Optimal. Leaf size=133 \[ \frac {2 \sqrt {2 a^2 p q-b^2} \tan ^{-1}\left (\frac {x \sqrt {2 a^2 p q-b^2}}{a \sqrt {p^2 x^4+q^2}+a p x^2+a q+b x}\right )}{a^2}-\frac {b \log \left (\sqrt {p^2 x^4+q^2}+p x^2+q\right )}{a^2}+\frac {b \log (x)}{a^2}+\frac {\sqrt {p^2 x^4+q^2}}{a x} \]
________________________________________________________________________________________
Rubi [C] time = 7.11, antiderivative size = 1209, normalized size of antiderivative = 9.09, number of steps used = 42, number of rules used = 20, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {6728, 277, 305, 220, 1196, 266, 50, 63, 208, 1729, 1209, 1198, 1217, 1707, 1248, 735, 844, 217, 206, 725} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {p^2 x^4+q^2}}{q}\right ) b}{2 a^2}+\frac {\left (b-\sqrt {b^2-4 a^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}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2}}+\frac {\left (b+\sqrt {b^2-4 a^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}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2}}+\frac {\sqrt {p^2 x^4+q^2} b}{2 a^2 q}+\frac {\sqrt {2 a^2 p q-b^2} \tan ^{-1}\left (\frac {\sqrt {2 a^2 p q-b^2} x}{a \sqrt {p^2 x^4+q^2}}\right )}{a^2}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )}{4 a^2}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )}{4 a^2}+\frac {\sqrt {b^2-2 a^2 p q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {-2 p q a^2+b^2-b \sqrt {b^2-4 a^2 p q}} \tanh ^{-1}\left (\frac {p \left (4 a^2 q^2+\left (b-\sqrt {b^2-4 a^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {b^2-2 a^2 p q} \sqrt {-2 p q a^2+b^2-b \sqrt {b^2-4 a^2 p q}} \sqrt {p^2 x^4+q^2}}\right )}{4 \sqrt {2} a^4 p q}+\frac {\sqrt {b^2-2 a^2 p q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {-2 p q a^2+b^2+b \sqrt {b^2-4 a^2 p q}} \tanh ^{-1}\left (\frac {p \left (4 a^2 q^2+\left (b+\sqrt {b^2-4 a^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {b^2-2 a^2 p q} \sqrt {-2 p q a^2+b^2+b \sqrt {b^2-4 a^2 p q}} \sqrt {p^2 x^4+q^2}}\right )}{4 \sqrt {2} a^4 p q}-\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 )}{a \sqrt {p^2 x^4+q^2}}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p^2 x^4+q^2}}{4 a^2 q}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p^2 x^4+q^2}}{4 a^2 q}+\frac {\sqrt {p^2 x^4+q^2}}{a x}-\frac {\left (b^2-2 a^2 p q\right ) \left (b-\sqrt {b^2-4 a^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 )}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2} b}-\frac {\left (b^2-2 a^2 p q\right ) \left (b+\sqrt {b^2-4 a^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 )}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2} b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 206
Rule 208
Rule 217
Rule 220
Rule 266
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 ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx &=\int \left (-\frac {\sqrt {q^2+p^2 x^4}}{a x^2}+\frac {b \sqrt {q^2+p^2 x^4}}{a^2 q x}+\frac {\left (-b^2+2 a^2 p q-a b p x\right ) \sqrt {q^2+p^2 x^4}}{a^2 q \left (a q+b x+a p x^2\right )}\right ) \, dx\\ &=-\frac {\int \frac {\sqrt {q^2+p^2 x^4}}{x^2} \, dx}{a}+\frac {\int \frac {\left (-b^2+2 a^2 p q-a b p x\right ) \sqrt {q^2+p^2 x^4}}{a q+b x+a p x^2} \, dx}{a^2 q}+\frac {b \int \frac {\sqrt {q^2+p^2 x^4}}{x} \, dx}{a^2 q}\\ &=\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {\left (2 p^2\right ) \int \frac {x^2}{\sqrt {q^2+p^2 x^4}} \, dx}{a}+\frac {\int \left (\frac {\left (-a b p-a p \sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{b-\sqrt {b^2-4 a^2 p q}+2 a p x}+\frac {\left (-a b p+a p \sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{b+\sqrt {b^2-4 a^2 p q}+2 a p x}\right ) \, dx}{a^2 q}+\frac {b \operatorname {Subst}\left (\int \frac {\sqrt {q^2+p^2 x}}{x} \, dx,x,x^4\right )}{4 a^2 q}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}+\frac {(b q) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {q^2+p^2 x}} \, dx,x,x^4\right )}{4 a^2}-\frac {(2 p q) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{a}+\frac {(2 p q) \int \frac {1-\frac {p x^2}{q}}{\sqrt {q^2+p^2 x^4}} \, dx}{a}-\frac {\left (p \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{b+\sqrt {b^2-4 a^2 p q}+2 a p x} \, dx}{a q}-\frac {\left (p \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{b-\sqrt {b^2-4 a^2 p q}+2 a p x} \, dx}{a q}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {2 \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 )}{a \sqrt {q^2+p^2 x^4}}-\frac {\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 )}{a \sqrt {q^2+p^2 x^4}}-\left (4 a p^2\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2} \, dx-\left (4 a p^2\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2} \, dx+\frac {(b q) \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 a^2 p^2}+\frac {\left (2 p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {x \sqrt {q^2+p^2 x^4}}{\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2} \, dx}{q}+\frac {\left (2 p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {x \sqrt {q^2+p^2 x^4}}{\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2} \, dx}{q}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 a^2}+\frac {2 \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 )}{a \sqrt {q^2+p^2 x^4}}-\frac {\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 )}{a \sqrt {q^2+p^2 x^4}}+\frac {\int \frac {p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )^2+4 a^2 p^4 x^2}{\sqrt {q^2+p^2 x^4}} \, dx}{4 a^3 p^2}+\frac {\int \frac {p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )^2+4 a^2 p^4 x^2}{\sqrt {q^2+p^2 x^4}} \, dx}{4 a^3 p^2}+\frac {\left (p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {q^2+p^2 x^2}}{\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x} \, dx,x,x^2\right )}{q}+\frac {\left (p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {q^2+p^2 x^2}}{\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x} \, dx,x,x^2\right )}{q}-\frac {1}{4} \left (a p^2 \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1}{\left (\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx-\frac {1}{4} \left (a p^2 \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1}{\left (\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 a^2}+\frac {2 \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 )}{a \sqrt {q^2+p^2 x^4}}-\frac {\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 )}{a \sqrt {q^2+p^2 x^4}}-2 \frac {(p q) \int \frac {1-\frac {p x^2}{q}}{\sqrt {q^2+p^2 x^4}} \, dx}{a}+\frac {\left (b \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{2 a^3}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \operatorname {Subst}\left (\int \frac {-4 a^2 p^2 q^2-p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )^2 x}{\left (\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{4 a^2 q}+\frac {\left (b \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{2 a^3}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \operatorname {Subst}\left (\int \frac {-4 a^2 p^2 q^2-p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )^2 x}{\left (\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{4 a^2 q}-\frac {\left (a p^2 \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{8 b \left (b-\sqrt {b^2-4 a^2 p q}\right )}-\frac {\left (a^3 p^3 q \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1+\frac {p x^2}{q}}{\left (\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx}{2 b \left (b-\sqrt {b^2-4 a^2 p q}\right )}-\frac {\left (a p^2 \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{8 b \left (b+\sqrt {b^2-4 a^2 p q}\right )}-\frac {\left (a^3 p^3 q \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1+\frac {p x^2}{q}}{\left (\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx}{2 b \left (b+\sqrt {b^2-4 a^2 p q}\right )}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\sqrt {-b^2+2 a^2 p q} \tan ^{-1}\left (\frac {\sqrt {-b^2+2 a^2 p q} x}{a \sqrt {q^2+p^2 x^4}}\right )}{a^2}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 a^2}+\frac {2 \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 )}{a \sqrt {q^2+p^2 x^4}}-2 \left (-\frac {p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\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 )}{a \sqrt {q^2+p^2 x^4}}\right )-\frac {\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 )}{a \sqrt {q^2+p^2 x^4}}+\frac {b \left (b-\sqrt {b^2-4 a^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 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}+\frac {b \left (b+\sqrt {b^2-4 a^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 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q-b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q+b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}-\frac {\left (p \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{4 a^2}+\frac {\left (p \left (b^2-2 a^2 p q\right ) \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{a^2}-\frac {\left (p \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{4 a^2}+\frac {\left (p \left (b^2-2 a^2 p q\right ) \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{a^2}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\sqrt {-b^2+2 a^2 p q} \tan ^{-1}\left (\frac {\sqrt {-b^2+2 a^2 p q} x}{a \sqrt {q^2+p^2 x^4}}\right )}{a^2}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 a^2}+\frac {2 \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 )}{a \sqrt {q^2+p^2 x^4}}-2 \left (-\frac {p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\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 )}{a \sqrt {q^2+p^2 x^4}}\right )-\frac {\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 )}{a \sqrt {q^2+p^2 x^4}}+\frac {b \left (b-\sqrt {b^2-4 a^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 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}+\frac {b \left (b+\sqrt {b^2-4 a^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 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q-b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q+b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}-\frac {\left (p \left (b-\sqrt {b^2-4 a^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 )}{4 a^2}-\frac {\left (p \left (b^2-2 a^2 p q\right ) \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 a^4 p^4 q^2+p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )^4-x^2} \, dx,x,\frac {-4 a^2 p^2 q^2-p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )^2 x^2}{\sqrt {q^2+p^2 x^4}}\right )}{a^2}-\frac {\left (p \left (b+\sqrt {b^2-4 a^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 )}{4 a^2}-\frac {\left (p \left (b^2-2 a^2 p q\right ) \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 a^4 p^4 q^2+p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )^4-x^2} \, dx,x,\frac {-4 a^2 p^2 q^2-p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )^2 x^2}{\sqrt {q^2+p^2 x^4}}\right )}{a^2}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\sqrt {-b^2+2 a^2 p q} \tan ^{-1}\left (\frac {\sqrt {-b^2+2 a^2 p q} x}{a \sqrt {q^2+p^2 x^4}}\right )}{a^2}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )}{4 a^2}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )}{4 a^2}+\frac {\sqrt {b^2-2 a^2 p q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p \left (4 a^2 q^2+\left (b-\sqrt {b^2-4 a^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {b^2-2 a^2 p q} \sqrt {b^2-2 a^2 p q-b \sqrt {b^2-4 a^2 p q}} \sqrt {q^2+p^2 x^4}}\right )}{2 \sqrt {2} a^2 \sqrt {b^2-2 a^2 p q-b \sqrt {b^2-4 a^2 p q}}}+\frac {\sqrt {b^2-2 a^2 p q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p \left (4 a^2 q^2+\left (b+\sqrt {b^2-4 a^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {b^2-2 a^2 p q} \sqrt {b^2-2 a^2 p q+b \sqrt {b^2-4 a^2 p q}} \sqrt {q^2+p^2 x^4}}\right )}{2 \sqrt {2} a^2 \sqrt {b^2-2 a^2 p q+b \sqrt {b^2-4 a^2 p q}}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 a^2}+\frac {2 \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 )}{a \sqrt {q^2+p^2 x^4}}-2 \left (-\frac {p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\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 )}{a \sqrt {q^2+p^2 x^4}}\right )-\frac {\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 )}{a \sqrt {q^2+p^2 x^4}}+\frac {b \left (b-\sqrt {b^2-4 a^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 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}+\frac {b \left (b+\sqrt {b^2-4 a^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 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q-b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q+b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^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 {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 8.74, size = 3835, normalized size = 28.83 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.56, size = 133, normalized size = 1.00 \begin {gather*} \frac {\sqrt {q^2+p^2 x^4}}{a x}+\frac {2 \sqrt {-b^2+2 a^2 p q} \tan ^{-1}\left (\frac {\sqrt {-b^2+2 a^2 p q} x}{a q+b x+a p x^2+a \sqrt {q^2+p^2 x^4}}\right )}{a^2}+\frac {b \log (x)}{a^2}-\frac {b \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (p x^{2} - q\right )}}{{\left (a p x^{2} + a q + b x\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.67, size = 6680, normalized size = 50.23
method | result | size |
risch | \(\text {Expression too large to display}\) | \(6680\) |
default | \(\text {Expression too large to display}\) | \(6985\) |
elliptic | \(\text {Expression too large to display}\) | \(9277\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (p x^{2} - q\right )}}{{\left (a p x^{2} + a q + b x\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\sqrt {p^2\,x^4+q^2}\,\left (q-p\,x^2\right )}{x^2\,\left (a\,p\,x^2+b\,x+a\,q\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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}}}{x^{2} \left (a p x^{2} + a q + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________