5/2)/(a*d^2+b*c^2)^(5/4)+1/2*c*(4*a*d^2+3*b*c^2)* 
arctanh(d^(1/2)*(b*x^2+a)^(1/4)/(a*d^2+b*c^2)^(1/4))/d^(5/2)/(a*d^2+b*c^2) 
^(5/4)-a^(1/2)*(2*a*d^2+3*b*c^2)*((b*x^2+a)/a)^(1/4)*EllipticE(sin(1/2*arc 
tan(b^(1/2)*x/a^(1/2))),2^(1/2))/b^(1/2)/d^2/(a*d^2+b*c^2)/(b*x^2+a)^(1/4) 
+1/2*a^(1/4)*c^2*(4*a*d^2+3*b*c^2)*(-b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^( 
1/4)/a^(1/4),-a^(1/2)/(a*d^2+b*c^2)^(1/2)*d,I)/d^3/(a*d^2+b*c^2)^(3/2)/x-1 
/2*a^(1/4)*c^2*(4*a*d^2+3*b*c^2)*(-b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^(1/ 
4)/a^(1/4),a^(1/2)/(a*d^2+b*c^2)^(1/2)*d,I)/d^3/(a*d^2+b*c^2)^(3/2)/x
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {x^2}{(c+d x)^2 \sqrt [4]{a+b x^2}} \, dx=\int \frac {x^2}{(c+d x)^2 \sqrt [4]{a+b x^2}} \, dx \] Input:

Integrate[x^2/((c + d*x)^2*(a + b*x^2)^(1/4)),x]
 

Output:

Integrate[x^2/((c + d*x)^2*(a + b*x^2)^(1/4)), x]
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 440, normalized size of antiderivative = 0.86, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {603, 27, 719, 227, 225, 212, 504, 310, 353, 73, 27, 827, 218, 221, 993, 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.

\(\displaystyle \int \frac {x^2}{\sqrt [4]{a+b x^2} (c+d x)^2} \, dx\)

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 227

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

\(\Big \downarrow \) 225

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

\(\Big \downarrow \) 212

\(\displaystyle -\frac {c \left (4 a+\frac {3 b c^2}{d^2}\right ) \int \frac {1}{(c+d x) \sqrt [4]{b x^2+a}}dx-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 504

\(\displaystyle -\frac {c \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (c \int \frac {1}{\sqrt [4]{b x^2+a} \left (c^2-d^2 x^2\right )}dx-d \int \frac {x}{\sqrt [4]{b x^2+a} \left (c^2-d^2 x^2\right )}dx\right )-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 310

\(\displaystyle -\frac {c \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 c \sqrt {-\frac {b x^2}{a}} \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c^2+a d^2-d^2 \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{x}-d \int \frac {x}{\sqrt [4]{b x^2+a} \left (c^2-d^2 x^2\right )}dx\right )-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 353

\(\displaystyle -\frac {c \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 c \sqrt {-\frac {b x^2}{a}} \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c^2+a d^2-d^2 \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{x}-\frac {1}{2} d \int \frac {1}{\sqrt [4]{b x^2+a} \left (c^2-d^2 x^2\right )}dx^2\right )-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {c \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 c \sqrt {-\frac {b x^2}{a}} \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c^2+a d^2-d^2 \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{x}-\frac {2 d \int \frac {b x^4}{b \left (c^2+\frac {a d^2}{b}\right )-d^2 x^8}d\sqrt [4]{b x^2+a}}{b}\right )-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 c \sqrt {-\frac {b x^2}{a}} \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c^2+a d^2-d^2 \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{x}-2 d \int \frac {x^4}{-d^2 x^8+b c^2+a d^2}d\sqrt [4]{b x^2+a}\right )-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 827

\(\displaystyle -\frac {c \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 c \sqrt {-\frac {b x^2}{a}} \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c^2+a d^2-d^2 \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{x}-2 d \left (\frac {\int \frac {1}{\sqrt {b c^2+a d^2}-d x^4}d\sqrt [4]{b x^2+a}}{2 d}-\frac {\int \frac {1}{d x^4+\sqrt {b c^2+a d^2}}d\sqrt [4]{b x^2+a}}{2 d}\right )\right )-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {c \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 c \sqrt {-\frac {b x^2}{a}} \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c^2+a d^2-d^2 \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{x}-2 d \left (\frac {\int \frac {1}{\sqrt {b c^2+a d^2}-d x^4}d\sqrt [4]{b x^2+a}}{2 d}-\frac {\arctan \left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{a d^2+b c^2}}\right )}{2 d^{3/2} \sqrt [4]{a d^2+b c^2}}\right )\right )-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {c \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 c \sqrt {-\frac {b x^2}{a}} \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c^2+a d^2-d^2 \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{x}-2 d \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{a d^2+b c^2}}\right )}{2 d^{3/2} \sqrt [4]{a d^2+b c^2}}-\frac {\arctan \left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{a d^2+b c^2}}\right )}{2 d^{3/2} \sqrt [4]{a d^2+b c^2}}\right )\right )-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 993

\(\displaystyle -\frac {c \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 c \sqrt {-\frac {b x^2}{a}} \left (\frac {\int \frac {1}{\left (\sqrt {b c^2+a d^2}-d \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 d}-\frac {\int \frac {1}{\left (\sqrt {b x^2+a} d+\sqrt {b c^2+a d^2}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 d}\right )}{x}-2 d \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{a d^2+b c^2}}\right )}{2 d^{3/2} \sqrt [4]{a d^2+b c^2}}-\frac {\arctan \left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{a d^2+b c^2}}\right )}{2 d^{3/2} \sqrt [4]{a d^2+b c^2}}\right )\right )-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 1542

\(\displaystyle -\frac {c \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 c \sqrt {-\frac {b x^2}{a}} \left (\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} d}{\sqrt {b c^2+a d^2}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 d \sqrt {a d^2+b c^2}}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b c^2+a d^2}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 d \sqrt {a d^2+b c^2}}\right )}{x}-2 d \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{a d^2+b c^2}}\right )}{2 d^{3/2} \sqrt [4]{a d^2+b c^2}}-\frac {\arctan \left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{a d^2+b c^2}}\right )}{2 d^{3/2} \sqrt [4]{a d^2+b c^2}}\right )\right )-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{3/4}}{d (c+d x) \left (a d^2+b c^2\right )}\)

Input:

Int[x^2/((c + d*x)^2*(a + b*x^2)^(1/4)),x]
 

Output:

-((c^2*(a + b*x^2)^(3/4))/(d*(b*c^2 + a*d^2)*(c + d*x))) - (-(((2*a + (3*b 
*c^2)/d^2)*(1 + (b*x^2)/a)^(1/4)*((2*x)/(1 + (b*x^2)/a)^(1/4) - (2*Sqrt[a] 
*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/Sqrt[b]))/(a + b*x^2)^(1/4)) 
 + c*(4*a + (3*b*c^2)/d^2)*(-2*d*(-1/2*ArcTan[(Sqrt[d]*(a + b*x^2)^(1/4))/ 
(b*c^2 + a*d^2)^(1/4)]/(d^(3/2)*(b*c^2 + a*d^2)^(1/4)) + ArcTanh[(Sqrt[d]* 
(a + b*x^2)^(1/4))/(b*c^2 + a*d^2)^(1/4)]/(2*d^(3/2)*(b*c^2 + a*d^2)^(1/4) 
)) + (2*c*Sqrt[-((b*x^2)/a)]*(-1/2*(a^(1/4)*EllipticPi[-((Sqrt[a]*d)/Sqrt[ 
b*c^2 + a*d^2]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(d*Sqrt[b*c^2 + a 
*d^2]) + (a^(1/4)*EllipticPi[(Sqrt[a]*d)/Sqrt[b*c^2 + a*d^2], ArcSin[(a + 
b*x^2)^(1/4)/a^(1/4)], -1])/(2*d*Sqrt[b*c^2 + a*d^2])))/x))/(2*(b*c^2 + a* 
d^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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 212

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) 
)*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 225

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) 
, x] - Simp[a   Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ 
a, 0] && PosQ[b/a]
 

rule 227

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

rule 310

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

rule 353

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

rule 504

Int[((a_) + (b_.)*(x_)^2)^(p_)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[c   I 
nt[(a + b*x^2)^p/(c^2 - d^2*x^2), x], x] - Simp[d   Int[x*((a + b*x^2)^p/(c 
^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, p}, x]
 

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 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 993

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

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]
 

Maple [F]

\[\int \frac {x^{2}}{\left (d x +c \right )^{2} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}d x\]

Input:

int(x^2/(d*x+c)^2/(b*x^2+a)^(1/4),x)
 

Output:

int(x^2/(d*x+c)^2/(b*x^2+a)^(1/4),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2}{(c+d x)^2 \sqrt [4]{a+b x^2}} \, dx=\text {Timed out} \] Input:

integrate(x^2/(d*x+c)^2/(b*x^2+a)^(1/4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^2}{(c+d x)^2 \sqrt [4]{a+b x^2}} \, dx=\int \frac {x^{2}}{\sqrt [4]{a + b x^{2}} \left (c + d x\right )^{2}}\, dx \] Input:

integrate(x**2/(d*x+c)**2/(b*x**2+a)**(1/4),x)
 

Output:

Integral(x**2/((a + b*x**2)**(1/4)*(c + d*x)**2), x)
 

Maxima [F]

\[ \int \frac {x^2}{(c+d x)^2 \sqrt [4]{a+b x^2}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{2}} \,d x } \] Input:

integrate(x^2/(d*x+c)^2/(b*x^2+a)^(1/4),x, algorithm="maxima")
 

Output:

integrate(x^2/((b*x^2 + a)^(1/4)*(d*x + c)^2), x)
 

Giac [F]

\[ \int \frac {x^2}{(c+d x)^2 \sqrt [4]{a+b x^2}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{2}} \,d x } \] Input:

integrate(x^2/(d*x+c)^2/(b*x^2+a)^(1/4),x, algorithm="giac")
 

Output:

integrate(x^2/((b*x^2 + a)^(1/4)*(d*x + c)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{(c+d x)^2 \sqrt [4]{a+b x^2}} \, dx=\int \frac {x^2}{{\left (b\,x^2+a\right )}^{1/4}\,{\left (c+d\,x\right )}^2} \,d x \] Input:

int(x^2/((a + b*x^2)^(1/4)*(c + d*x)^2),x)
 

Output:

int(x^2/((a + b*x^2)^(1/4)*(c + d*x)^2), x)
 

Reduce [F]

\[ \int \frac {x^2}{(c+d x)^2 \sqrt [4]{a+b x^2}} \, dx=\int \frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} c^{2}+2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} c d x +\left (b \,x^{2}+a \right )^{\frac {1}{4}} d^{2} x^{2}}d x \] Input:

int(x^2/(d*x+c)^2/(b*x^2+a)^(1/4),x)
                                                                                    
                                                                                    
 

Output:

int(x**2/((a + b*x**2)**(1/4)*c**2 + 2*(a + b*x**2)**(1/4)*c*d*x + (a + b* 
x**2)**(1/4)*d**2*x**2),x)