Optimal. Leaf size=191 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8 (-d)+2 \text {$\#$1}^4 a d-a^2 d+b^2 c\& ,\frac {-\text {$\#$1}^4 a d \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 a d \log (x)+a^2 d \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )-b^2 c \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )-a^2 d \log (x)+b^2 c \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{2 d^2}+\frac {4 \sqrt [4]{a x^4-b x^3}}{d x} \]
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Rubi [B] time = 1.46, antiderivative size = 427, normalized size of antiderivative = 2.24, number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2056, 908, 37, 6725, 93, 298, 205, 208} \begin {gather*} \frac {\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 )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}+\frac {\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 )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}-\frac {\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 )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}-\frac {\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 )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}+\frac {4 \sqrt [4]{a x^4-b x^3}}{d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 93
Rule 205
Rule 208
Rule 298
Rule 908
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\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}}{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 \frac {-a d+b c x}{\sqrt [4]{x} (-b+a x)^{3/4} \left (-d+c x^2\right )} \, dx}{d x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (b \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{x^{5/4} (-b+a x)^{3/4}} \, dx}{d x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}-\frac {\sqrt [4]{-b x^3+a x^4} \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}{d x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {4 \sqrt [4]{-b x^3+a x^4}}{d 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}{2 d 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}{2 d x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {4 \sqrt [4]{-b x^3+a x^4}}{d 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 {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 )}{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 {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 )}{d x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}+\frac {\left (\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}} d 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 ) \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}} d x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\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 )}{d x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (\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 )}{d x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}+\frac {\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 )}{d^{9/8} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\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 )}{d^{9/8} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\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 )}{d^{9/8} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\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 )}{d^{9/8} x^{3/4} \sqrt [4]{-b+a x}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 144, normalized size = 0.75 \begin {gather*} \frac {2 x^2 \left (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} (b-a x)\right )}{3 d^{3/2} \left (x^3 (a x-b)\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.63, size = 190, normalized size = 0.99 \begin {gather*} \frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}+\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 ]}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 739, normalized size = 3.87 \begin {gather*} -\frac {4 \, d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (d^{8} x \sqrt {\frac {b^{2} c}{d^{9}}} - a d^{4} x\right )} \sqrt {\frac {d^{2} x^{2} \sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {3}{4}} - {\left (d^{8} \sqrt {\frac {b^{2} c}{d^{9}}} - a d^{4}\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {3}{4}}}{{\left (b^{2} c - a^{2} d\right )} x}\right ) - 4 \, d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (d^{8} x \sqrt {\frac {b^{2} c}{d^{9}}} + a d^{4} x\right )} \sqrt {\frac {d^{2} x^{2} \sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {3}{4}} - {\left (d^{8} \sqrt {\frac {b^{2} c}{d^{9}}} + a d^{4}\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {3}{4}}}{{\left (b^{2} c - a^{2} d\right )} x}\right ) + d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 8 \, {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{2 \, d 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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-b \,x^{3}\right )^{\frac {1}{4}}}{x^{2} \left (c \,x^{2}-d \right )}\, dx\]
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 )} 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 = -0.01 \begin {gather*} -\int \frac {{\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 )}}{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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