Optimal. Leaf size=318 \[ \frac {\left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^3}-\frac {\left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^3}-\frac {\left (\sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^3}+\frac {\left (\sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^3}-\frac {c \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d^3} \]
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Rubi [A] time = 0.31, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {371, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ \frac {\left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^3}-\frac {\left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^3}-\frac {\left (\sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^3}+\frac {\left (\sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^3}-\frac {c \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d^3} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 275
Rule 371
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 1876
Rubi steps
\begin {align*} \int \frac {x^2}{a+b (c+d x)^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-c+x)^2}{a+b x^4} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {2 c x}{a+b x^4}+\frac {c^2+x^2}{a+b x^4}\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {c^2+x^2}{a+b x^4} \, dx,x,c+d x\right )}{d^3}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {x}{a+b x^4} \, dx,x,c+d x\right )}{d^3}\\ &=-\frac {c \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,(c+d x)^2\right )}{d^3}-\frac {\left (1-\frac {\sqrt {b} c^2}{\sqrt {a}}\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^3}+\frac {\left (1+\frac {\sqrt {b} c^2}{\sqrt {a}}\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^3}\\ &=-\frac {c \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d^3}+\frac {\left (\sqrt {a}-\sqrt {b} c^2\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^3}+\frac {\left (\sqrt {a}-\sqrt {b} c^2\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^3}+\frac {\left (1+\frac {\sqrt {b} c^2}{\sqrt {a}}\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^3}+\frac {\left (1+\frac {\sqrt {b} c^2}{\sqrt {a}}\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^3}\\ &=-\frac {c \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d^3}+\frac {\left (\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^3}-\frac {\left (\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^3}+\frac {\left (\sqrt {a}+\sqrt {b} c^2\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^3}-\frac {\left (\sqrt {a}+\sqrt {b} c^2\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^3}\\ &=-\frac {c \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d^3}-\frac {\left (\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^3}+\frac {\left (\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^3}+\frac {\left (\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^3}-\frac {\left (\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^3}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 106, normalized size = 0.33 \[ \frac {\text {RootSum}\left [\text {$\#$1}^4 b d^4+4 \text {$\#$1}^3 b c d^3+6 \text {$\#$1}^2 b c^2 d^2+4 \text {$\#$1} b c^3 d+a+b c^4\& ,\frac {\text {$\#$1}^2 \log (x-\text {$\#$1})}{\text {$\#$1}^3 d^3+3 \text {$\#$1}^2 c d^2+3 \text {$\#$1} c^2 d+c^3}\& \right ]}{4 b d} \]
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 \[ \int \frac {x^{2}}{{\left (d x + c\right )}^{4} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 97, normalized size = 0.31 \[ \frac {\RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{2} \ln \left (-\RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )+x \right )}{4 b d \left (d^{3} \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{3}+3 \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{2} c \,d^{2}+3 \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right ) c^{2} d +c^{3}\right )} \]
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 \[ \int \frac {x^{2}}{{\left (d x + c\right )}^{4} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 625, normalized size = 1.97 \[ \sum _{k=1}^4\ln \left (-b\,d^4\,\left (a+b\,c^4+4\,b\,c^3\,d\,x+\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )\,b^2\,c^5\,d^3\,4+\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )\,b^2\,c^4\,d^4\,x\,4-\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )\,a\,b\,c\,d^3\,20-\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )\,a\,b\,d^4\,x\,4+{\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )}^2\,a\,b^2\,c^2\,d^6\,48+{\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )}^2\,a\,b^2\,c\,d^7\,x\,32\right )\right )\,\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.67, size = 274, normalized size = 0.86 \[ \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{3} d^{12} + 192 t^{2} a^{2} b^{2} c^{2} d^{6} + t \left (- 32 a^{2} b c d^{3} + 32 a b^{2} c^{5} d^{3}\right ) + a^{2} + 2 a b c^{4} + b^{2} c^{8}, \left (t \mapsto t \log {\left (x + \frac {64 t^{3} a^{4} b^{2} d^{9} + 448 t^{3} a^{3} b^{3} c^{4} d^{9} + 160 t^{2} a^{3} b^{2} c^{3} d^{6} - 32 t^{2} a^{2} b^{3} c^{7} d^{6} + 60 t a^{3} b c^{2} d^{3} + 256 t a^{2} b^{2} c^{6} d^{3} + 4 t a b^{3} c^{10} d^{3} - 5 a^{3} c - 9 a^{2} b c^{5} - 3 a b^{2} c^{9} + b^{3} c^{13}}{a^{3} d - 33 a^{2} b c^{4} d - 33 a b^{2} c^{8} d + b^{3} c^{12} d} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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