Optimal. Leaf size=352 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8 a-8 \text {$\#$1}^6 a \sqrt {p} \sqrt {q}+24 \text {$\#$1}^4 a p q+16 \text {$\#$1}^4 b-32 \text {$\#$1}^2 a p^{3/2} q^{3/2}+16 a p^2 q^2\& ,\frac {-\text {$\#$1}^6 \log \left (-\text {$\#$1} x+\sqrt {p x^4+q}+\sqrt {p} x^2+\sqrt {q}\right )+\text {$\#$1}^6 \log (x)+2 \text {$\#$1}^4 \sqrt {p} \sqrt {q} \log \left (-\text {$\#$1} x+\sqrt {p x^4+q}+\sqrt {p} x^2+\sqrt {q}\right )-2 \text {$\#$1}^4 \sqrt {p} \sqrt {q} \log (x)+4 \text {$\#$1}^2 p q \log \left (-\text {$\#$1} x+\sqrt {p x^4+q}+\sqrt {p} x^2+\sqrt {q}\right )-4 \text {$\#$1}^2 p q \log (x)-8 p^{3/2} q^{3/2} \log \left (-\text {$\#$1} x+\sqrt {p x^4+q}+\sqrt {p} x^2+\sqrt {q}\right )+8 p^{3/2} q^{3/2} \log (x)}{\text {$\#$1}^7 (-a)+6 \text {$\#$1}^5 a \sqrt {p} \sqrt {q}-12 \text {$\#$1}^3 a p q-8 \text {$\#$1}^3 b+8 \text {$\#$1} a p^{3/2} q^{3/2}}\& \right ] \]
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Rubi [A] time = 0.43, antiderivative size = 247, normalized size of antiderivative = 0.70, number of steps used = 10, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {6712, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^4+q}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^4+q}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {p x^4+q}}+\sqrt {a}+\frac {\sqrt {b} x^2}{p x^4+q}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {p x^4+q}}+\sqrt {a}+\frac {\sqrt {b} x^2}{p x^4+q}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6712
Rubi steps
\begin {align*} \int \frac {\left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^4+a \left (q+p x^4\right )^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\frac {x}{\sqrt {q+p x^4}}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\frac {x}{\sqrt {q+p x^4}}\right )}{2 \sqrt {a}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\frac {x}{\sqrt {q+p x^4}}\right )}{2 \sqrt {a}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x}{\sqrt {q+p x^4}}\right )}{4 \sqrt {a} \sqrt {b}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x}{\sqrt {q+p x^4}}\right )}{4 \sqrt {a} \sqrt {b}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x}{\sqrt {q+p x^4}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x}{\sqrt {q+p x^4}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}\\ &=\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^2}{q+p x^4}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {q+p x^4}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^2}{q+p x^4}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {q+p x^4}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {q+p x^4}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {q+p x^4}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}\\ &=\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {q+p x^4}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {q+p x^4}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^2}{q+p x^4}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {q+p x^4}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^2}{q+p x^4}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {q+p x^4}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}\\ \end {align*}
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Mathematica [C] time = 1.93, size = 420, normalized size = 1.19 \begin {gather*} -\frac {i \sqrt {\frac {p x^4}{q}+1} \left (-\Pi \left (-\frac {i \sqrt {2} \sqrt {q}}{\sqrt {p} \sqrt {-\frac {b-\sqrt {b+4 a p q} \sqrt {b}+2 a p q}{a p^2}}};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} x\right )\right |-1\right )-\Pi \left (\frac {i \sqrt {2} \sqrt {q}}{\sqrt {p} \sqrt {-\frac {b-\sqrt {b+4 a p q} \sqrt {b}+2 a p q}{a p^2}}};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} x\right )\right |-1\right )-\Pi \left (-\frac {i \sqrt {2} \sqrt {q}}{\sqrt {p} \sqrt {-\frac {b+\sqrt {b+4 a p q} \sqrt {b}+2 a p q}{a p^2}}};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} x\right )\right |-1\right )-\Pi \left (\frac {i \sqrt {2} \sqrt {q}}{\sqrt {p} \sqrt {-\frac {b+\sqrt {b+4 a p q} \sqrt {b}+2 a p q}{a p^2}}};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} x\right )\right |-1\right )+2 F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} x\right )\right |-1\right )\right )}{2 a \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} \sqrt {p x^4+q}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.63, size = 171, normalized size = 0.49 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x \sqrt {q+p x^4}}{\sqrt {a} q-\sqrt {b} x^2+\sqrt {a} p x^4}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^2}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{a} p x^4}{\sqrt {2} \sqrt [4]{b}}}{x \sqrt {q+p x^4}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 10.70, size = 496, normalized size = 1.41 \begin {gather*} -\frac {1}{2} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \arctan \left (-\frac {2 \, {\left (a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{5} + a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{4} + q} - {\left ({\left (a^{4} b p^{2} x^{8} + a^{4} b q^{2} + {\left (2 \, a^{4} b p q - a^{3} b^{2}\right )} x^{4}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}} + 2 \, {\left (a^{2} b p x^{6} + a^{2} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}}}{a p^{2} x^{8} + {\left (2 \, a p q + b\right )} x^{4} + a q^{2}}\right ) + \frac {1}{8} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {2 \, {\left (a^{2} b p x^{6} + a^{2} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}} + 2 \, {\left (p x^{5} - a b x^{3} \sqrt {-\frac {1}{a^{3} b}} + q x\right )} \sqrt {p x^{4} + q} - {\left (a p^{2} x^{8} + {\left (2 \, a p q - b\right )} x^{4} + a q^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}}}{2 \, {\left (a p^{2} x^{8} + {\left (2 \, a p q + b\right )} x^{4} + a q^{2}\right )}}\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, {\left (a^{2} b p x^{6} + a^{2} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}} - 2 \, {\left (p x^{5} - a b x^{3} \sqrt {-\frac {1}{a^{3} b}} + q x\right )} \sqrt {p x^{4} + q} - {\left (a p^{2} x^{8} + {\left (2 \, a p q - b\right )} x^{4} + a q^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}}}{2 \, {\left (a p^{2} x^{8} + {\left (2 \, a p q + b\right )} x^{4} + a q^{2}\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 0.20, size = 191, normalized size = 0.54
method | result | size |
default | \(\frac {\left (\frac {\ln \left (\frac {\frac {p \,x^{4}+q}{2 x^{2}}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{4}+q}\, \sqrt {2}}{2 x}+\frac {\sqrt {\frac {b}{a}}}{2}}{\frac {p \,x^{4}+q}{2 x^{2}}+\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{4}+q}\, \sqrt {2}}{2 x}+\frac {\sqrt {\frac {b}{a}}}{2}}\right )}{4 a \left (\frac {b}{a}\right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x}+1\right )}{2 a \left (\frac {b}{a}\right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x}-1\right )}{2 a \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(191\) |
elliptic | \(\frac {\left (\frac {\ln \left (\frac {\frac {p \,x^{4}+q}{2 x^{2}}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{4}+q}\, \sqrt {2}}{2 x}+\frac {\sqrt {\frac {b}{a}}}{2}}{\frac {p \,x^{4}+q}{2 x^{2}}+\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{4}+q}\, \sqrt {2}}{2 x}+\frac {\sqrt {\frac {b}{a}}}{2}}\right )}{4 a \left (\frac {b}{a}\right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x}+1\right )}{2 a \left (\frac {b}{a}\right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x}-1\right )}{2 a \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(191\) |
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 x^{4} + q} {\left (p x^{4} - q\right )}}{b x^{4} + {\left (p x^{4} + q\right )}^{2} a}\,{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\,x^4+q}\,\left (q-p\,x^4\right )}{a\,{\left (p\,x^4+q\right )}^2+b\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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