Optimal. Leaf size=732 \[ \frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) \left (-\sqrt {a} \sqrt {c} (B d-A e)+a B e+A c d\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt {a+c x^4} \left (a e^2+c d^2\right )}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} (B d-A e) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a+c x^4} \left (a e^2+c d^2\right )}+\frac {a^{3/4} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 (B d-A e) \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{c} d \sqrt {a+c x^4} \left (c^2 d^4-a^2 e^4\right )}-\frac {\sqrt {c} x \sqrt {a+c x^4} (B d-A e)}{2 a \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (a e^2+c d^2\right )}+\frac {x \left (a B e+c x^2 (B d-A e)+A c d\right )}{2 a \sqrt {a+c x^4} \left (a e^2+c d^2\right )}-\frac {\sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} (B d-A e) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+c d^2\right )}-\frac {e^{3/2} (B d-A e) \tan ^{-1}\left (\frac {x \sqrt {a e^2+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (a e^2+c d^2\right )^{3/2}} \]
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Rubi [A] time = 0.78, antiderivative size = 732, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1721, 1179, 1198, 220, 1196, 1217, 1707} \[ \frac {a^{3/4} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 (B d-A e) \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{c} d \sqrt {a+c x^4} \left (c^2 d^4-a^2 e^4\right )}+\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) \left (-\sqrt {a} \sqrt {c} (B d-A e)+a B e+A c d\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt {a+c x^4} \left (a e^2+c d^2\right )}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} (B d-A e) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a+c x^4} \left (a e^2+c d^2\right )}-\frac {\sqrt {c} x \sqrt {a+c x^4} (B d-A e)}{2 a \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (a e^2+c d^2\right )}+\frac {x \left (a B e+c x^2 (B d-A e)+A c d\right )}{2 a \sqrt {a+c x^4} \left (a e^2+c d^2\right )}-\frac {e^{3/2} (B d-A e) \tan ^{-1}\left (\frac {x \sqrt {a e^2+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (a e^2+c d^2\right )^{3/2}}-\frac {\sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} (B d-A e) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 220
Rule 1179
Rule 1196
Rule 1198
Rule 1217
Rule 1707
Rule 1721
Rubi steps
\begin {align*} \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+c x^4\right )^{3/2}} \, dx &=\int \left (\frac {A c d+a B e+c (B d-A e) x^2}{\left (c d^2+a e^2\right ) \left (a+c x^4\right )^{3/2}}+\frac {e (-B d+A e)}{\left (c d^2+a e^2\right ) \left (d+e x^2\right ) \sqrt {a+c x^4}}\right ) \, dx\\ &=\frac {\int \frac {A c d+a B e+c (B d-A e) x^2}{\left (a+c x^4\right )^{3/2}} \, dx}{c d^2+a e^2}-\frac {(e (B d-A e)) \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{c d^2+a e^2}\\ &=\frac {x \left (A c d+a B e+c (B d-A e) x^2\right )}{2 a \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {\int \frac {-A c d-a B e+c (B d-A e) x^2}{\sqrt {a+c x^4}} \, dx}{2 a \left (c d^2+a e^2\right )}-\frac {\left (\sqrt {c} e (B d-A e)\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{\left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )}+\frac {\left (\sqrt {a} e^2 (B d-A e)\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{\left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )}\\ &=\frac {x \left (A c d+a B e+c (B d-A e) x^2\right )}{2 a \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {e^{3/2} (B d-A e) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (c d^2+a e^2\right )^{3/2}}-\frac {\sqrt [4]{c} e (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}+\frac {\left (\sqrt {c} (B d-A e)\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{2 \sqrt {a} \left (c d^2+a e^2\right )}+\frac {\left (A c d+a B e-\sqrt {a} \sqrt {c} (B d-A e)\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac {x \left (A c d+a B e+c (B d-A e) x^2\right )}{2 a \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {\sqrt {c} (B d-A e) x \sqrt {a+c x^4}}{2 a \left (c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^{3/2} (B d-A e) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (c d^2+a e^2\right )^{3/2}}+\frac {\sqrt [4]{c} (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}+\frac {\left (A c d+a B e-\sqrt {a} \sqrt {c} (B d-A e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{5/4} \sqrt [4]{c} \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.79, size = 432, normalized size = 0.59 \[ \frac {d \sqrt {\frac {c x^4}{a}+1} \left (\sqrt {a} B-i A \sqrt {c}\right ) \left (\sqrt {c} d-i \sqrt {a} e\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-\sqrt {a} \sqrt {c} d \sqrt {\frac {c x^4}{a}+1} (B d-A e) E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+A c d^2 x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}-2 i a A e^2 \sqrt {\frac {c x^4}{a}+1} \Pi \left (-\frac {i \sqrt {a} e}{\sqrt {c} d};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-A c d e x^3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}+B c d^2 x^3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}+2 i a B d e \sqrt {\frac {c x^4}{a}+1} \Pi \left (-\frac {i \sqrt {a} e}{\sqrt {c} d};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+a B d e x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}{2 a d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \sqrt {a+c x^4} \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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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 \[ \int \frac {B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 564, normalized size = 0.77 \[ \frac {\left (\frac {x}{2 \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}\, a}+\frac {\sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{2 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, a}\right ) B}{e}+\frac {\left (A e -B d \right ) \left (-\frac {i \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {c}\, e \EllipticE \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{2 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {a}}+\frac {i \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {c}\, e \EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{2 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {a}}+\frac {\sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, c d \EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{2 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, a}+\frac {\sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, e^{2} \EllipticPi \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, d}-\frac {2 \left (\frac {e \,x^{3}}{4 \left (a \,e^{2}+c \,d^{2}\right ) a}-\frac {d x}{4 \left (a \,e^{2}+c \,d^{2}\right ) a}\right ) c}{\sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}\right )}{e} \]
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 \[ \int \frac {B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}}\,{d x} \]
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 \[ \int \frac {B\,x^2+A}{{\left (c\,x^4+a\right )}^{3/2}\,\left (e\,x^2+d\right )} \,d x \]
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 \[ \int \frac {A + B x^{2}}{\left (a + c x^{4}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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