Optimal. Leaf size=356 \[ -\frac {c \left (3 \sqrt {a}-\sqrt {b} c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^4}+\frac {c \left (3 \sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^4}+\frac {c \left (3 \sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^4}-\frac {c \left (3 \sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^4}+\frac {3 c^2 \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} d^4}+\frac {\log \left (a+b (c+d x)^4\right )}{4 b d^4} \]
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Rubi [A] time = 0.43, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {371, 1876, 1248, 635, 205, 260, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {c \left (3 \sqrt {a}-\sqrt {b} c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^4}+\frac {c \left (3 \sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^4}+\frac {c \left (3 \sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^4}-\frac {c \left (3 \sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^4}+\frac {3 c^2 \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} d^4}+\frac {\log \left (a+b (c+d x)^4\right )}{4 b d^4} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 260
Rule 371
Rule 617
Rule 628
Rule 635
Rule 1162
Rule 1165
Rule 1168
Rule 1248
Rule 1876
Rubi steps
\begin {align*} \int \frac {x^3}{a+b (c+d x)^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-c+x)^3}{a+b x^4} \, dx,x,c+d x\right )}{d^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {x \left (3 c^2+x^2\right )}{a+b x^4}+\frac {-c^3-3 c x^2}{a+b x^4}\right ) \, dx,x,c+d x\right )}{d^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x \left (3 c^2+x^2\right )}{a+b x^4} \, dx,x,c+d x\right )}{d^4}+\frac {\operatorname {Subst}\left (\int \frac {-c^3-3 c x^2}{a+b x^4} \, dx,x,c+d x\right )}{d^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {3 c^2+x}{a+b x^2} \, dx,x,(c+d x)^2\right )}{2 d^4}+\frac {\left (c \left (3-\frac {\sqrt {b} c^2}{\sqrt {a}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 b d^4}-\frac {\left (c \left (3+\frac {\sqrt {b} c^2}{\sqrt {a}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 b d^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,(c+d x)^2\right )}{2 d^4}+\frac {\left (3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,(c+d x)^2\right )}{2 d^4}-\frac {\left (c \left (3 \sqrt {a}-\sqrt {b} c^2\right )\right ) \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,c+d x\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^4}-\frac {\left (c \left (3 \sqrt {a}-\sqrt {b} c^2\right )\right ) \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,c+d x\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^4}-\frac {\left (c \left (3+\frac {\sqrt {b} c^2}{\sqrt {a}}\right )\right ) \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,c+d x\right )}{4 b d^4}-\frac {\left (c \left (3+\frac {\sqrt {b} c^2}{\sqrt {a}}\right )\right ) \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,c+d x\right )}{4 b d^4}\\ &=\frac {3 c^2 \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} d^4}-\frac {c \left (3 \sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^4}+\frac {c \left (3 \sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^4}+\frac {\log \left (a+b (c+d x)^4\right )}{4 b d^4}-\frac {\left (c \left (3 \sqrt {a}+\sqrt {b} c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^4}+\frac {\left (c \left (3 \sqrt {a}+\sqrt {b} c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^4}\\ &=\frac {3 c^2 \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} d^4}+\frac {c \left (3 \sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^4}-\frac {c \left (3 \sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^4}-\frac {c \left (3 \sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^4}+\frac {c \left (3 \sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^4}+\frac {\log \left (a+b (c+d x)^4\right )}{4 b d^4}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 106, normalized size = 0.30 \[ \frac {\text {RootSum}\left [\text {$\#$1}^4 b d^4+4 \text {$\#$1}^3 b c d^3+6 \text {$\#$1}^2 b c^2 d^2+4 \text {$\#$1} b c^3 d+a+b c^4\& ,\frac {\text {$\#$1}^3 \log (x-\text {$\#$1})}{\text {$\#$1}^3 d^3+3 \text {$\#$1}^2 c d^2+3 \text {$\#$1} c^2 d+c^3}\& \right ]}{4 b d} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x^{3}}{{\left (d x + c\right )}^{4} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 97, normalized size = 0.27 \[ \frac {\RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{3} \ln \left (-\RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )+x \right )}{4 b d \left (d^{3} \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{3}+3 \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{2} c \,d^{2}+3 \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right ) c^{2} d +c^{3}\right )} \]
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 \[ \int \frac {x^{3}}{{\left (d x + c\right )}^{4} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.69, size = 1003, normalized size = 2.82 \[ \sum _{k=1}^4\ln \left (b\,c^2\,d\,\left (2\,a\,c+2\,b\,c^5-3\,a\,d\,x+5\,b\,c^4\,d\,x-\mathrm {root}\left (256\,a^3\,b^4\,d^{16}\,z^4-256\,a^3\,b^3\,d^{12}\,z^3+480\,a^2\,b^3\,c^4\,d^8\,z^2+96\,a^3\,b^2\,d^8\,z^2+192\,a^2\,b^2\,c^4\,d^4\,z-48\,a\,b^3\,c^8\,d^4\,z-16\,a^3\,b\,d^4\,z+3\,a\,b^2\,c^8+3\,a^2\,b\,c^4+b^3\,c^{12}+a^3,z,k\right )\,b^2\,c^5\,d^4\,2+{\mathrm {root}\left (256\,a^3\,b^4\,d^{16}\,z^4-256\,a^3\,b^3\,d^{12}\,z^3+480\,a^2\,b^3\,c^4\,d^8\,z^2+96\,a^3\,b^2\,d^8\,z^2+192\,a^2\,b^2\,c^4\,d^4\,z-48\,a\,b^3\,c^8\,d^4\,z-16\,a^3\,b\,d^4\,z+3\,a\,b^2\,c^8+3\,a^2\,b\,c^4+b^3\,c^{12}+a^3,z,k\right )}^2\,a\,b^2\,c\,d^8\,32+{\mathrm {root}\left (256\,a^3\,b^4\,d^{16}\,z^4-256\,a^3\,b^3\,d^{12}\,z^3+480\,a^2\,b^3\,c^4\,d^8\,z^2+96\,a^3\,b^2\,d^8\,z^2+192\,a^2\,b^2\,c^4\,d^4\,z-48\,a\,b^3\,c^8\,d^4\,z-16\,a^3\,b\,d^4\,z+3\,a\,b^2\,c^8+3\,a^2\,b\,c^4+b^3\,c^{12}+a^3,z,k\right )}^2\,a\,b^2\,d^9\,x\,24-\mathrm {root}\left (256\,a^3\,b^4\,d^{16}\,z^4-256\,a^3\,b^3\,d^{12}\,z^3+480\,a^2\,b^3\,c^4\,d^8\,z^2+96\,a^3\,b^2\,d^8\,z^2+192\,a^2\,b^2\,c^4\,d^4\,z-48\,a\,b^3\,c^8\,d^4\,z-16\,a^3\,b\,d^4\,z+3\,a\,b^2\,c^8+3\,a^2\,b\,c^4+b^3\,c^{12}+a^3,z,k\right )\,b^2\,c^4\,d^5\,x\,2+\mathrm {root}\left (256\,a^3\,b^4\,d^{16}\,z^4-256\,a^3\,b^3\,d^{12}\,z^3+480\,a^2\,b^3\,c^4\,d^8\,z^2+96\,a^3\,b^2\,d^8\,z^2+192\,a^2\,b^2\,c^4\,d^4\,z-48\,a\,b^3\,c^8\,d^4\,z-16\,a^3\,b\,d^4\,z+3\,a\,b^2\,c^8+3\,a^2\,b\,c^4+b^3\,c^{12}+a^3,z,k\right )\,a\,b\,c\,d^4\,38+\mathrm {root}\left (256\,a^3\,b^4\,d^{16}\,z^4-256\,a^3\,b^3\,d^{12}\,z^3+480\,a^2\,b^3\,c^4\,d^8\,z^2+96\,a^3\,b^2\,d^8\,z^2+192\,a^2\,b^2\,c^4\,d^4\,z-48\,a\,b^3\,c^8\,d^4\,z-16\,a^3\,b\,d^4\,z+3\,a\,b^2\,c^8+3\,a^2\,b\,c^4+b^3\,c^{12}+a^3,z,k\right )\,a\,b\,d^5\,x\,6\right )\,2\right )\,\mathrm {root}\left (256\,a^3\,b^4\,d^{16}\,z^4-256\,a^3\,b^3\,d^{12}\,z^3+480\,a^2\,b^3\,c^4\,d^8\,z^2+96\,a^3\,b^2\,d^8\,z^2+192\,a^2\,b^2\,c^4\,d^4\,z-48\,a\,b^3\,c^8\,d^4\,z-16\,a^3\,b\,d^4\,z+3\,a\,b^2\,c^8+3\,a^2\,b\,c^4+b^3\,c^{12}+a^3,z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.74, size = 374, normalized size = 1.05 \[ \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{4} d^{16} - 256 t^{3} a^{3} b^{3} d^{12} + t^{2} \left (96 a^{3} b^{2} d^{8} + 480 a^{2} b^{3} c^{4} d^{8}\right ) + t \left (- 16 a^{3} b d^{4} + 192 a^{2} b^{2} c^{4} d^{4} - 48 a b^{3} c^{8} d^{4}\right ) + a^{3} + 3 a^{2} b c^{4} + 3 a b^{2} c^{8} + b^{3} c^{12}, \left (t \mapsto t \log {\left (x + \frac {- 1728 t^{3} a^{4} b^{3} d^{12} - 960 t^{3} a^{3} b^{4} c^{4} d^{12} + 1296 t^{2} a^{4} b^{2} d^{8} + 2016 t^{2} a^{3} b^{3} c^{4} d^{8} - 48 t^{2} a^{2} b^{4} c^{8} d^{8} - 324 t a^{4} b d^{4} - 4716 t a^{3} b^{2} c^{4} d^{4} - 1452 t a^{2} b^{3} c^{8} d^{4} - 4 t a b^{4} c^{12} d^{4} + 27 a^{4} - 390 a^{3} b c^{4} - 444 a^{2} b^{2} c^{8} - 26 a b^{3} c^{12} + b^{4} c^{16}}{729 a^{3} b c^{3} d - 1053 a^{2} b^{2} c^{7} d - 117 a b^{3} c^{11} d + b^{4} c^{15} d} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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