Optimal. Leaf size=200 \[ \frac {\sqrt {p^2 x^4+q^2} \left (a c p x^2+a c q-2 a d x+2 b c x\right )}{2 c^2 x^2}+\frac {\log \left (\sqrt {p^2 x^4+q^2}+p x^2+q\right ) \left (-a c^2 p q+a d^2-b c d\right )}{c^3}-\frac {2 (a d-b c) \sqrt {2 c^2 p q-d^2} \tan ^{-1}\left (\frac {x \sqrt {2 c^2 p q-d^2}}{c \sqrt {p^2 x^4+q^2}+c p x^2+c q+d x}\right )}{c^3}+\frac {\log (x) \left (a c^2 p q-a d^2+b c d\right )}{c^3} \]
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Rubi [C] time = 9.28, antiderivative size = 1405, normalized size of antiderivative = 7.02, number of steps used = 46, number of rules used = 21, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6728, 275, 277, 217, 206, 305, 220, 1196, 266, 50, 63, 208, 1729, 1209, 1198, 1217, 1707, 1248, 735, 844, 725} \begin {gather*} \frac {\sqrt {2 c^2 p q-d^2} \tan ^{-1}\left (\frac {\sqrt {2 c^2 p q-d^2} x}{c \sqrt {p^2 x^4+q^2}}\right ) (b c-a d)}{c^3}-\frac {\left (d-\sqrt {d^2-4 c^2 p q}\right ) \tanh ^{-1}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right ) (b c-a d)}{4 c^3}-\frac {\left (d+\sqrt {d^2-4 c^2 p q}\right ) \tanh ^{-1}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right ) (b c-a d)}{4 c^3}+\frac {\sqrt {d^2-2 c^2 p q} \left (d+\sqrt {d^2-4 c^2 p q}\right ) \sqrt {-2 p q c^2+d^2-d \sqrt {d^2-4 c^2 p q}} \tanh ^{-1}\left (\frac {p \left (4 c^2 q^2+\left (d-\sqrt {d^2-4 c^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {d^2-2 c^2 p q} \sqrt {-2 p q c^2+d^2-d \sqrt {d^2-4 c^2 p q}} \sqrt {p^2 x^4+q^2}}\right ) (b c-a d)}{4 \sqrt {2} c^5 p q}+\frac {\sqrt {d^2-2 c^2 p q} \left (d-\sqrt {d^2-4 c^2 p q}\right ) \sqrt {-2 p q c^2+d^2+d \sqrt {d^2-4 c^2 p q}} \tanh ^{-1}\left (\frac {p \left (4 c^2 q^2+\left (d+\sqrt {d^2-4 c^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {d^2-2 c^2 p q} \sqrt {-2 p q c^2+d^2+d \sqrt {d^2-4 c^2 p q}} \sqrt {p^2 x^4+q^2}}\right ) (b c-a d)}{4 \sqrt {2} c^5 p q}-\frac {\left (d^2-2 c^2 p q\right ) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right ) (b c-a d)}{4 c^4 d \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2}}+\frac {d \left (d-\sqrt {d^2-4 c^2 p q}\right ) \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right ) (b c-a d)}{4 c^4 \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2}}-\frac {\left (d^2-2 c^2 p q\right ) \left (d+\sqrt {d^2-4 c^2 p q}\right ) \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right ) (b c-a d)}{4 c^4 d \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2}}+\frac {d \left (d+\sqrt {d^2-4 c^2 p q}\right ) \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right ) (b c-a d)}{4 c^4 \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2}}-\frac {\sqrt {p} \sqrt {q} \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right ) (b c-a d)}{c^2 \sqrt {p^2 x^4+q^2}}-\frac {\left (d-\sqrt {d^2-4 c^2 p q}\right ) \sqrt {p^2 x^4+q^2} (b c-a d)}{4 c^3 q}-\frac {\left (d+\sqrt {d^2-4 c^2 p q}\right ) \sqrt {p^2 x^4+q^2} (b c-a d)}{4 c^3 q}+\frac {\sqrt {p^2 x^4+q^2} (b c-a d)}{c^2 x}-\frac {a p q \tanh ^{-1}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )}{2 c}-\frac {\left (a p q c^2+b d c-a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {p^2 x^4+q^2}}{q}\right )}{2 c^3}+\frac {\left (a p q c^2+b d c-a d^2\right ) \sqrt {p^2 x^4+q^2}}{2 c^3 q}+\frac {a q \sqrt {p^2 x^4+q^2}}{2 c x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 206
Rule 208
Rule 217
Rule 220
Rule 266
Rule 275
Rule 277
Rule 305
Rule 725
Rule 735
Rule 844
Rule 1196
Rule 1198
Rule 1209
Rule 1217
Rule 1248
Rule 1707
Rule 1729
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-q+p x^2\right ) \left (a q+b x+a p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^3 \left (c q+d x+c p x^2\right )} \, dx &=\int \left (-\frac {a q \sqrt {q^2+p^2 x^4}}{c x^3}+\frac {(-b c+a d) \sqrt {q^2+p^2 x^4}}{c^2 x^2}+\frac {\left (b c d-a d^2+a c^2 p q\right ) \sqrt {q^2+p^2 x^4}}{c^3 q x}+\frac {(b c-a d) \left (-d^2+2 c^2 p q-c d p x\right ) \sqrt {q^2+p^2 x^4}}{c^3 q \left (c q+d x+c p x^2\right )}\right ) \, dx\\ &=-\frac {(b c-a d) \int \frac {\sqrt {q^2+p^2 x^4}}{x^2} \, dx}{c^2}+\frac {(b c-a d) \int \frac {\left (-d^2+2 c^2 p q-c d p x\right ) \sqrt {q^2+p^2 x^4}}{c q+d x+c p x^2} \, dx}{c^3 q}-\frac {(a q) \int \frac {\sqrt {q^2+p^2 x^4}}{x^3} \, dx}{c}+\frac {\left (b c d-a d^2+a c^2 p q\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{x} \, dx}{c^3 q}\\ &=\frac {(b c-a d) \sqrt {q^2+p^2 x^4}}{c^2 x}-\frac {\left (2 (b c-a d) p^2\right ) \int \frac {x^2}{\sqrt {q^2+p^2 x^4}} \, dx}{c^2}+\frac {(b c-a d) \int \left (\frac {\left (-c d p-c p \sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{d-\sqrt {d^2-4 c^2 p q}+2 c p x}+\frac {\left (-c d p+c p \sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{d+\sqrt {d^2-4 c^2 p q}+2 c p x}\right ) \, dx}{c^3 q}-\frac {(a q) \operatorname {Subst}\left (\int \frac {\sqrt {q^2+p^2 x^2}}{x^2} \, dx,x,x^2\right )}{2 c}+\frac {\left (b c d-a d^2+a c^2 p q\right ) \operatorname {Subst}\left (\int \frac {\sqrt {q^2+p^2 x}}{x} \, dx,x,x^4\right )}{4 c^3 q}\\ &=\frac {\left (b c d-a d^2+a c^2 p q\right ) \sqrt {q^2+p^2 x^4}}{2 c^3 q}+\frac {a q \sqrt {q^2+p^2 x^4}}{2 c x^2}+\frac {(b c-a d) \sqrt {q^2+p^2 x^4}}{c^2 x}-\frac {(2 (b c-a d) p q) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{c^2}+\frac {(2 (b c-a d) p q) \int \frac {1-\frac {p x^2}{q}}{\sqrt {q^2+p^2 x^4}} \, dx}{c^2}-\frac {\left (a p^2 q\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{2 c}+\frac {\left (q \left (b c d-a d^2+a c^2 p q\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {q^2+p^2 x}} \, dx,x,x^4\right )}{4 c^3}-\frac {\left ((b c-a d) p \left (d-\sqrt {d^2-4 c^2 p q}\right )\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{d+\sqrt {d^2-4 c^2 p q}+2 c p x} \, dx}{c^2 q}-\frac {\left ((b c-a d) p \left (d+\sqrt {d^2-4 c^2 p q}\right )\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{d-\sqrt {d^2-4 c^2 p q}+2 c p x} \, dx}{c^2 q}\\ &=\frac {\left (b c d-a d^2+a c^2 p q\right ) \sqrt {q^2+p^2 x^4}}{2 c^3 q}+\frac {a q \sqrt {q^2+p^2 x^4}}{2 c x^2}+\frac {(b c-a d) \sqrt {q^2+p^2 x^4}}{c^2 x}-\frac {2 (b c-a d) p x \sqrt {q^2+p^2 x^4}}{c^2 \left (q+p x^2\right )}+\frac {2 (b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}-\frac {(b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}-\left (4 (b c-a d) p^2\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{\left (d-\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x^2} \, dx-\left (4 (b c-a d) p^2\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{\left (d+\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x^2} \, dx-\frac {\left (a p^2 q\right ) \operatorname {Subst}\left (\int \frac {1}{1-p^2 x^2} \, dx,x,\frac {x^2}{\sqrt {q^2+p^2 x^4}}\right )}{2 c}+\frac {\left (q \left (b c d-a d^2+a c^2 p q\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {q^2}{p^2}+\frac {x^2}{p^2}} \, dx,x,\sqrt {q^2+p^2 x^4}\right )}{2 c^3 p^2}+\frac {\left (2 (b c-a d) p^2 \left (d-\sqrt {d^2-4 c^2 p q}\right )\right ) \int \frac {x \sqrt {q^2+p^2 x^4}}{\left (d+\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x^2} \, dx}{c q}+\frac {\left (2 (b c-a d) p^2 \left (d+\sqrt {d^2-4 c^2 p q}\right )\right ) \int \frac {x \sqrt {q^2+p^2 x^4}}{\left (d-\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x^2} \, dx}{c q}\\ &=\frac {\left (b c d-a d^2+a c^2 p q\right ) \sqrt {q^2+p^2 x^4}}{2 c^3 q}+\frac {a q \sqrt {q^2+p^2 x^4}}{2 c x^2}+\frac {(b c-a d) \sqrt {q^2+p^2 x^4}}{c^2 x}-\frac {2 (b c-a d) p x \sqrt {q^2+p^2 x^4}}{c^2 \left (q+p x^2\right )}-\frac {a p q \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )}{2 c}-\frac {\left (b c d-a d^2+a c^2 p q\right ) \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 c^3}+\frac {2 (b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}-\frac {(b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}+\frac {(b c-a d) \int \frac {p^2 \left (d-\sqrt {d^2-4 c^2 p q}\right )^2+4 c^2 p^4 x^2}{\sqrt {q^2+p^2 x^4}} \, dx}{4 c^4 p^2}+\frac {(b c-a d) \int \frac {p^2 \left (d+\sqrt {d^2-4 c^2 p q}\right )^2+4 c^2 p^4 x^2}{\sqrt {q^2+p^2 x^4}} \, dx}{4 c^4 p^2}+\frac {\left ((b c-a d) p^2 \left (d-\sqrt {d^2-4 c^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {q^2+p^2 x^2}}{\left (d+\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x} \, dx,x,x^2\right )}{c q}+\frac {\left ((b c-a d) p^2 \left (d+\sqrt {d^2-4 c^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {q^2+p^2 x^2}}{\left (d-\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x} \, dx,x,x^2\right )}{c q}-\frac {1}{4} \left ((b c-a d) p^2 \left (16 q^2+\frac {\left (d-\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right )\right ) \int \frac {1}{\left (\left (d-\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx-\frac {1}{4} \left ((b c-a d) p^2 \left (16 q^2+\frac {\left (d+\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right )\right ) \int \frac {1}{\left (\left (d+\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx\\ &=\frac {\left (b c d-a d^2+a c^2 p q\right ) \sqrt {q^2+p^2 x^4}}{2 c^3 q}-\frac {(b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 c^3 q}-\frac {(b c-a d) \left (d+\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 c^3 q}+\frac {a q \sqrt {q^2+p^2 x^4}}{2 c x^2}+\frac {(b c-a d) \sqrt {q^2+p^2 x^4}}{c^2 x}-\frac {2 (b c-a d) p x \sqrt {q^2+p^2 x^4}}{c^2 \left (q+p x^2\right )}-\frac {a p q \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )}{2 c}-\frac {\left (b c d-a d^2+a c^2 p q\right ) \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 c^3}+\frac {2 (b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}-\frac {(b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}-2 \frac {((b c-a d) p q) \int \frac {1-\frac {p x^2}{q}}{\sqrt {q^2+p^2 x^4}} \, dx}{c^2}+\frac {\left (d (b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{2 c^4}-\frac {\left ((b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {-4 c^2 p^2 q^2-p^2 \left (d+\sqrt {d^2-4 c^2 p q}\right )^2 x}{\left (\left (d+\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{4 c^3 q}+\frac {\left (d (b c-a d) \left (d+\sqrt {d^2-4 c^2 p q}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{2 c^4}-\frac {\left ((b c-a d) \left (d+\sqrt {d^2-4 c^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {-4 c^2 p^2 q^2-p^2 \left (d-\sqrt {d^2-4 c^2 p q}\right )^2 x}{\left (\left (d-\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{4 c^3 q}-\frac {\left ((b c-a d) p^2 \left (16 q^2+\frac {\left (d-\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{8 d \left (d-\sqrt {d^2-4 c^2 p q}\right )}-\frac {\left (c^2 (b c-a d) p^3 q \left (16 q^2+\frac {\left (d-\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right )\right ) \int \frac {1+\frac {p x^2}{q}}{\left (\left (d-\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx}{2 d \left (d-\sqrt {d^2-4 c^2 p q}\right )}-\frac {\left ((b c-a d) p^2 \left (16 q^2+\frac {\left (d+\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{8 d \left (d+\sqrt {d^2-4 c^2 p q}\right )}-\frac {\left (c^2 (b c-a d) p^3 q \left (16 q^2+\frac {\left (d+\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right )\right ) \int \frac {1+\frac {p x^2}{q}}{\left (\left (d+\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx}{2 d \left (d+\sqrt {d^2-4 c^2 p q}\right )}\\ &=\frac {\left (b c d-a d^2+a c^2 p q\right ) \sqrt {q^2+p^2 x^4}}{2 c^3 q}-\frac {(b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 c^3 q}-\frac {(b c-a d) \left (d+\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 c^3 q}+\frac {a q \sqrt {q^2+p^2 x^4}}{2 c x^2}+\frac {(b c-a d) \sqrt {q^2+p^2 x^4}}{c^2 x}-\frac {2 (b c-a d) p x \sqrt {q^2+p^2 x^4}}{c^2 \left (q+p x^2\right )}+\frac {(b c-a d) \sqrt {-d^2+2 c^2 p q} \tan ^{-1}\left (\frac {\sqrt {-d^2+2 c^2 p q} x}{c \sqrt {q^2+p^2 x^4}}\right )}{c^3}-\frac {a p q \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )}{2 c}-\frac {\left (b c d-a d^2+a c^2 p q\right ) \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 c^3}+\frac {2 (b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}-2 \left (-\frac {(b c-a d) p x \sqrt {q^2+p^2 x^4}}{c^2 \left (q+p x^2\right )}+\frac {(b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}\right )-\frac {(b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}+\frac {d (b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 c^4 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}+\frac {d (b c-a d) \left (d+\sqrt {d^2-4 c^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 c^4 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}-\frac {(b c-a d) p^{3/2} \left (16 q^2+\frac {\left (d-\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d-\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}-\frac {(b c-a d) p^{3/2} \left (16 q^2+\frac {\left (d+\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d+\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}+\frac {(b c-a d) p^{3/2} \left (d^2-4 c^2 p q-d \sqrt {d^2-4 c^2 p q}\right ) \left (16 q^2+\frac {\left (d-\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {d^2}{4 c^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d-\sqrt {d^2-4 c^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}+\frac {(b c-a d) p^{3/2} \left (d^2-4 c^2 p q+d \sqrt {d^2-4 c^2 p q}\right ) \left (16 q^2+\frac {\left (d+\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {d^2}{4 c^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d+\sqrt {d^2-4 c^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}-\frac {\left ((b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right )^2 \left (d+\sqrt {d^2-4 c^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{16 c^5 q}-\frac {\left ((b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \left (d+\sqrt {d^2-4 c^2 p q}\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{16 c^5 q}+\frac {\left ((b c-a d) \left (d+\sqrt {d^2-4 c^2 p q}\right ) \left (16 c^4 p^2 q^2+\left (d-\sqrt {d^2-4 c^2 p q}\right )^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (d-\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{16 c^5 q}+\frac {\left ((b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \left (16 c^4 p^2 q^2+\left (d+\sqrt {d^2-4 c^2 p q}\right )^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (d+\sqrt {d^2-4 c^2 p q}\right )^2-4 c^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{16 c^5 q}\\ &=\frac {\left (b c d-a d^2+a c^2 p q\right ) \sqrt {q^2+p^2 x^4}}{2 c^3 q}-\frac {(b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 c^3 q}-\frac {(b c-a d) \left (d+\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 c^3 q}+\frac {a q \sqrt {q^2+p^2 x^4}}{2 c x^2}+\frac {(b c-a d) \sqrt {q^2+p^2 x^4}}{c^2 x}-\frac {2 (b c-a d) p x \sqrt {q^2+p^2 x^4}}{c^2 \left (q+p x^2\right )}+\frac {(b c-a d) \sqrt {-d^2+2 c^2 p q} \tan ^{-1}\left (\frac {\sqrt {-d^2+2 c^2 p q} x}{c \sqrt {q^2+p^2 x^4}}\right )}{c^3}-\frac {a p q \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )}{2 c}-\frac {\left (b c d-a d^2+a c^2 p q\right ) \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 c^3}+\frac {2 (b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}-2 \left (-\frac {(b c-a d) p x \sqrt {q^2+p^2 x^4}}{c^2 \left (q+p x^2\right )}+\frac {(b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}\right )-\frac {(b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}+\frac {d (b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 c^4 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}+\frac {d (b c-a d) \left (d+\sqrt {d^2-4 c^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 c^4 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}-\frac {(b c-a d) p^{3/2} \left (16 q^2+\frac {\left (d-\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d-\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}-\frac {(b c-a d) p^{3/2} \left (16 q^2+\frac {\left (d+\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d+\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}+\frac {(b c-a d) p^{3/2} \left (d^2-4 c^2 p q-d \sqrt {d^2-4 c^2 p q}\right ) \left (16 q^2+\frac {\left (d-\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {d^2}{4 c^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d-\sqrt {d^2-4 c^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}+\frac {(b c-a d) p^{3/2} \left (d^2-4 c^2 p q+d \sqrt {d^2-4 c^2 p q}\right ) \left (16 q^2+\frac {\left (d+\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {d^2}{4 c^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d+\sqrt {d^2-4 c^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}-\frac {\left ((b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right )^2 \left (d+\sqrt {d^2-4 c^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-p^2 x^2} \, dx,x,\frac {x^2}{\sqrt {q^2+p^2 x^4}}\right )}{16 c^5 q}-\frac {\left ((b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \left (d+\sqrt {d^2-4 c^2 p q}\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-p^2 x^2} \, dx,x,\frac {x^2}{\sqrt {q^2+p^2 x^4}}\right )}{16 c^5 q}-\frac {\left ((b c-a d) \left (d+\sqrt {d^2-4 c^2 p q}\right ) \left (16 c^4 p^2 q^2+\left (d-\sqrt {d^2-4 c^2 p q}\right )^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 c^4 p^4 q^2+p^2 \left (d-\sqrt {d^2-4 c^2 p q}\right )^4-x^2} \, dx,x,\frac {-4 c^2 p^2 q^2-p^2 \left (d-\sqrt {d^2-4 c^2 p q}\right )^2 x^2}{\sqrt {q^2+p^2 x^4}}\right )}{16 c^5 q}-\frac {\left ((b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \left (16 c^4 p^2 q^2+\left (d+\sqrt {d^2-4 c^2 p q}\right )^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 c^4 p^4 q^2+p^2 \left (d+\sqrt {d^2-4 c^2 p q}\right )^4-x^2} \, dx,x,\frac {-4 c^2 p^2 q^2-p^2 \left (d+\sqrt {d^2-4 c^2 p q}\right )^2 x^2}{\sqrt {q^2+p^2 x^4}}\right )}{16 c^5 q}\\ &=\frac {\left (b c d-a d^2+a c^2 p q\right ) \sqrt {q^2+p^2 x^4}}{2 c^3 q}-\frac {(b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 c^3 q}-\frac {(b c-a d) \left (d+\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 c^3 q}+\frac {a q \sqrt {q^2+p^2 x^4}}{2 c x^2}+\frac {(b c-a d) \sqrt {q^2+p^2 x^4}}{c^2 x}-\frac {2 (b c-a d) p x \sqrt {q^2+p^2 x^4}}{c^2 \left (q+p x^2\right )}+\frac {(b c-a d) \sqrt {-d^2+2 c^2 p q} \tan ^{-1}\left (\frac {\sqrt {-d^2+2 c^2 p q} x}{c \sqrt {q^2+p^2 x^4}}\right )}{c^3}-\frac {a p q \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )}{2 c}-\frac {(b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right )^2 \left (d+\sqrt {d^2-4 c^2 p q}\right ) \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )}{16 c^5 p q}-\frac {(b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \left (d+\sqrt {d^2-4 c^2 p q}\right )^2 \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )}{16 c^5 p q}+\frac {(b c-a d) \sqrt {d^2-2 c^2 p q} \left (d+\sqrt {d^2-4 c^2 p q}\right ) \sqrt {d^2-2 c^2 p q-d \sqrt {d^2-4 c^2 p q}} \tanh ^{-1}\left (\frac {p \left (4 c^2 q^2+\left (d-\sqrt {d^2-4 c^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {d^2-2 c^2 p q} \sqrt {d^2-2 c^2 p q-d \sqrt {d^2-4 c^2 p q}} \sqrt {q^2+p^2 x^4}}\right )}{4 \sqrt {2} c^5 p q}+\frac {(b c-a d) \sqrt {d^2-2 c^2 p q} \left (d-\sqrt {d^2-4 c^2 p q}\right ) \sqrt {d^2-2 c^2 p q+d \sqrt {d^2-4 c^2 p q}} \tanh ^{-1}\left (\frac {p \left (4 c^2 q^2+\left (d+\sqrt {d^2-4 c^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {d^2-2 c^2 p q} \sqrt {d^2-2 c^2 p q+d \sqrt {d^2-4 c^2 p q}} \sqrt {q^2+p^2 x^4}}\right )}{4 \sqrt {2} c^5 p q}-\frac {\left (b c d-a d^2+a c^2 p q\right ) \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 c^3}+\frac {2 (b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}-2 \left (-\frac {(b c-a d) p x \sqrt {q^2+p^2 x^4}}{c^2 \left (q+p x^2\right )}+\frac {(b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}\right )-\frac {(b c-a d) \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{c^2 \sqrt {q^2+p^2 x^4}}+\frac {d (b c-a d) \left (d-\sqrt {d^2-4 c^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 c^4 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}+\frac {d (b c-a d) \left (d+\sqrt {d^2-4 c^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 c^4 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}-\frac {(b c-a d) p^{3/2} \left (16 q^2+\frac {\left (d-\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d-\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}-\frac {(b c-a d) p^{3/2} \left (16 q^2+\frac {\left (d+\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d+\sqrt {d^2-4 c^2 p q}\right ) \sqrt {q^2+p^2 x^4}}+\frac {(b c-a d) p^{3/2} \left (d^2-4 c^2 p q-d \sqrt {d^2-4 c^2 p q}\right ) \left (16 q^2+\frac {\left (d-\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {d^2}{4 c^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d-\sqrt {d^2-4 c^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}+\frac {(b c-a d) p^{3/2} \left (d^2-4 c^2 p q+d \sqrt {d^2-4 c^2 p q}\right ) \left (16 q^2+\frac {\left (d+\sqrt {d^2-4 c^2 p q}\right )^4}{c^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {d^2}{4 c^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 d \sqrt {q} \left (d+\sqrt {d^2-4 c^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 9.15, size = 7717, normalized size = 38.58 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 3.94, size = 200, normalized size = 1.00 \begin {gather*} \frac {\left (a c q+2 b c x-2 a d x+a c p x^2\right ) \sqrt {q^2+p^2 x^4}}{2 c^2 x^2}-\frac {2 (-b c+a d) \sqrt {-d^2+2 c^2 p q} \tan ^{-1}\left (\frac {\sqrt {-d^2+2 c^2 p q} x}{c q+d x+c p x^2+c \sqrt {q^2+p^2 x^4}}\right )}{c^3}+\frac {\left (b c d-a d^2+a c^2 p q\right ) \log (x)}{c^3}+\frac {\left (-b c d+a d^2-a c^2 p q\right ) \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right )}{c^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (a p x^{2} + a q + b x\right )} {\left (p x^{2} - q\right )}}{{\left (c p x^{2} + c q + d x\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.92, size = 13530, normalized size = 67.65
method | result | size |
risch | \(\text {Expression too large to display}\) | \(13530\) |
default | \(\text {Expression too large to display}\) | \(13966\) |
elliptic | \(\text {Expression too large to display}\) | \(30360\) |
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*} \int \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (a p x^{2} + a q + b x\right )} {\left (p x^{2} - q\right )}}{{\left (c p x^{2} + c q + d x\right )} x^{3}}\,{d x} \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 {\sqrt {p^2\,x^4+q^2}\,\left (q-p\,x^2\right )\,\left (a\,p\,x^2+b\,x+a\,q\right )}{x^3\,\left (c\,p\,x^2+d\,x+c\,q\right )} \,d x \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 (p x^{2} - q\right ) \sqrt {p^{2} x^{4} + q^{2}} \left (a p x^{2} + a q + b x\right )}{x^{3} \left (c p x^{2} + c q + d x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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