Optimal. Leaf size=371 \[ \frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {\sqrt {b} \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 a^{3/2} c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{a} d^{3/2} (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c (-b c+a d)^{7/2} x}+\frac {\sqrt [4]{a} d^{3/2} (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c (-b c+a d)^{7/2} x} \]
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Rubi [A]
time = 0.39, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {425, 541, 544,
235, 233, 202, 408, 504, 1232} \begin {gather*} \frac {\sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} \left (-5 a^2 d^2-52 a b c d+12 b^2 c^2\right ) E\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 a^{3/2} c \sqrt [4]{a+b x^2} (b c-a d)^3}-\frac {\sqrt [4]{a} d^{3/2} \sqrt {-\frac {b x^2}{a}} (11 b c-2 a d) \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 x (a d-b c)^{7/2}}+\frac {\sqrt [4]{a} d^{3/2} \sqrt {-\frac {b x^2}{a}} (11 b c-2 a d) \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 x (a d-b c)^{7/2}}-\frac {d x}{2 c \left (a+b x^2\right )^{5/4} \left (c+d x^2\right ) (b c-a d)}+\frac {b x (5 a d+4 b c)}{10 a c \left (a+b x^2\right )^{5/4} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 233
Rule 235
Rule 408
Rule 425
Rule 504
Rule 541
Rule 544
Rule 1232
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx &=-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {\int \frac {2 b c-a d-\frac {7}{2} b d x^2}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=\frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-6 b^2 c^2+20 a b c d-5 a^2 d^2\right )-\frac {3}{4} b d (4 b c+5 a d) x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx}{5 a c (b c-a d)^2}\\ &=\frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}+\frac {b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x}{10 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {2 \int \frac {\frac {1}{4} \left (-6 b^3 c^3+26 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right )-\frac {1}{8} b d \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x^2}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{5 a^2 c (b c-a d)^3}\\ &=\frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}+\frac {b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x}{10 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {\left (d^2 (11 b c-2 a d)\right ) \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{4 c (b c-a d)^3}-\frac {\left (b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{20 a^2 c (b c-a d)^3}\\ &=\frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}+\frac {b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x}{10 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {\left (d^2 (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {x^2}{\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)^3 x}-\frac {\left (b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}\\ &=\frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}-\frac {\left (d^{3/2} (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-b c+a d}-\sqrt {d} x^2\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^3 x}+\frac {\left (d^{3/2} (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-b c+a d}+\sqrt {d} x^2\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^3 x}+\frac {\left (b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}\\ &=\frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {\sqrt {b} \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 a^{3/2} c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{a} d^{3/2} (11 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)^{7/2} x}+\frac {\sqrt [4]{a} d^{3/2} (11 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)^{7/2} 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 = 10.76, size = 536, normalized size = 1.44 \begin {gather*} \frac {b d \left (-12 b^2 c^2+52 a b c d+5 a^2 d^2\right ) x^3 \sqrt [4]{1+\frac {b x^2}{a}} F_1\left (\frac {3}{2};\frac {1}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {6 c \left (-6 a c x \left (10 a^4 d^3+15 a^3 b d^2 \left (-2 c+d x^2\right )-6 b^4 c^2 x^2 \left (c+2 d x^2\right )+a^2 b^2 d \left (30 c^2+26 c d x^2+5 d^2 x^4\right )+2 a b^3 c \left (-5 c^2+5 c d x^2+26 d^2 x^4\right )\right ) F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^3 \left (5 a^4 d^3+10 a^3 b d^3 x^2-12 b^4 c^2 x^2 \left (c+d x^2\right )+a^2 b^2 d \left (56 c^2+56 c d x^2+5 d^2 x^4\right )+4 a b^3 c \left (-4 c^2+9 c d x^2+13 d^2 x^4\right )\right ) \left (4 a d F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 a c F_1\left (\frac {1}{2};\frac {1}{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 {1}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}}{60 a^2 c^2 (b c-a d)^3 \sqrt [4]{a+b x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {9}{4}} \left (d \,x^{2}+c \right )^{2}}\, dx\]
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 {1}{\left (a + b x^{2}\right )^{\frac {9}{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 {1}{{\left (b\,x^2+a\right )}^{9/4}\,{\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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