Optimal. Leaf size=124 \[ \frac {\left (4 a b d+3 b^2 c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{8 a^{7/4}}+\frac {\left (-4 a b d-3 b^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{8 a^{7/4}}+\frac {\sqrt [4]{a x^4+b x^3} (4 a c x-4 a d+b c)}{4 a} \]
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Rubi [A] time = 0.29, antiderivative size = 193, normalized size of antiderivative = 1.56, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2039, 2021, 2032, 63, 331, 298, 203, 206} \begin {gather*} \frac {b x^{9/4} (a x+b)^{3/4} (4 a d+3 b c) \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{8 a^{7/4} \left (a x^4+b x^3\right )^{3/4}}-\frac {b x^{9/4} (a x+b)^{3/4} (4 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{8 a^{7/4} \left (a x^4+b x^3\right )^{3/4}}-\frac {\sqrt [4]{a x^4+b x^3} (4 a d+3 b c)}{4 a}+\frac {c \left (a x^4+b x^3\right )^{5/4}}{a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 298
Rule 331
Rule 2021
Rule 2032
Rule 2039
Rubi steps
\begin {align*} \int \frac {(-d+2 c x) \sqrt [4]{b x^3+a x^4}}{x} \, dx &=\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {\left (\frac {3 b c}{2}+2 a d\right ) \int \frac {\sqrt [4]{b x^3+a x^4}}{x} \, dx}{2 a}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {(b (3 b c+4 a d)) \int \frac {x^2}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{16 a}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {\left (b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{16 a \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {\left (b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{4 a \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {\left (b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4 a \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {\left (b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{3/2} \left (b x^3+a x^4\right )^{3/4}}+\frac {\left (b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{3/2} \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}+\frac {b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{7/4} \left (b x^3+a x^4\right )^{3/4}}-\frac {b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{7/4} \left (b x^3+a x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 81, normalized size = 0.65 \begin {gather*} \frac {\sqrt [4]{x^3 (a x+b)} \left (3 c (a x+b) \sqrt [4]{\frac {a x}{b}+1}-(4 a d+3 b c) \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-\frac {a x}{b}\right )\right )}{3 a \sqrt [4]{\frac {a x}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.67, size = 124, normalized size = 1.00 \begin {gather*} \frac {(b c-4 a d+4 a c x) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {\left (3 b^2 c+4 a b d\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{8 a^{7/4}}+\frac {\left (-3 b^2 c-4 a b d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{8 a^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 771, normalized size = 6.22 \begin {gather*} \frac {4 \, a \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {1}{4}} \arctan \left (\frac {a^{5} x \sqrt {\frac {a^{4} x^{2} \sqrt {\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}} + {\left (9 \, b^{4} c^{2} + 24 \, a b^{3} c d + 16 \, a^{2} b^{2} d^{2}\right )} \sqrt {a x^{4} + b x^{3}}}{x^{2}}} \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {3}{4}} - {\left (3 \, a^{5} b^{2} c + 4 \, a^{6} b d\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {3}{4}}}{{\left (81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}\right )} x}\right ) - a \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (3 \, b^{2} c + 4 \, a b d\right )}}{x}\right ) + a \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{2} x \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (3 \, b^{2} c + 4 \, a b d\right )}}{x}\right ) + 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (4 \, a c x + b c - 4 \, a d\right )}}{16 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 322, normalized size = 2.60 \begin {gather*} \frac {\frac {2 \, \sqrt {2} {\left (3 \, b^{3} c + 4 \, a b^{2} d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {2 \, \sqrt {2} {\left (3 \, b^{3} c + 4 \, a b^{2} d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {\sqrt {2} {\left (3 \, b^{3} c + 4 \, a b^{2} d\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a} - \frac {\sqrt {2} {\left (3 \, b^{3} c + 4 \, a b^{2} d\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {8 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} b^{3} c + 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a b^{3} c - 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a b^{2} d + 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{2} b^{2} d\right )} x^{2}}{a b^{2}}}{32 \, b} \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 \[\int \frac {\left (2 c x -d \right ) \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{x}\, dx\]
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 (2 \, c x - d\right )}}{x}\,{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}\,\left (d-2\,c\,x\right )}{x} \,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 (2 c x - d\right )}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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