Optimal. Leaf size=241 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8 (-d)+2 \text {$\#$1}^4 a d-a^2 d+b^2 c\& ,\frac {-\text {$\#$1}^4 a d \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 a d \log (x)+a^2 d \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-b^2 c \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-a^2 d \log (x)+b^2 c \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{d}+\frac {4 \sqrt [4]{a x^4+b x^3}}{x}-2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )+2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right ) \]
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Rubi [B] time = 1.76, antiderivative size = 519, normalized size of antiderivative = 2.15, number of steps used = 21, number of rules used = 13, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2056, 6725, 47, 63, 331, 298, 203, 206, 908, 37, 93, 205, 208} \begin {gather*} \frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}+\frac {4 \sqrt [4]{a x^4+b x^3}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 63
Rule 93
Rule 203
Rule 205
Rule 206
Rule 208
Rule 298
Rule 331
Rule 908
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {\sqrt [4]{b+a x} \left (d+c x^2\right )}{x^{5/4} \left (-d+c x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {\sqrt [4]{b+a x}}{x^{5/4}}+\frac {2 d \sqrt [4]{b+a x}}{x^{5/4} \left (-d+c x^2\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {\sqrt [4]{b+a x}}{x^{5/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 d \sqrt [4]{b x^3+a x^4}\right ) \int \frac {\sqrt [4]{b+a x}}{x^{5/4} \left (-d+c x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {-a d-b c x}{\sqrt [4]{x} (b+a x)^{3/4} \left (-d+c x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{x^{5/4} (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \int \left (-\frac {-b \sqrt {c} d-a d^{3/2}}{2 d \sqrt [4]{x} (b+a x)^{3/4} \left (\sqrt {d}-\sqrt {c} x\right )}-\frac {b \sqrt {c} d-a d^{3/2}}{2 d \sqrt [4]{x} (b+a x)^{3/4} \left (\sqrt {d}+\sqrt {c} x\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 a \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (4 a \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (b \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4} \left (\sqrt {d}+\sqrt {c} x\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (b \sqrt {c}+a \sqrt {d}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4} \left (\sqrt {d}-\sqrt {c} x\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (2 \sqrt {a} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 \sqrt {a} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 \left (b \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {d}-\left (-b \sqrt {c}+a \sqrt {d}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 \left (b \sqrt {c}+a \sqrt {d}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {d}-\left (b \sqrt {c}+a \sqrt {d}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (b \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{d}-\sqrt {-b \sqrt {c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {-b \sqrt {c}+a \sqrt {d}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 \left (b \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{d}+\sqrt {-b \sqrt {c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {-b \sqrt {c}+a \sqrt {d}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 \sqrt {b \sqrt {c}+a \sqrt {d}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{d}-\sqrt {b \sqrt {c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \sqrt {b \sqrt {c}+a \sqrt {d}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{d}+\sqrt {b \sqrt {c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{-b \sqrt {c}+a \sqrt {d}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-b \sqrt {c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{b+a x}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{b \sqrt {c}+a \sqrt {d}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{b \sqrt {c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{b+a x}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \sqrt [4]{-b \sqrt {c}+a \sqrt {d}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-b \sqrt {c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{b+a x}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \sqrt [4]{b \sqrt {c}+a \sqrt {d}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{b \sqrt {c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{b+a x}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 177, normalized size = 0.73 \begin {gather*} \frac {4 \sqrt [4]{x^3 (a x+b)} \left (\frac {x \left (b \sqrt {c}-a \sqrt {d}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (a-\frac {b \sqrt {c}}{\sqrt {d}}\right ) x}{b+a x}\right )-x \left (a \sqrt {d}+b \sqrt {c}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (a+\frac {b \sqrt {c}}{\sqrt {d}}\right ) x}{b+a x}\right )+6 \sqrt {d} (a x+b)}{\sqrt {d} (a x+b)}-\frac {3 \, _2F_1\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};-\frac {a x}{b}\right )}{\sqrt [4]{\frac {a x}{b}+1}}\right )}{3 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 240, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{b x^3+a x^4}}{x}-2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\frac {\text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {-b^2 c \log (x)+a^2 d \log (x)+b^2 c \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-a^2 d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-a d \log (x) \text {$\#$1}^4+a d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{d} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 687, normalized size = 2.85 \begin {gather*} -\frac {4 \, {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \arctan \left (\frac {{\left (a d x - \sqrt {\frac {b^{2} c}{d}} d x\right )} {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {3}{4}} \sqrt {\frac {\sqrt {a + \sqrt {\frac {b^{2} c}{d}}} x^{2} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a d - \sqrt {\frac {b^{2} c}{d}} d\right )} {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {3}{4}}}{{\left (b^{2} c - a^{2} d\right )} x}\right ) - 4 \, {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \arctan \left (-\frac {{\left (a d x + \sqrt {\frac {b^{2} c}{d}} d x\right )} {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {3}{4}} \sqrt {\frac {\sqrt {a - \sqrt {\frac {b^{2} c}{d}}} x^{2} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a d + \sqrt {\frac {b^{2} c}{d}} d\right )} {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {3}{4}}}{{\left (b^{2} c - a^{2} d\right )} x}\right ) + 4 \, a^{\frac {1}{4}} x \arctan \left (\frac {a^{\frac {3}{4}} x \sqrt {\frac {\sqrt {a} x^{2} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} a^{\frac {3}{4}}}{a x}\right ) + {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - a^{\frac {1}{4}} x \log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + a^{\frac {1}{4}} x \log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x} \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*} \mathit {sage}_{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+d \right ) \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{x^{2} \left (c \,x^{2}-d \right )}\, dx\]
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 {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (c x^{2} + d\right )}}{{\left (c x^{2} - d\right )} x^{2}}\,{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 {\left (c\,x^2+d\right )\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x^2\,\left (d-c\,x^2\right )} \,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 {\sqrt [4]{x^{3} \left (a x + b\right )} \left (c x^{2} + d\right )}{x^{2} \left (c x^{2} - d\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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