Optimal. Leaf size=167 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x \sqrt {p x^3+q}}{\sqrt {a} p x^3+\sqrt {a} q-\sqrt {b} x^2}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{a} p x^3}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^2}{\sqrt {2} \sqrt [4]{a}}}{x \sqrt {p x^3+q}}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}} \]
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Rubi [A] time = 0.46, antiderivative size = 242, normalized size of antiderivative = 1.45, 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^3+q}}\right )}{\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^3+q}}+1\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {p x^3+q}}+\sqrt {a}+\frac {\sqrt {b} x^2}{p x^3+q}\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^3+q}}+\sqrt {a}+\frac {\sqrt {b} x^2}{p x^3+q}\right )}{2 \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 (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^4+a \left (q+p x^3\right )^2} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\frac {x}{\sqrt {q+p x^3}}\right )\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^3}}\right )}{\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^3}}\right )}{\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^3}}\right )}{2 \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^3}}\right )}{2 \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^3}}\right )}{2 \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^3}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}\\ &=\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^2}{q+p x^3}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {q+p x^3}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^2}{q+p x^3}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {q+p x^3}}\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^3}}\right )}{\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^3}}\right )}{\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^3}}\right )}{\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^3}}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^2}{q+p x^3}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {q+p x^3}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^2}{q+p x^3}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {q+p x^3}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}\\ \end {align*}
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Mathematica [C] time = 6.51, size = 13567, normalized size = 81.24 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.19, size = 167, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x \sqrt {p x^3+q}}{\sqrt {a} p x^3+\sqrt {a} q-\sqrt {b} x^2}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{a} p x^3}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^2}{\sqrt {2} \sqrt [4]{a}}}{x \sqrt {p x^3+q}}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.19, size = 354, normalized size = 2.12 \begin {gather*} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \arctan \left (\frac {a^{2} b x \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}}{\sqrt {p x^{3} + q}}\right ) + \frac {1}{4} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{6} + 2 \, a p q x^{3} - b x^{4} + a q^{2} + 2 \, {\left (a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{4} + a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{3} + q} - 2 \, {\left (a^{2} b p x^{5} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}}\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{6} + 2 \, a p q x^{3} - b x^{4} + a q^{2} - 2 \, {\left (a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{4} + a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{3} + q} - 2 \, {\left (a^{2} b p x^{5} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}}\right ) \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 x^{3} + q} {\left (p x^{3} - 2 \, q\right )}}{b x^{4} + {\left (p x^{3} + q\right )}^{2} a}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 1164, normalized size = 6.97
result too large to display
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^{3} + q} {\left (p x^{3} - 2 \, q\right )}}{b x^{4} + {\left (p x^{3} + 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 = -0.01 \begin {gather*} \int -\frac {\sqrt {p\,x^3+q}\,\left (2\,q-p\,x^3\right )}{a\,{\left (p\,x^3+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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (p x^{3} - 2 q\right ) \sqrt {p x^{3} + q}}{a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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