Optimal. Leaf size=274 \[ -\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 d \left (a+b x^4\right )^{3/4}}-\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d) \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}-\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d) \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}+\frac {b x \sqrt [4]{a+b x^4}}{2 d} \]
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Rubi [A] time = 0.16, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {408, 195, 237, 335, 275, 231, 407, 409, 1218} \[ -\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 d \left (a+b x^4\right )^{3/4}}-\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d) \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}-\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d) \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}+\frac {b x \sqrt [4]{a+b x^4}}{2 d} \]
Antiderivative was successfully verified.
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Rule 195
Rule 231
Rule 237
Rule 275
Rule 335
Rule 407
Rule 408
Rule 409
Rule 1218
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{5/4}}{c+d x^4} \, dx &=\frac {b \int \sqrt [4]{a+b x^4} \, dx}{d}-\frac {(b c-a d) \int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{d}\\ &=\frac {b x \sqrt [4]{a+b x^4}}{2 d}+\frac {(a b) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{2 d}-\frac {\left ((b c-a d) \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}\\ &=\frac {b x \sqrt [4]{a+b x^4}}{2 d}+\frac {\left (a b \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}{2 d \left (a+b x^4\right )^{3/4}}-\frac {\left ((b c-a d) \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}-\frac {\left ((b c-a d) \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}\\ &=\frac {b x \sqrt [4]{a+b x^4}}{2 d}-\frac {(b c-a d) \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}-\frac {(b c-a d) \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}-\frac {\left (a b \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 )}{2 d \left (a+b x^4\right )^{3/4}}\\ &=\frac {b x \sqrt [4]{a+b x^4}}{2 d}-\frac {(b c-a d) \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}-\frac {(b c-a d) \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}-\frac {\left (a b \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 )}{4 d \left (a+b x^4\right )^{3/4}}\\ &=\frac {b x \sqrt [4]{a+b x^4}}{2 d}-\frac {\sqrt {a} b^{3/2} \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 )}{2 d \left (a+b x^4\right )^{3/4}}-\frac {(b c-a d) \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}-\frac {(b c-a d) \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}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 346, normalized size = 1.26 \[ \frac {x \left (\frac {5 \left (b x^4 \left (a+b x^4\right ) \left (c+d x^4\right ) \left (4 a d F_1\left (\frac {5}{4};\frac {3}{4},2;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c F_1\left (\frac {5}{4};\frac {7}{4},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )-5 a c \left (2 a^2 d+a b d x^4+b^2 x^4 \left (c+d x^4\right )\right ) F_1\left (\frac {1}{4};\frac {3}{4},1;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )}{\left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac {5}{4};\frac {3}{4},2;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c F_1\left (\frac {5}{4};\frac {7}{4},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )-5 a c F_1\left (\frac {1}{4};\frac {3}{4},1;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )}+\frac {b x^4 \left (\frac {b x^4}{a}+1\right )^{3/4} (3 a d-2 b c) F_1\left (\frac {5}{4};\frac {3}{4},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c}\right )}{10 d \left (a+b x^4\right )^{3/4}} \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{d x^{4} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.57, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{d \,x^{4}+c}\, dx \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{d x^{4} + c}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^4+a\right )}^{5/4}}{d\,x^4+c} \,d x \]
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 \[ \int \frac {\left (a + b x^{4}\right )^{\frac {5}{4}}}{c + d x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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