Integrand size = 22, antiderivative size = 1153 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 (d+e x) (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}-\frac {3 \left (4 b^2 d e+4 a c d e-5 b \left (c d^2+a e^2\right )-\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {3 \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (b+2 c x)}{2 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac {3^{3/4} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right ),-7-4 \sqrt {3}\right )}{2^{5/6} c^{2/3} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}} \]
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Time = 0.91 (sec) , antiderivative size = 1153, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {752, 652, 637, 309, 224, 1891} \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (b+2 c x)}{2 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}-\frac {3 \left (4 d e b^2-5 \left (c d^2+a e^2\right ) b+4 a c d e-\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{c x^2+b x+a}}-\frac {3 (d+e x) (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^{5/3} \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} (b+2 c x)}-\frac {3^{3/4} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right ),-7-4 \sqrt {3}\right )}{2^{5/6} c^{2/3} \left (b^2-4 a c\right )^{5/3} \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} (b+2 c x)} \]
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Rule 224
Rule 309
Rule 637
Rule 652
Rule 752
Rule 1891
Rubi steps \begin{align*} \text {integral}& = -\frac {3 (d+e x) (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}-\frac {3 \int \frac {\frac {2}{3} \left (5 c d^2-e (4 b d-3 a e)\right )+\frac {2}{3} e (2 c d-b e) x}{\left (a+b x+c x^2\right )^{4/3}} \, dx}{4 \left (b^2-4 a c\right )} \\ & = -\frac {3 (d+e x) (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}-\frac {3 \left (4 b^2 d e+4 a c d e-5 b \left (c d^2+a e^2\right )-\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \int \frac {1}{\sqrt [3]{a+b x+c x^2}} \, dx}{2 \left (b^2-4 a c\right )^2} \\ & = -\frac {3 (d+e x) (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}-\frac {3 \left (4 b^2 d e+4 a c d e-5 b \left (c d^2+a e^2\right )-\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {\left (3 \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{2 \left (b^2-4 a c\right )^2 (b+2 c x)} \\ & = -\frac {3 (d+e x) (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}-\frac {3 \left (4 b^2 d e+4 a c d e-5 b \left (c d^2+a e^2\right )-\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {\left (3 \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} x}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{2\ 2^{2/3} \sqrt [3]{c} \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac {\left (3 \left (1-\sqrt {3}\right ) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{2\ 2^{2/3} \sqrt [3]{c} \left (b^2-4 a c\right )^{5/3} (b+2 c x)} \\ & = -\frac {3 (d+e x) (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}-\frac {3 \left (4 b^2 d e+4 a c d e-5 b \left (c d^2+a e^2\right )-\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {3 \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (b+2 c x)}{2 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac {3^{3/4} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right )|-7-4 \sqrt {3}\right )}{2^{5/6} c^{2/3} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.62 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.26 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\frac {3 c \left (-b^3 \left (d^2+8 d e x-3 e^2 x^2\right )+b^2 \left (2 a e (-3 d+7 e x)+2 c x \left (4 d^2-15 d e x+e^2 x^2\right )\right )+4 c \left (5 c^2 d^2 x^3+a^2 e (-4 d+e x)+a c x \left (7 d^2+3 e^2 x^2\right )\right )+2 b \left (5 a^2 e^2+5 c^2 d x^2 (3 d-2 e x)+a c \left (7 d^2-14 d e x+9 e^2 x^2\right )\right )\right )-2^{2/3} \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right ) (b+2 c x) (a+x (b+c x)) \sqrt [3]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{4 c \left (b^2-4 a c\right )^2 (a+x (b+c x))^{4/3}} \]
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\[\int \frac {\left (e x +d \right )^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{3}}}d x\]
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\[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \]
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\[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (a + b x + c x^{2}\right )^{\frac {7}{3}}}\, dx \]
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\[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \]
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\[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{{\left (c\,x^2+b\,x+a\right )}^{7/3}} \,d x \]
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