Optimal. Leaf size=119 \[ \frac {\sqrt {p^2 x^4+q^2} \left (6 a p^4 x^8+20 a p^3 q x^6+12 a p^2 q^2 x^4+20 a p q^3 x^2+6 a q^4+15 b p x^5+15 b q x^3\right )}{30 x^5}-b p q \log \left (\sqrt {p^2 x^4+q^2}+p x^2+q\right )+b p q \log (x) \]
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Rubi [A] time = 0.49, antiderivative size = 163, normalized size of antiderivative = 1.37, number of steps used = 16, number of rules used = 13, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {1833, 1584, 1252, 813, 844, 217, 206, 266, 63, 208, 1835, 1586, 449} \begin {gather*} \frac {a p^2 \left (p^2 x^4+q^2\right )^{3/2}}{5 x}+\frac {a q^2 \left (p^2 x^4+q^2\right )^{3/2}}{5 x^5}+\frac {2 a p q \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.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 449
Rule 813
Rule 844
Rule 1252
Rule 1584
Rule 1586
Rule 1833
Rule 1835
Rubi steps
\begin {align*} \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4} \left (b x^3+a \left (q+p x^2\right )^3\right )}{x^6} \, dx &=\int \left (\frac {\left (-b q x^2+b p x^4\right ) \sqrt {q^2+p^2 x^4}}{x^5}+\frac {\sqrt {q^2+p^2 x^4} \left (-a q^4-2 a p q^3 x^2+2 a p^3 q x^6+a p^4 x^8\right )}{x^6}\right ) \, dx\\ &=\int \frac {\left (-b q x^2+b p x^4\right ) \sqrt {q^2+p^2 x^4}}{x^5} \, dx+\int \frac {\sqrt {q^2+p^2 x^4} \left (-a q^4-2 a p q^3 x^2+2 a p^3 q x^6+a p^4 x^8\right )}{x^6} \, dx\\ &=\frac {a q^2 \left (q^2+p^2 x^4\right )^{3/2}}{5 x^5}-\frac {\int \frac {\sqrt {q^2+p^2 x^4} \left (20 a p q^5 x+2 a p^2 q^4 x^3-20 a p^3 q^3 x^5-10 a p^4 q^2 x^7\right )}{x^5} \, dx}{10 q^2}+\int \frac {\left (-b q+b p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^3} \, dx\\ &=\frac {a q^2 \left (q^2+p^2 x^4\right )^{3/2}}{5 x^5}+\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 {\int \frac {\sqrt {q^2+p^2 x^4} \left (20 a p q^5+2 a p^2 q^4 x^2-20 a p^3 q^3 x^4-10 a p^4 q^2 x^6\right )}{x^4} \, dx}{10 q^2}\\ &=\frac {b \left (q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{2 x^2}+\frac {a q^2 \left (q^2+p^2 x^4\right )^{3/2}}{5 x^5}+\frac {2 a p q \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 {\int \frac {\sqrt {q^2+p^2 x^4} \left (-12 a p^2 q^6 x+60 a p^4 q^4 x^5\right )}{x^3} \, dx}{60 q^4}\\ &=\frac {b \left (q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{2 x^2}+\frac {a q^2 \left (q^2+p^2 x^4\right )^{3/2}}{5 x^5}+\frac {2 a p q \left (q^2+p^2 x^4\right )^{3/2}}{3 x^3}+\frac {\int \frac {\sqrt {q^2+p^2 x^4} \left (-12 a p^2 q^6+60 a p^4 q^4 x^4\right )}{x^2} \, dx}{60 q^4}-\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 q^2 \left (q^2+p^2 x^4\right )^{3/2}}{5 x^5}+\frac {2 a p q \left (q^2+p^2 x^4\right )^{3/2}}{3 x^3}+\frac {a p^2 \left (q^2+p^2 x^4\right )^{3/2}}{5 x}-\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 q^2 \left (q^2+p^2 x^4\right )^{3/2}}{5 x^5}+\frac {2 a p q \left (q^2+p^2 x^4\right )^{3/2}}{3 x^3}+\frac {a p^2 \left (q^2+p^2 x^4\right )^{3/2}}{5 x}-\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 q^2 \left (q^2+p^2 x^4\right )^{3/2}}{5 x^5}+\frac {2 a p q \left (q^2+p^2 x^4\right )^{3/2}}{3 x^3}+\frac {a p^2 \left (q^2+p^2 x^4\right )^{3/2}}{5 x}-\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*}
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Mathematica [C] time = 0.27, size = 354, normalized size = 2.97 \begin {gather*} \frac {6 a q^4 \sqrt {p^2 x^4+q^2} \, _2F_1\left (-\frac {5}{4},-\frac {1}{2};-\frac {1}{4};-\frac {p^2 x^4}{q^2}\right )+20 a p q^3 x^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 )+10 a p^4 x^8 \sqrt {p^2 x^4+q^2} \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {p^2 x^4}{q^2}\right )+60 a p^3 q x^6 \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 )+15 b p x^5 \sqrt {p^2 x^4+q^2} \sqrt {\frac {p^2 x^4}{q^2}+1}-15 b p q x^5 \sqrt {\frac {p^2 x^4}{q^2}+1} \tanh ^{-1}\left (\frac {\sqrt {p^2 x^4+q^2}}{q}\right )+15 b q x^3 \sqrt {p^2 x^4+q^2} \sqrt {\frac {p^2 x^4}{q^2}+1}-15 b p x^5 \sqrt {p^2 x^4+q^2} \sinh ^{-1}\left (\frac {p x^2}{q}\right )}{30 x^5 \sqrt {\frac {p^2 x^4}{q^2}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.58, size = 119, normalized size = 1.00 \begin {gather*} \frac {\sqrt {q^2+p^2 x^4} \left (6 a q^4+20 a p q^3 x^2+15 b q x^3+12 a p^2 q^2 x^4+15 b p x^5+20 a p^3 q x^6+6 a p^4 x^8\right )}{30 x^5}+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.
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fricas [A] time = 0.73, size = 117, normalized size = 0.98 \begin {gather*} \frac {30 \, b p q x^{5} \log \left (\frac {p x^{2} + q - \sqrt {p^{2} x^{4} + q^{2}}}{x}\right ) + {\left (6 \, a p^{4} x^{8} + 20 \, a p^{3} q x^{6} + 12 \, a p^{2} q^{2} x^{4} + 20 \, a p q^{3} x^{2} + 15 \, b p x^{5} + 6 \, a q^{4} + 15 \, b q x^{3}\right )} \sqrt {p^{2} x^{4} + q^{2}}}{30 \, x^{5}} \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 ({\left (p x^{2} + q\right )}^{3} a + b x^{3}\right )} {\left (p x^{2} - q\right )}}{x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 172, normalized size = 1.45
method | result | size |
elliptic | \(\frac {p b \sqrt {p^{2} x^{4}+q^{2}}}{2}-\frac {b \,p^{2} q \ln \left (\frac {p^{2} x^{2}}{\sqrt {p^{2}}}+\sqrt {p^{2} x^{4}+q^{2}}\right )}{2 \sqrt {p^{2}}}+\frac {b q \sqrt {p^{2} x^{4}+q^{2}}}{2 x^{2}}-\frac {p b \,q^{2} \ln \left (\frac {2 q^{2}+2 \sqrt {q^{2}}\, \sqrt {p^{2} x^{4}+q^{2}}}{x^{2}}\right )}{2 \sqrt {q^{2}}}+4 a \left (\frac {p q \left (p^{2} x^{4}+q^{2}\right )^{\frac {3}{2}} \sqrt {2}}{12 x^{3}}+\frac {\left (p^{2} x^{4}+q^{2}\right )^{\frac {5}{2}} \sqrt {2}}{40 x^{5}}\right ) \sqrt {2}\) | \(172\) |
risch | \(\frac {\sqrt {p^{2} x^{4}+q^{2}}\, q \left (12 a \,p^{2} q \,x^{4}+20 a p \,q^{2} x^{2}+6 a \,q^{3}+15 b \,x^{3}\right )}{30 x^{5}}+\frac {a \,p^{4} x^{3} \sqrt {p^{2} x^{4}+q^{2}}}{5}+\frac {2 a \,p^{3} q x \sqrt {p^{2} x^{4}+q^{2}}}{3}+\frac {p b \sqrt {p^{2} x^{4}+q^{2}}}{2}-\frac {b \,p^{2} q \ln \left (\frac {p^{2} x^{2}}{\sqrt {p^{2}}}+\sqrt {p^{2} x^{4}+q^{2}}\right )}{2 \sqrt {p^{2}}}-\frac {p b \,q^{2} \ln \left (\frac {2 q^{2}+2 \sqrt {q^{2}}\, \sqrt {p^{2} x^{4}+q^{2}}}{x^{2}}\right )}{2 \sqrt {q^{2}}}\) | \(196\) |
default | \(a \,p^{4} \left (\frac {x^{3} \sqrt {p^{2} x^{4}+q^{2}}}{5}+\frac {2 i q^{3} \sqrt {1-\frac {i p \,x^{2}}{q}}\, \sqrt {1+\frac {i p \,x^{2}}{q}}\, \left (\EllipticF \left (x \sqrt {\frac {i p}{q}}, i\right )-\EllipticE \left (x \sqrt {\frac {i p}{q}}, i\right )\right )}{5 \sqrt {\frac {i p}{q}}\, \sqrt {p^{2} x^{4}+q^{2}}\, p}\right )+2 q a \,p^{3} \left (\frac {x \sqrt {p^{2} x^{4}+q^{2}}}{3}+\frac {2 q^{2} \sqrt {1-\frac {i p \,x^{2}}{q}}\, \sqrt {1+\frac {i p \,x^{2}}{q}}\, \EllipticF \left (x \sqrt {\frac {i p}{q}}, i\right )}{3 \sqrt {\frac {i p}{q}}\, \sqrt {p^{2} x^{4}+q^{2}}}\right )-2 q^{3} a p \left (-\frac {\sqrt {p^{2} x^{4}+q^{2}}}{3 x^{3}}+\frac {2 p^{2} \sqrt {1-\frac {i p \,x^{2}}{q}}\, \sqrt {1+\frac {i p \,x^{2}}{q}}\, \EllipticF \left (x \sqrt {\frac {i p}{q}}, i\right )}{3 \sqrt {\frac {i p}{q}}\, \sqrt {p^{2} x^{4}+q^{2}}}\right )+p b \left (\frac {\sqrt {p^{2} x^{4}+q^{2}}}{2}-\frac {q^{2} \ln \left (\frac {2 q^{2}+2 \sqrt {q^{2}}\, \sqrt {p^{2} x^{4}+q^{2}}}{x^{2}}\right )}{2 \sqrt {q^{2}}}\right )-q b \left (-\frac {\left (p^{2} x^{4}+q^{2}\right )^{\frac {3}{2}}}{2 q^{2} x^{2}}+\frac {p^{2} x^{2} \sqrt {p^{2} x^{4}+q^{2}}}{2 q^{2}}+\frac {p^{2} \ln \left (\frac {p^{2} x^{2}}{\sqrt {p^{2}}}+\sqrt {p^{2} x^{4}+q^{2}}\right )}{2 \sqrt {p^{2}}}\right )-a \,q^{4} \left (-\frac {\sqrt {p^{2} x^{4}+q^{2}}}{5 x^{5}}-\frac {2 p^{2} \sqrt {p^{2} x^{4}+q^{2}}}{5 q^{2} x}+\frac {2 i p^{3} \sqrt {1-\frac {i p \,x^{2}}{q}}\, \sqrt {1+\frac {i p \,x^{2}}{q}}\, \left (\EllipticF \left (x \sqrt {\frac {i p}{q}}, i\right )-\EllipticE \left (x \sqrt {\frac {i p}{q}}, i\right )\right )}{5 q \sqrt {\frac {i p}{q}}\, \sqrt {p^{2} x^{4}+q^{2}}}\right )\) | \(590\) |
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 ({\left (p x^{2} + q\right )}^{3} a + b x^{3}\right )} {\left (p x^{2} - q\right )}}{x^{6}}\,{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 = -0.01 \begin {gather*} -\int \frac {\sqrt {p^2\,x^4+q^2}\,\left (q-p\,x^2\right )\,\left (a\,{\left (p\,x^2+q\right )}^3+b\,x^3\right )}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 8.30, size = 323, normalized size = 2.71 \begin {gather*} \frac {a p^{4} q x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {p^{2} x^{4} e^{i \pi }}{q^{2}}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {a p^{3} q^{2} 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 )}}{2 \Gamma \left (\frac {5}{4}\right )} - \frac {a p q^{4} \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 )}}{2 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {a q^{5} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {p^{2} x^{4} e^{i \pi }}{q^{2}}} \right )}}{4 x^{5} \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.
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