Optimal. Leaf size=859 \[ \frac {x \left (A c \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {B e^3 x \sqrt {a+b x^2+c x^4}}{3 c^2}+\frac {\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x \sqrt {a+b x^2+c x^4}}{3 a c^{5/2} \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} c^{11/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (3 A c^3 d^3-5 a^2 B c e^3-3 \sqrt {a} c^{5/2} d^2 (B d+3 A e)+a e (3 c d-2 b e) (3 B c d-4 b B e+3 A c e)+3 a^{3/2} \sqrt {c} e^2 (9 B c d-4 b B e+3 A c e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) c^{11/4} \sqrt {a+b x^2+c x^4}} \]
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Rubi [A]
time = 0.89, antiderivative size = 859, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1692, 1693,
1211, 1117, 1209} \begin {gather*} \frac {B x \sqrt {c x^4+b x^2+a} e^3}{3 c^2}-\frac {\left (a B \left (6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c \left (c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} c^{11/4} \left (b^2-4 a c\right ) \sqrt {c x^4+b x^2+a}}-\frac {\left (3 A c^3 d^3-3 \sqrt {a} c^{5/2} (B d+3 A e) d^2-5 a^2 B c e^3+a e (3 c d-2 b e) (3 B c d-4 b B e+3 A c e)+3 a^{3/2} \sqrt {c} e^2 (9 B c d-4 b B e+3 A c e)\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) c^{11/4} \sqrt {c x^4+b x^2+a}}+\frac {\left (a B \left (6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c \left (c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) x \sqrt {c x^4+b x^2+a}}{3 a c^{5/2} \left (b^2-4 a c\right ) \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {x \left (-\left (\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c \left (3 c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2\right )+A c \left (b^2 c d^3-2 a c \left (c d^2-3 a e^2\right ) d-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c \left (3 c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {c x^4+b x^2+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1117
Rule 1209
Rule 1211
Rule 1692
Rule 1693
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac {x \left (A c \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\int \frac {\frac {a \left (a b^2 B e^3-b c \left (B c d^3+3 A c d^2 e+3 a B d e^2+a A e^3\right )+2 c \left (a B e \left (3 c d^2-a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right )\right )}{c^2}-\frac {\left (a B \left (2 c^3 d^3-2 b^3 e^3-3 c^2 d e (b d+6 a e)+b c e^2 (6 b d+7 a e)\right )+A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2}{c^2}+a B \left (4 a-\frac {b^2}{c}\right ) e^3 x^4}{\sqrt {a+b x^2+c x^4}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac {x \left (A c \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {B e^3 x \sqrt {a+b x^2+c x^4}}{3 c^2}-\frac {\int \frac {\frac {a \left (4 a b^2 B e^3-3 b c \left (B c d^3+3 A c d^2 e+3 a B d e^2+a A e^3\right )+2 c \left (a B e \left (9 c d^2-5 a e^2\right )+3 A c d \left (c d^2+3 a e^2\right )\right )\right )}{c}-\frac {\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2}{c}}{\sqrt {a+b x^2+c x^4}} \, dx}{3 a c \left (b^2-4 a c\right )}\\ &=\frac {x \left (A c \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {B e^3 x \sqrt {a+b x^2+c x^4}}{3 c^2}-\frac {\left (3 A c^3 d^3-5 a^2 B c e^3-3 \sqrt {a} c^{5/2} d^2 (B d+3 A e)+a e (3 c d-2 b e) (3 B c d-4 b B e+3 A c e)+3 a^{3/2} \sqrt {c} e^2 (9 B c d-4 b B e+3 A c e)\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{3 \sqrt {a} \left (b-2 \sqrt {a} \sqrt {c}\right ) c^{5/2}}-\frac {\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{3 \sqrt {a} c^{5/2} \left (b^2-4 a c\right )}\\ &=\frac {x \left (A c \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {B e^3 x \sqrt {a+b x^2+c x^4}}{3 c^2}+\frac {\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x \sqrt {a+b x^2+c x^4}}{3 a c^{5/2} \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} c^{11/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (3 A c^3 d^3-5 a^2 B c e^3-3 \sqrt {a} c^{5/2} d^2 (B d+3 A e)+a e (3 c d-2 b e) (3 B c d-4 b B e+3 A c e)+3 a^{3/2} \sqrt {c} e^2 (9 B c d-4 b B e+3 A c e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) c^{11/4} \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 14.74, size = 1058, normalized size = 1.23 \begin {gather*} \frac {-4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (3 A c \left (b^2 \left (c d^3-a e^3 x^2\right )+b \left (-a^2 e^3+c^2 d^3 x^2-3 a c d e \left (d-e x^2\right )\right )+2 a c \left (a e^2 \left (3 d+e x^2\right )-c d^2 \left (d+3 e x^2\right )\right )\right )+a B \left (4 b^3 e^3 x^2+b^2 e^2 \left (4 a e-9 c d x^2+c e x^4\right )-b c \left (3 c d^2 \left (d-3 e x^2\right )+a e^2 \left (9 d+13 e x^2\right )\right )-2 c \left (5 a^2 e^3+3 c^2 d^3 x^2+a c e \left (-9 d^2-9 d e x^2+2 e^2 x^4\right )\right )\right )\right )+i \left (-b+\sqrt {b^2-4 a c}\right ) \left (a B \left (-6 c^3 d^3+8 b^3 e^3+9 c^2 d e (b d+6 a e)-b c e^2 (18 b d+29 a e)\right )+3 A c \left (-2 a b^2 e^3+6 a c e \left (-c d^2+a e^2\right )+b c d \left (c d^2+3 a e^2\right )\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (a B \left (8 b^3 \left (-b+\sqrt {b^2-4 a c}\right ) e^3-6 c^3 d^2 \left (\sqrt {b^2-4 a c} d-6 a e\right )+b c e^2 \left (18 b^2 d-18 b \sqrt {b^2-4 a c} d+37 a b e-29 a \sqrt {b^2-4 a c} e\right )+c^2 e \left (-9 b^2 d^2+2 a e \left (27 \sqrt {b^2-4 a c} d-10 a e\right )+9 b d \left (\sqrt {b^2-4 a c} d-8 a e\right )\right )\right )+3 A c \left (2 a b^3 e^3-b^2 \left (c^2 d^3+3 a c d e^2+2 a \sqrt {b^2-4 a c} e^3\right )+b c \left (c \sqrt {b^2-4 a c} d^3+a e^2 \left (3 \sqrt {b^2-4 a c} d-8 a e\right )\right )+2 a c \left (2 c^2 d^3+3 a \sqrt {b^2-4 a c} e^3-3 c d e \left (\sqrt {b^2-4 a c} d-2 a e\right )\right )\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{12 a c^3 \left (-b^2+4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2444\) vs.
\(2(841)=1682\).
time = 0.16, size = 2445, normalized size = 2.85
method | result | size |
elliptic | \(-\frac {2 c \left (\frac {\left (2 A \,a^{2} c^{2} e^{3}-A a \,b^{2} c \,e^{3}+3 A a b \,c^{2} d \,e^{2}-6 A a \,c^{3} d^{2} e +A b \,c^{3} d^{3}-3 B \,a^{2} b c \,e^{3}+6 B \,a^{2} c^{2} d \,e^{2}+B a \,b^{3} e^{3}-3 B a \,b^{2} c d \,e^{2}+3 B a b \,c^{2} d^{2} e -2 B a \,c^{3} d^{3}\right ) x^{3}}{2 c^{3} a \left (4 a c -b^{2}\right )}-\frac {\left (A \,a^{2} b c \,e^{3}-6 A \,a^{2} c^{2} d \,e^{2}+3 A a b \,c^{2} d^{2} e +2 A a \,c^{3} d^{3}-A \,b^{2} c^{2} d^{3}+2 B \,a^{3} c \,e^{3}-B \,a^{2} b^{2} e^{3}+3 B \,a^{2} b c d \,e^{2}-6 B \,a^{2} c^{2} d^{2} e +B a b \,c^{2} d^{3}\right ) x}{2 c^{3} \left (4 a c -b^{2}\right ) a}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {B \,e^{3} x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c^{2}}+\frac {\left (-\frac {e \left (A b c \,e^{2}-3 c^{2} d e A +B a c \,e^{2}-B \,b^{2} e^{2}+3 B b c d e -3 B \,c^{2} d^{2}\right )}{c^{3}}+\frac {A a b c \,e^{3}-3 A a \,c^{2} d \,e^{2}+A \,c^{3} d^{3}+B \,a^{2} c \,e^{3}-B a \,b^{2} e^{3}+3 B a b c d \,e^{2}-3 B a \,c^{2} d^{2} e}{c^{3} a}-\frac {A \,a^{2} b c \,e^{3}-6 A \,a^{2} c^{2} d \,e^{2}+3 A a b \,c^{2} d^{2} e +2 A a \,c^{3} d^{3}-A \,b^{2} c^{2} d^{3}+2 B \,a^{3} c \,e^{3}-B \,a^{2} b^{2} e^{3}+3 B \,a^{2} b c d \,e^{2}-6 B \,a^{2} c^{2} d^{2} e +B a b \,c^{2} d^{3}}{a \,c^{2} \left (4 a c -b^{2}\right )}-\frac {a B \,e^{3}}{3 c^{2}}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {e^{2} \left (A c e -B b e +3 B c d \right )}{c^{2}}+\frac {2 A \,a^{2} c^{2} e^{3}-A a \,b^{2} c \,e^{3}+3 A a b \,c^{2} d \,e^{2}-6 A a \,c^{3} d^{2} e +A b \,c^{3} d^{3}-3 B \,a^{2} b c \,e^{3}+6 B \,a^{2} c^{2} d \,e^{2}+B a \,b^{3} e^{3}-3 B a \,b^{2} c d \,e^{2}+3 B a b \,c^{2} d^{2} e -2 B a \,c^{3} d^{3}}{a \,c^{2} \left (4 a c -b^{2}\right )}-\frac {2 e^{3} B b}{3 c^{2}}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(1141\) |
default | \(\text {Expression too large to display}\) | \(2445\) |
risch | \(\text {Expression too large to display}\) | \(2482\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 ) \left (d + e x^{2}\right )^{3}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{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 )\,{\left (e\,x^2+d\right )}^3}{{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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