Optimal. Leaf size=496 \[ -\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )+\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )+\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {\sqrt {b} c d \left (a-b c^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a+b c^4\right )^2}-\frac {1}{x \left (a+b c^4\right )}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2} \]
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Rubi [A] time = 0.89, antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {371, 6725, 1876, 1248, 635, 205, 260, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} d \left (\sqrt {b} c^2 \left (3 a-b c^4\right )+\sqrt {a} \left (a-3 b c^4\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} d \left (\sqrt {b} c^2 \left (3 a-b c^4\right )+\sqrt {a} \left (a-3 b c^4\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2}-\frac {\sqrt {b} c d \left (a-b c^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a+b c^4\right )^2}-\frac {1}{x \left (a+b c^4\right )} \]
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
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx &=d \operatorname {Subst}\left (\int \frac {1}{(-c+x)^2 \left (a+b x^4\right )} \, dx,x,c+d x\right )\\ &=d \operatorname {Subst}\left (\int \left (\frac {1}{\left (a+b c^4\right ) (c-x)^2}+\frac {4 b c^3}{\left (a+b c^4\right )^2 (c-x)}+\frac {b \left (-c^2 \left (3 a-b c^4\right )-2 c \left (a-b c^4\right ) x-\left (a-3 b c^4\right ) x^2+4 b c^3 x^3\right )}{\left (a+b c^4\right )^2 \left (a+b x^4\right )}\right ) \, dx,x,c+d x\right )\\ &=-\frac {1}{\left (a+b c^4\right ) x}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {(b d) \operatorname {Subst}\left (\int \frac {-c^2 \left (3 a-b c^4\right )-2 c \left (a-b c^4\right ) x-\left (a-3 b c^4\right ) x^2+4 b c^3 x^3}{a+b x^4} \, dx,x,c+d x\right )}{\left (a+b c^4\right )^2}\\ &=-\frac {1}{\left (a+b c^4\right ) x}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {(b d) \operatorname {Subst}\left (\int \left (\frac {x \left (-2 c \left (a-b c^4\right )+4 b c^3 x^2\right )}{a+b x^4}+\frac {-c^2 \left (3 a-b c^4\right )+\left (-a+3 b c^4\right ) x^2}{a+b x^4}\right ) \, dx,x,c+d x\right )}{\left (a+b c^4\right )^2}\\ &=-\frac {1}{\left (a+b c^4\right ) x}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {(b d) \operatorname {Subst}\left (\int \frac {x \left (-2 c \left (a-b c^4\right )+4 b c^3 x^2\right )}{a+b x^4} \, dx,x,c+d x\right )}{\left (a+b c^4\right )^2}+\frac {(b d) \operatorname {Subst}\left (\int \frac {-c^2 \left (3 a-b c^4\right )+\left (-a+3 b c^4\right ) x^2}{a+b x^4} \, dx,x,c+d x\right )}{\left (a+b c^4\right )^2}\\ &=-\frac {1}{\left (a+b c^4\right ) x}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {(b d) \operatorname {Subst}\left (\int \frac {-2 c \left (a-b c^4\right )+4 b c^3 x}{a+b x^2} \, dx,x,(c+d x)^2\right )}{2 \left (a+b c^4\right )^2}+\frac {\left (\left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \left (a+b c^4\right )^2}-\frac {\left (\left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \left (a+b c^4\right )^2}\\ &=-\frac {1}{\left (a+b c^4\right ) x}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {\left (2 b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,(c+d x)^2\right )}{\left (a+b c^4\right )^2}-\frac {\left (b c \left (a-b c^4\right ) d\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,(c+d x)^2\right )}{\left (a+b c^4\right )^2}-\frac {\left (\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\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} \sqrt [4]{a} \left (a+b c^4\right )^2}-\frac {\left (\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\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} \sqrt [4]{a} \left (a+b c^4\right )^2}-\frac {\left (\left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\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 \left (a+b c^4\right )^2}-\frac {\left (\left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\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 \left (a+b c^4\right )^2}\\ &=-\frac {1}{\left (a+b c^4\right ) x}-\frac {\sqrt {b} c \left (a-b c^4\right ) d \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a+b c^4\right )^2}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \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} \sqrt [4]{a} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \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} \sqrt [4]{a} \left (a+b c^4\right )^2}+\frac {b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2}-\frac {\left (\sqrt [4]{b} \left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\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} \sqrt [4]{a} \left (a+b c^4\right )^2}+\frac {\left (\sqrt [4]{b} \left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\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} \sqrt [4]{a} \left (a+b c^4\right )^2}\\ &=-\frac {1}{\left (a+b c^4\right ) x}-\frac {\sqrt {b} c \left (a-b c^4\right ) d \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} \left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} \left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (a+b c^4\right )^2}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \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} \sqrt [4]{a} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \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} \sqrt [4]{a} \left (a+b c^4\right )^2}+\frac {b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 238, normalized size = 0.48 \[ \frac {d x \text {RootSum}\left [\text {$\#$1}^4 b d^4+4 \text {$\#$1}^3 b c d^3+6 \text {$\#$1}^2 b c^2 d^2+4 \text {$\#$1} b c^3 d+a+b c^4\& ,\frac {4 \text {$\#$1}^3 b c^3 d^3 \log (x-\text {$\#$1})-\text {$\#$1}^2 a d^2 \log (x-\text {$\#$1})+15 \text {$\#$1}^2 b c^4 d^2 \log (x-\text {$\#$1})-6 a c^2 \log (x-\text {$\#$1})-4 \text {$\#$1} a c d \log (x-\text {$\#$1})+10 b c^6 \log (x-\text {$\#$1})+20 \text {$\#$1} b c^5 d \log (x-\text {$\#$1})}{\text {$\#$1}^3 d^3+3 \text {$\#$1}^2 c d^2+3 \text {$\#$1} c^2 d+c^3}\& \right ]-4 \left (a+b c^4+4 b c^3 d x \log (x)\right )}{4 x \left (a+b c^4\right )^2} \]
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 \[ \int \frac {1}{{\left ({\left (d x + c\right )}^{4} b + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 188, normalized size = 0.38 \[ -\frac {4 b \,c^{3} d \ln \relax (x )}{\left (b \,c^{4}+a \right )^{2}}+\frac {d \left (4 \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{3} b \,c^{3} d^{3}+10 b \,c^{6}+\left (15 b \,c^{4}-a \right ) \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{2} d^{2}+4 \left (5 b \,c^{4}-a \right ) \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right ) c d -6 a \,c^{2}\right ) \ln \left (-\RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )+x \right )}{4 \left (b \,c^{4}+a \right )^{2} \left (d^{3} \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{3}+3 \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{2} c \,d^{2}+3 \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right ) c^{2} d +c^{3}\right )}-\frac {1}{\left (b \,c^{4}+a \right ) x} \]
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 \[ -\frac {4 \, b c^{3} d \log \relax (x)}{b^{2} c^{8} + 2 \, a b c^{4} + a^{2}} + \frac {b d^{2} \int \frac {4 \, b c^{3} d^{3} x^{3} + 10 \, b c^{6} + {\left (15 \, b c^{4} - a\right )} d^{2} x^{2} - 6 \, a c^{2} + 4 \, {\left (5 \, b c^{5} - a c\right )} d x}{b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a}\,{d x}}{b^{2} c^{8} + 2 \, a b c^{4} + a^{2}} - \frac {1}{{\left (b c^{4} + a\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.48, size = 2440, normalized size = 4.92 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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