Optimal. Leaf size=422 \[ \frac{x^2 \left (560 a^2 c^2-568 a b^2 c+99 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{b x \left (2416 a^2 c^2-1560 a b^2 c+231 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{\left (18896 a^2 b^2 c^2-6720 a^3 c^3-8988 a b^4 c+1155 b^6\right ) \sqrt{a x^2+b x^3+c x^4}}{286720 c^5}-\frac{b \left (81648 a^2 b^2 c^2-58816 a^3 c^3-30660 a b^4 c+3465 b^6\right ) \sqrt{a x^2+b x^3+c x^4}}{573440 c^6 x}+\frac{3 x \left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{13/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{x^3 \left (10 c x \left (11 b^2-28 a c\right )+b \left (68 a c+11 b^2\right )\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c} \]
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Rubi [A] time = 1.20339, antiderivative size = 422, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {1919, 1945, 1949, 12, 1914, 621, 206} \[ \frac{x^2 \left (560 a^2 c^2-568 a b^2 c+99 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{b x \left (2416 a^2 c^2-1560 a b^2 c+231 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{\left (18896 a^2 b^2 c^2-6720 a^3 c^3-8988 a b^4 c+1155 b^6\right ) \sqrt{a x^2+b x^3+c x^4}}{286720 c^5}-\frac{b \left (81648 a^2 b^2 c^2-58816 a^3 c^3-30660 a b^4 c+3465 b^6\right ) \sqrt{a x^2+b x^3+c x^4}}{573440 c^6 x}+\frac{3 x \left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{13/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{x^3 \left (10 c x \left (11 b^2-28 a c\right )+b \left (68 a c+11 b^2\right )\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c} \]
Antiderivative was successfully verified.
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Rule 1919
Rule 1945
Rule 1949
Rule 12
Rule 1914
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx &=\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac{3 \int x^2 \left (-4 a b-\frac{1}{2} \left (11 b^2-28 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4} \, dx}{112 c}\\ &=-\frac{x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac{\int \frac{x^4 \left (2 a b \left (11 b^2-52 a c\right )+\frac{1}{4} \left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{2240 c^2}\\ &=\frac{\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}-\frac{\int \frac{x^3 \left (\frac{3}{4} a \left (99 b^4-568 a b^2 c+560 a^2 c^2\right )+\frac{3}{8} b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{8960 c^3}\\ &=-\frac{b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac{\int \frac{x^2 \left (\frac{3}{4} a b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right )+\frac{3}{16} \left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{26880 c^4}\\ &=\frac{\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt{a x^2+b x^3+c x^4}}{286720 c^5}-\frac{b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}-\frac{\int \frac{x \left (\frac{3}{16} a \left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right )+\frac{3}{32} b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{53760 c^5}\\ &=\frac{\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt{a x^2+b x^3+c x^4}}{286720 c^5}-\frac{b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) \sqrt{a x^2+b x^3+c x^4}}{573440 c^6 x}-\frac{b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac{\int \frac{315 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) x}{64 \sqrt{a x^2+b x^3+c x^4}} \, dx}{53760 c^6}\\ &=\frac{\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt{a x^2+b x^3+c x^4}}{286720 c^5}-\frac{b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) \sqrt{a x^2+b x^3+c x^4}}{573440 c^6 x}-\frac{b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac{\left (3 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right )\right ) \int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{32768 c^6}\\ &=\frac{\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt{a x^2+b x^3+c x^4}}{286720 c^5}-\frac{b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) \sqrt{a x^2+b x^3+c x^4}}{573440 c^6 x}-\frac{b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac{\left (3 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{32768 c^6 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt{a x^2+b x^3+c x^4}}{286720 c^5}-\frac{b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) \sqrt{a x^2+b x^3+c x^4}}{573440 c^6 x}-\frac{b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac{\left (3 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) x \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{16384 c^6 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt{a x^2+b x^3+c x^4}}{286720 c^5}-\frac{b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) \sqrt{a x^2+b x^3+c x^4}}{573440 c^6 x}-\frac{b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac{3 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{13/2} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.426287, size = 236, normalized size = 0.56 \[ \frac{\left (x^2 (a+x (b+c x))\right )^{3/2} \left (\frac{\left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{4096 c^{11/2} x^3 (a+x (b+c x))^{3/2}}+\frac{\left (372 a b c-280 a c^2 x+330 b^2 c x-231 b^3\right ) (a+x (b+c x))}{560 c^3 x^3}-\frac{11 b (a+x (b+c x))}{14 c x}+a+b x+c x^2\right )}{8 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 649, normalized size = 1.5 \begin{align*}{\frac{1}{1146880\,{x}^{3}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 26880\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{4}{c}^{5}+3465\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{8}c+143360\,{x}^{3} \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{13/2}-59136\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{7/2}{b}^{3}+18480\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{5/2}{b}^{5}-6930\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}{b}^{7}-112640\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{11/2}{x}^{2}b-71680\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{11/2}xa+84480\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{9/2}x{b}^{2}+95232\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{9/2}ab+17920\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{11/2}x{a}^{2}-134400\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{3}{b}^{2}{c}^{4}+117600\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}{b}^{4}{c}^{3}-35280\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{6}{c}^{2}-40320\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}a{b}^{3}+26880\,\sqrt{c{x}^{2}+bx+a}{c}^{11/2}x{a}^{3}-13860\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}x{b}^{6}+13440\,\sqrt{c{x}^{2}+bx+a}{c}^{9/2}{a}^{3}b-63840\,\sqrt{c{x}^{2}+bx+a}{c}^{7/2}{a}^{2}{b}^{3}+42840\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}a{b}^{5}+36960\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}x{b}^{4}+8960\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{9/2}{a}^{2}b-80640\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{9/2}xa{b}^{2}-127680\,\sqrt{c{x}^{2}+bx+a}{c}^{9/2}x{a}^{2}{b}^{2}+85680\,\sqrt{c{x}^{2}+bx+a}{c}^{7/2}xa{b}^{4} \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{15}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03112, size = 1600, normalized size = 3.79 \begin{align*} \left [\frac{105 \,{\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt{c} x \log \left (-\frac{8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{c} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \,{\left (71680 \, c^{8} x^{7} + 87040 \, b c^{7} x^{6} - 3465 \, b^{7} c + 30660 \, a b^{5} c^{2} - 81648 \, a^{2} b^{3} c^{3} + 58816 \, a^{3} b c^{4} + 1280 \,{\left (b^{2} c^{6} + 84 \, a c^{7}\right )} x^{5} - 128 \,{\left (11 \, b^{3} c^{5} - 52 \, a b c^{6}\right )} x^{4} + 16 \,{\left (99 \, b^{4} c^{4} - 568 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{3} - 8 \,{\left (231 \, b^{5} c^{3} - 1560 \, a b^{3} c^{4} + 2416 \, a^{2} b c^{5}\right )} x^{2} + 2 \,{\left (1155 \, b^{6} c^{2} - 8988 \, a b^{4} c^{3} + 18896 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{2293760 \, c^{7} x}, -\frac{105 \,{\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \,{\left (71680 \, c^{8} x^{7} + 87040 \, b c^{7} x^{6} - 3465 \, b^{7} c + 30660 \, a b^{5} c^{2} - 81648 \, a^{2} b^{3} c^{3} + 58816 \, a^{3} b c^{4} + 1280 \,{\left (b^{2} c^{6} + 84 \, a c^{7}\right )} x^{5} - 128 \,{\left (11 \, b^{3} c^{5} - 52 \, a b c^{6}\right )} x^{4} + 16 \,{\left (99 \, b^{4} c^{4} - 568 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{3} - 8 \,{\left (231 \, b^{5} c^{3} - 1560 \, a b^{3} c^{4} + 2416 \, a^{2} b c^{5}\right )} x^{2} + 2 \,{\left (1155 \, b^{6} c^{2} - 8988 \, a b^{4} c^{3} + 18896 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{1146880 \, c^{7} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29441, size = 703, normalized size = 1.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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