Optimal. Leaf size=292 \[ -\frac {\sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{2 c \left (a+b x^2\right )^{3/4} (b c-a d)}-\frac {d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}+\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (5 b c-2 a d) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2}+\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (5 b c-2 a d) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2} \]
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Rubi [A] time = 0.23, antiderivative size = 292, 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 = {414, 530, 233, 231, 401, 108, 409, 1218} \[ -\frac {d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}-\frac {\sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c \left (a+b x^2\right )^{3/4} (b c-a d)}+\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (5 b c-2 a d) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2}+\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (5 b c-2 a d) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 108
Rule 231
Rule 233
Rule 401
Rule 409
Rule 414
Rule 530
Rule 1218
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx &=-\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {\int \frac {2 b c-a d-\frac {1}{2} b d x^2}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=-\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {b \int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{4 c (b c-a d)}+\frac {(5 b c-2 a d) \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{4 c (b c-a d)}\\ &=-\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {\left ((5 b c-2 a d) \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 )}{8 c (b c-a 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 (b c-a d) \left (a+b x^2\right )^{3/4}}\\ &=-\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \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 (b c-a d) \left (a+b x^2\right )^{3/4}}-\frac {\left ((5 b c-2 a d) \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 )}{2 c (b c-a d) x}\\ &=-\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \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 (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac {\left ((5 b c-2 a d) \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 )}{4 c (b c-a d)^2 x}+\frac {\left ((5 b c-2 a d) \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 )}{4 c (b c-a d)^2 x}\\ &=-\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \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 (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} (5 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 (b c-a d)^2 x}+\frac {\sqrt [4]{a} (5 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 (b c-a d)^2 x}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 336, normalized size = 1.15 \[ \frac {x \left (\frac {b d x^2 \left (\frac {b x^2}{a}+1\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a d-b c}+\frac {c \left (36 a c \left (2 a d-2 b c+b d x^2\right ) F_1\left (\frac {1}{2};\frac {3}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-6 d x^2 \left (a+b x^2\right ) \left (4 a d F_1\left (\frac {3}{2};\frac {3}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c F_1\left (\frac {3}{2};\frac {7}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) (b c-a d) \left (x^2 \left (4 a d F_1\left (\frac {3}{2};\frac {3}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c F_1\left (\frac {3}{2};\frac {7}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )-6 a c F_1\left (\frac {1}{2};\frac {3}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}\right )}{12 c^2 \left (a+b x^2\right )^{3/4}} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (d \,x^{2}+c \right )^{2}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^2+a\right )}^{3/4}\,{\left (d\,x^2+c\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{4}} \left (c + d x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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