Optimal. Leaf size=316 \[ \frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 d^2 \left (a+b x^4\right )^{3/4}}-\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{12 d^2}+\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d)^2 \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d)^2 \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac {b x \left (a+b x^4\right )^{5/4}}{6 d} \]
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Rubi [A] time = 0.34, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {416, 528, 529, 237, 335, 275, 231, 407, 409, 1218} \[ \frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 d^2 \left (a+b x^4\right )^{3/4}}-\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{12 d^2}+\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d)^2 \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d)^2 \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac {b x \left (a+b x^4\right )^{5/4}}{6 d} \]
Antiderivative was successfully verified.
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Rule 231
Rule 237
Rule 275
Rule 335
Rule 407
Rule 409
Rule 416
Rule 528
Rule 529
Rule 1218
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{9/4}}{c+d x^4} \, dx &=\frac {b x \left (a+b x^4\right )^{5/4}}{6 d}+\frac {\int \frac {\sqrt [4]{a+b x^4} \left (-a (b c-6 a d)-b (6 b c-11 a d) x^4\right )}{c+d x^4} \, dx}{6 d}\\ &=-\frac {b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac {b x \left (a+b x^4\right )^{5/4}}{6 d}+\frac {\int \frac {a \left (6 b^2 c^2-13 a b c d+12 a^2 d^2\right )+b \left (12 b^2 c^2-30 a b c d+23 a^2 d^2\right ) x^4}{\left (a+b x^4\right )^{3/4} \left (c+d x^4\right )} \, dx}{12 d^2}\\ &=-\frac {b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {(a b (6 b c-11 a d)) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{12 d^2}+\frac {(b c-a d)^2 \int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{d^2}\\ &=-\frac {b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\left (a b (6 b c-11 a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{12 d^2 \left (a+b x^4\right )^{3/4}}+\frac {\left ((b c-a d)^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-b x^4} \left (c-(b c-a d) x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{d^2}\\ &=-\frac {b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac {b x \left (a+b x^4\right )^{5/4}}{6 d}+\frac {\left (a b (6 b c-11 a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{12 d^2 \left (a+b x^4\right )^{3/4}}+\frac {\left ((b c-a d)^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b c-a d} x^2}{\sqrt {c}}\right ) \sqrt {1-b x^4}} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{2 c d^2}+\frac {\left ((b c-a d)^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b c-a d} x^2}{\sqrt {c}}\right ) \sqrt {1-b x^4}} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{2 c d^2}\\ &=-\frac {b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac {b x \left (a+b x^4\right )^{5/4}}{6 d}+\frac {(b c-a d)^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac {(b c-a d)^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac {\left (a b (6 b c-11 a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{24 d^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac {b x \left (a+b x^4\right )^{5/4}}{6 d}+\frac {\sqrt {a} b^{3/2} (6 b c-11 a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 d^2 \left (a+b x^4\right )^{3/4}}+\frac {(b c-a d)^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac {(b c-a d)^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}\\ \end {align*}
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Mathematica [C] time = 0.70, size = 294, normalized size = 0.93 \[ \frac {x \left (\frac {b x^4 \left (\frac {b x^4}{a}+1\right )^{3/4} \left (23 a^2 d^2-30 a b c d+12 b^2 c^2\right ) F_1\left (\frac {5}{4};\frac {3}{4},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c}-\frac {25 a^2 c \left (12 a^2 d^2-13 a b c d+6 b^2 c^2\right ) F_1\left (\frac {1}{4};\frac {3}{4},1;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac {5}{4};\frac {3}{4},2;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c F_1\left (\frac {5}{4};\frac {7}{4},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )-5 a c F_1\left (\frac {1}{4};\frac {3}{4},1;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )}+5 b \left (a+b x^4\right ) \left (13 a d-6 b c+2 b d x^4\right )\right )}{60 d^2 \left (a+b x^4\right )^{3/4}} \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}}}{d x^{4} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}}}{d \,x^{4}+c}\, dx \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}}}{d x^{4} + c}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^4+a\right )}^{9/4}}{d\,x^4+c} \,d x \]
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 \[ \int \frac {\left (a + b x^{4}\right )^{\frac {9}{4}}}{c + d x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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