Optimal. Leaf size=278 \[ \frac {\sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{2 c d \left (a+b x^2\right )^{3/4}}+\frac {x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right )}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (b c-2 a d) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d x (b c-a d)}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (b c-2 a d) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d x (b c-a d)} \]
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Rubi [A] time = 0.21, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {412, 530, 233, 231, 401, 108, 409, 1218} \[ \frac {x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d \left (a+b x^2\right )^{3/4}}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (b c-2 a d) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d x (b c-a d)}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (b c-2 a d) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d x (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 108
Rule 231
Rule 233
Rule 401
Rule 409
Rule 412
Rule 530
Rule 1218
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a+b x^2}}{\left (c+d x^2\right )^2} \, dx &=\frac {x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right )}-\frac {\int \frac {-a-\frac {b x^2}{2}}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{2 c}\\ &=\frac {x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right )}+\frac {b \int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{4 c d}-\frac {\left (\frac {b c}{2}-a d\right ) \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{2 c d}\\ &=\frac {x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right )}-\frac {\left (\left (\frac {b c}{2}-a d\right ) \sqrt {-\frac {b x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {b x}{a}} (a+b x)^{3/4} (c+d x)} \, dx,x,x^2\right )}{4 c d x}+\frac {\left (b \left (1+\frac {b x^2}{a}\right )^{3/4}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \, dx}{4 c d \left (a+b x^2\right )^{3/4}}\\ &=\frac {x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d \left (a+b x^2\right )^{3/4}}+\frac {\left (\left (\frac {b c}{2}-a d\right ) \sqrt {-\frac {b x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{a}} \left (-b c+a d-d x^4\right )} \, dx,x,\sqrt [4]{a+b x^2}\right )}{c d x}\\ &=\frac {x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d \left (a+b x^2\right )^{3/4}}-\frac {\left (\left (\frac {b c}{2}-a d\right ) \sqrt {-\frac {b x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-b c+a d}}\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{2 c d (b c-a d) x}-\frac {\left (\left (\frac {b c}{2}-a d\right ) \sqrt {-\frac {b x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {-b c+a d}}\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{2 c d (b c-a d) x}\\ &=\frac {x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d \left (a+b x^2\right )^{3/4}}-\frac {\sqrt [4]{a} (b c-2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d (b c-a d) x}-\frac {\sqrt [4]{a} (b c-2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d (b c-a d) x}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 232, normalized size = 0.83 \[ \frac {x \left (\frac {6 \left (\frac {a+b x^2}{c}-\frac {6 a^2 F_1\left (\frac {1}{2};\frac {3}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac {3}{2};\frac {3}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c F_1\left (\frac {3}{2};\frac {7}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )-6 a c F_1\left (\frac {1}{2};\frac {3}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}\right )}{c+d x^2}+\frac {b x^2 \left (\frac {b x^2}{a}+1\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c^2}\right )}{12 \left (a+b x^2\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+a \right )^{\frac {1}{4}}}{\left (d \,x^{2}+c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^2+a\right )}^{1/4}}{{\left (d\,x^2+c\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [4]{a + b x^{2}}}{\left (c + d x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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