Optimal. Leaf size=709 \[ -\frac {\sqrt {e} \left (4 a c-b^2\right )^{3/4} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac {\sqrt {e} \left (4 a c-b^2\right )^{3/4} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac {\left (b^2-4 a c\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (-\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 1.57, antiderivative size = 709, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {749, 748, 747, 401, 108, 409, 1213, 537, 444, 63, 212, 208, 205} \[ -\frac {\sqrt {e} \left (4 a c-b^2\right )^{3/4} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac {\sqrt {e} \left (4 a c-b^2\right )^{3/4} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac {\left (b^2-4 a c\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (-\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 63
Rule 108
Rule 205
Rule 208
Rule 212
Rule 401
Rule 409
Rule 444
Rule 537
Rule 747
Rule 748
Rule 749
Rule 1213
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx &=\frac {\left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \int \frac {1}{(d+e x) \left (-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}\right )^{3/4}} \, dx}{\left (a+b x+c x^2\right )^{3/4}}\\ &=\frac {\left (2 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {c (2 c d-b e)}{b^2-4 a c}+e x\right ) \left (1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4}} \, dx,x,-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )}{\left (a+b x+c x^2\right )^{3/4}}\\ &=-\frac {\left (2 \sqrt {2} e \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )}{\left (a+b x+c x^2\right )^{3/4}}-\frac {\left (2 \sqrt {2} c (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/4}}\\ &=-\frac {\left (\sqrt {2} e \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) x}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x\right )} \, dx,x,\left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2\right )}{\left (a+b x+c x^2\right )^{3/4}}-\frac {\left (\sqrt {2} c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {\left (b^2-4 a c\right ) x}{c^2}} \left (1-\frac {\left (b^2-4 a c\right ) x}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x\right )} \, dx,x,\left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2\right )}{\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac {\left (4 \sqrt {2} c^2 e \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c^2 e^2}{b^2-4 a c}+\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}+\frac {c^2 e^2 x^4}{b^2-4 a c}} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (4 \sqrt {2} c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^4} \left (-e^2+\frac {(2 c d-b e)^2}{b^2-4 a c}+e^2 x^4\right )} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac {\left (\sqrt {2} \left (b^2-4 a c\right ) e \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}-\sqrt {-b^2+4 a c} e x^2} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\sqrt {c} \sqrt {c d^2-b d e+a e^2} \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (\sqrt {2} \left (b^2-4 a c\right ) e \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}+\sqrt {-b^2+4 a c} e x^2} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\sqrt {c} \sqrt {c d^2-b d e+a e^2} \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (2 \sqrt {2} c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {-b^2+4 a c} e x^2}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}}\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (-e^2+\frac {(2 c d-b e)^2}{b^2-4 a c}\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (2 \sqrt {2} c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {-b^2+4 a c} e x^2}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}}\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (-e^2+\frac {(2 c d-b e)^2}{b^2-4 a c}\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}\\ &=-\frac {\left (-b^2+4 a c\right )^{3/4} \sqrt {e} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} \left (c d^2-b d e+a e^2\right )^{3/4} \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (-b^2+4 a c\right )^{3/4} \sqrt {e} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} \left (c d^2-b d e+a e^2\right )^{3/4} \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (2 \sqrt {2} c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\frac {\sqrt {-b^2+4 a c} e x^2}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}}\right )} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (-e^2+\frac {(2 c d-b e)^2}{b^2-4 a c}\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (2 \sqrt {2} c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\frac {\sqrt {-b^2+4 a c} e x^2}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}}\right )} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (-e^2+\frac {(2 c d-b e)^2}{b^2-4 a c}\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}\\ &=-\frac {\left (-b^2+4 a c\right )^{3/4} \sqrt {e} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} \left (c d^2-b d e+a e^2\right )^{3/4} \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (-b^2+4 a c\right )^{3/4} \sqrt {e} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} \left (c d^2-b d e+a e^2\right )^{3/4} \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} c \left (c d^2-b d e+a e^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} c \left (c d^2-b d e+a e^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.93, size = 529, normalized size = 0.75 \[ -\frac {\sqrt [4]{a+x (b+c x)} \left (\sqrt {2} \sqrt [4]{c} \sqrt {e} (b+2 c x) \sqrt [4]{e (a e-b d)+c d^2} \left (\tan ^{-1}\left (\frac {\sqrt {e} \sqrt [4]{4 a c-b^2} \sqrt [4]{\frac {c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt [4]{c} \sqrt [4]{e (a e-b d)+c d^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt [4]{4 a c-b^2} \sqrt [4]{\frac {c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt [4]{c} \sqrt [4]{e (a e-b d)+c d^2}}\right )\right )+\sqrt [4]{4 a c-b^2} \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left (-\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2+e (a e-b d)}};\left .\sin ^{-1}\left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{4 a c-b^2}}\right )\right |-1\right )+\sqrt [4]{4 a c-b^2} \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left (\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2+e (a e-b d)}};\left .\sin ^{-1}\left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{4 a c-b^2}}\right )\right |-1\right )\right )}{\sqrt {2} \sqrt [4]{4 a c-b^2} (b+2 c x) \sqrt [4]{\frac {c (a+x (b+c x))}{4 a c-b^2}} \left (e (a e-b d)+c d^2\right )} \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.72, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/4}} \,d x \]
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 \[ \int \frac {1}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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