Optimal. Leaf size=279 \[ -\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} (3 b c+a d) \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d^2 \left (a+b x^2\right )^{3/4}}-\frac {\sqrt [4]{a} (3 b c+2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^2 x}-\frac {\sqrt [4]{a} (3 b c+2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^2 x} \]
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Rubi [A]
time = 0.18, antiderivative size = 279, 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 = {424, 544, 239,
237, 410, 109, 418, 1232} \begin {gather*} -\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (2 a d+3 b c) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^2 x}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (2 a d+3 b c) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^2 x}+\frac {\sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} (a d+3 b c) F\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d^2 \left (a+b x^2\right )^{3/4}}-\frac {x \sqrt [4]{a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 109
Rule 237
Rule 239
Rule 410
Rule 418
Rule 424
Rule 544
Rule 1232
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/4}}{\left (c+d x^2\right )^2} \, dx &=-\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {\int \frac {a (b c+a d)+\frac {1}{2} b (3 b c+a d) x^2}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{2 c d}\\ &=-\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {(b (3 b c+a d)) \int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{4 c d^2}-\frac {((b c-a d) (3 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 d^2}\\ &=-\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}-\frac {\left ((b c-a d) (3 b c+2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {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 d^2 x}+\frac {\left (b (3 b c+a d) \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^2 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} (3 b c+a d) \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^2 \left (a+b x^2\right )^{3/4}}+\frac {\left ((b c-a d) (3 b c+2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {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 d^2 x}\\ &=-\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} (3 b c+a d) \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^2 \left (a+b x^2\right )^{3/4}}-\frac {\left ((3 b c+2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {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 d^2 x}-\frac {\left ((3 b c+2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {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 d^2 x}\\ &=-\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} (3 b c+a d) \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^2 \left (a+b x^2\right )^{3/4}}-\frac {\sqrt [4]{a} (3 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^2 x}-\frac {\sqrt [4]{a} (3 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^2 x}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.24, size = 341, normalized size = 1.22 \begin {gather*} \frac {x \left (b (3 b c+a d) x^2 \left (1+\frac {b x^2}{a}\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 )+\frac {6 c \left (-6 a c \left (2 a^2 d-b^2 c x^2+a 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 )+(-b c+a 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 ) \left (-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 )+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 )\right )}\right )}{12 c^2 d \left (a+b x^2\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{4}}}{\left (d \,x^{2}+c \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{\frac {5}{4}}}{\left (c + d x^{2}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{5/4}}{{\left (d\,x^2+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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