Optimal. Leaf size=85 \[ \frac {\sqrt {p^2 x^4+q^2} \left (2 a p^2 x^4+2 a q^2+3 b p x^3+3 b q x\right )}{6 x^3}-b p q \log \left (\sqrt {p^2 x^4+q^2}+p x^2+q\right )+b p q \log (x) \]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 109, normalized size of antiderivative = 1.28, number of steps used = 11, number of rules used = 10, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {1833, 1252, 813, 844, 217, 206, 266, 63, 208, 449} \begin {gather*} \frac {a \left (p^2 x^4+q^2\right )^{3/2}}{3 x^3}-\frac {1}{2} b p q \tanh ^{-1}\left (\frac {\sqrt {p^2 x^4+q^2}}{q}\right )+\frac {b \left (p x^2+q\right ) \sqrt {p^2 x^4+q^2}}{2 x^2}-\frac {1}{2} b p q \tanh ^{-1}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 449
Rule 813
Rule 844
Rule 1252
Rule 1833
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^4} \, dx &=\int \left (\frac {\left (-b q+b p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^3}+\frac {\sqrt {q^2+p^2 x^4} \left (-a q^2+a p^2 x^4\right )}{x^4}\right ) \, dx\\ &=\int \frac {\left (-b q+b p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^3} \, dx+\int \frac {\sqrt {q^2+p^2 x^4} \left (-a q^2+a p^2 x^4\right )}{x^4} \, dx\\ &=\frac {a \left (q^2+p^2 x^4\right )^{3/2}}{3 x^3}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-b q+b p x) \sqrt {q^2+p^2 x^2}}{x^2} \, dx,x,x^2\right )\\ &=\frac {b \left (q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{2 x^2}+\frac {a \left (q^2+p^2 x^4\right )^{3/2}}{3 x^3}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {-2 b p q^2+2 b p^2 q x}{x \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )\\ &=\frac {b \left (q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{2 x^2}+\frac {a \left (q^2+p^2 x^4\right )^{3/2}}{3 x^3}-\frac {1}{2} \left (b p^2 q\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )+\frac {1}{2} \left (b p q^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )\\ &=\frac {b \left (q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{2 x^2}+\frac {a \left (q^2+p^2 x^4\right )^{3/2}}{3 x^3}-\frac {1}{2} \left (b 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 )+\frac {1}{4} \left (b p q^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {q^2+p^2 x}} \, dx,x,x^4\right )\\ &=\frac {b \left (q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{2 x^2}+\frac {a \left (q^2+p^2 x^4\right )^{3/2}}{3 x^3}-\frac {1}{2} b p q \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )+\frac {\left (b q^2\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 p}\\ &=\frac {b \left (q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{2 x^2}+\frac {a \left (q^2+p^2 x^4\right )^{3/2}}{3 x^3}-\frac {1}{2} b p q \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )-\frac {1}{2} b p q \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.28, size = 226, normalized size = 2.66 \begin {gather*} \frac {6 a p^2 x^4 \sqrt {p^2 x^4+q^2} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {p^2 x^4}{q^2}\right )+2 a q^2 \sqrt {p^2 x^4+q^2} \, _2F_1\left (-\frac {3}{4},-\frac {1}{2};\frac {1}{4};-\frac {p^2 x^4}{q^2}\right )+3 b p x^3 \sqrt {\frac {p^2 x^4}{q^2}+1} \left (\sqrt {p^2 x^4+q^2}-q \tanh ^{-1}\left (\frac {\sqrt {p^2 x^4+q^2}}{q}\right )\right )+3 b x \sqrt {p^2 x^4+q^2} \left (q \sqrt {\frac {p^2 x^4}{q^2}+1}-p x^2 \sinh ^{-1}\left (\frac {p x^2}{q}\right )\right )}{6 x^3 \sqrt {\frac {p^2 x^4}{q^2}+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.16, size = 85, normalized size = 1.00 \begin {gather*} \frac {\sqrt {q^2+p^2 x^4} \left (2 a q^2+3 b q x+3 b p x^3+2 a p^2 x^4\right )}{6 x^3}+b p q \log (x)-b p q \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.49, size = 83, normalized size = 0.98 \begin {gather*} \frac {6 \, b p q x^{3} \log \left (\frac {p x^{2} + q - \sqrt {p^{2} x^{4} + q^{2}}}{x}\right ) + {\left (2 \, a p^{2} x^{4} + 3 \, b p x^{3} + 2 \, a q^{2} + 3 \, b q x\right )} \sqrt {p^{2} x^{4} + q^{2}}}{6 \, x^{3}} \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 (a p x^{2} + a q + b x\right )} {\left (p x^{2} - q\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.25, size = 140, normalized size = 1.65
method | result | size |
elliptic | \(\frac {p b \sqrt {p^{2} x^{4}+q^{2}}}{2}-\frac {b \,p^{2} q \ln \left (\frac {p^{2} x^{2}}{\sqrt {p^{2}}}+\sqrt {p^{2} x^{4}+q^{2}}\right )}{2 \sqrt {p^{2}}}+\frac {b q \sqrt {p^{2} x^{4}+q^{2}}}{2 x^{2}}-\frac {p b \,q^{2} \ln \left (\frac {2 q^{2}+2 \sqrt {q^{2}}\, \sqrt {p^{2} x^{4}+q^{2}}}{x^{2}}\right )}{2 \sqrt {q^{2}}}+\frac {a \left (p^{2} x^{4}+q^{2}\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(140\) |
risch | \(\frac {\sqrt {p^{2} x^{4}+q^{2}}\, q \left (2 a q +3 b x \right )}{6 x^{3}}+\frac {a \,p^{2} x \sqrt {p^{2} x^{4}+q^{2}}}{3}+\frac {p b \sqrt {p^{2} x^{4}+q^{2}}}{2}-\frac {b \,p^{2} q \ln \left (\frac {p^{2} x^{2}}{\sqrt {p^{2}}}+\sqrt {p^{2} x^{4}+q^{2}}\right )}{2 \sqrt {p^{2}}}-\frac {p b \,q^{2} \ln \left (\frac {2 q^{2}+2 \sqrt {q^{2}}\, \sqrt {p^{2} x^{4}+q^{2}}}{x^{2}}\right )}{2 \sqrt {q^{2}}}\) | \(149\) |
default | \(a \,p^{2} \left (\frac {x \sqrt {p^{2} x^{4}+q^{2}}}{3}+\frac {2 q^{2} \sqrt {1-\frac {i p \,x^{2}}{q}}\, \sqrt {1+\frac {i p \,x^{2}}{q}}\, \EllipticF \left (x \sqrt {\frac {i p}{q}}, i\right )}{3 \sqrt {\frac {i p}{q}}\, \sqrt {p^{2} x^{4}+q^{2}}}\right )-a \,q^{2} \left (-\frac {\sqrt {p^{2} x^{4}+q^{2}}}{3 x^{3}}+\frac {2 p^{2} \sqrt {1-\frac {i p \,x^{2}}{q}}\, \sqrt {1+\frac {i p \,x^{2}}{q}}\, \EllipticF \left (x \sqrt {\frac {i p}{q}}, i\right )}{3 \sqrt {\frac {i p}{q}}\, \sqrt {p^{2} x^{4}+q^{2}}}\right )+p b \left (\frac {\sqrt {p^{2} x^{4}+q^{2}}}{2}-\frac {q^{2} \ln \left (\frac {2 q^{2}+2 \sqrt {q^{2}}\, \sqrt {p^{2} x^{4}+q^{2}}}{x^{2}}\right )}{2 \sqrt {q^{2}}}\right )-q b \left (-\frac {\left (p^{2} x^{4}+q^{2}\right )^{\frac {3}{2}}}{2 q^{2} x^{2}}+\frac {p^{2} x^{2} \sqrt {p^{2} x^{4}+q^{2}}}{2 q^{2}}+\frac {p^{2} \ln \left (\frac {p^{2} x^{2}}{\sqrt {p^{2}}}+\sqrt {p^{2} x^{4}+q^{2}}\right )}{2 \sqrt {p^{2}}}\right )\) | \(334\) |
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 (a p x^{2} + a q + b x\right )} {\left (p x^{2} - q\right )}}{x^{4}}\,{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 )\,\left (a\,p\,x^2+b\,x+a\,q\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 5.45, size = 223, normalized size = 2.62 \begin {gather*} \frac {a p^{2} q x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {p^{2} x^{4} e^{i \pi }}{q^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {a q^{3} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {p^{2} x^{4} e^{i \pi }}{q^{2}}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {b p^{2} x^{2}}{2 \sqrt {\frac {p^{2} x^{4}}{q^{2}} + 1}} + \frac {b p^{2} x^{2}}{2 \sqrt {1 + \frac {q^{2}}{p^{2} x^{4}}}} - \frac {b p q \operatorname {asinh}{\left (\frac {q}{p x^{2}} \right )}}{2} - \frac {b p q \operatorname {asinh}{\left (\frac {p x^{2}}{q} \right )}}{2} + \frac {b q^{2}}{2 x^{2} \sqrt {\frac {p^{2} x^{4}}{q^{2}} + 1}} + \frac {b q^{2}}{2 x^{2} \sqrt {1 + \frac {q^{2}}{p^{2} x^{4}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________