Optimal. Leaf size=187 \[ \frac {4 \sqrt [4]{b x^3+a x^4}}{d x}+\frac {\text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\& ,\frac {b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\& \right ]}{2 d^2} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(406\) vs. \(2(187)=374\).
time = 0.94, antiderivative size = 406, normalized size of antiderivative = 2.17, number of steps
used = 13, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2081, 922, 37,
6857, 95, 304, 211, 214} \begin {gather*} \frac {\sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \text {ArcTan}\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}} \text {ArcTan}\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 95
Rule 211
Rule 214
Rule 304
Rule 922
Rule 2081
Rule 6857
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.00, size = 198, normalized size = 1.06 \begin {gather*} \frac {32 d x^2 (b+a x)-x^{9/4} (b+a x)^{3/4} \text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-4 b^2 c \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{8 d^2 \left (x^3 (b+a x)\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, 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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 0.38, size = 730, normalized size = 3.90 \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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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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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 102.30, size = 329, normalized size = 1.76 \begin {gather*} -\frac {2 \, \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + 2 \, \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) + \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, d} + \frac {4 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}}{d} \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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