Optimal. Leaf size=223 \[ -\frac {3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4096 c^{11/2}}+\frac {3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{2048 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {\left (-16 a c+21 b^2-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c} \]
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Rubi [A] time = 0.21, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1114, 742, 779, 612, 621, 206} \[ \frac {\left (-16 a c+21 b^2-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{2048 c^5}-\frac {3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4096 c^{11/2}}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 742
Rule 779
Rule 1114
Rubi steps
\begin {align*} \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^3 \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\operatorname {Subst}\left (\int x \left (-2 a-\frac {9 b x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{14 c}\\ &=\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}-\frac {\left (b \left (3 b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{32 c^3}\\ &=-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}+\frac {\left (3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^2\right )}{512 c^4}\\ &=\frac {3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{2048 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{4096 c^5}\\ &=\frac {3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{2048 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right )}{2048 c^5}\\ &=\frac {3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{2048 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}-\frac {3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4096 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 192, normalized size = 0.86 \[ \frac {-\frac {\left (16 a c-21 b^2+30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{40 c^2}+\frac {7 \left (4 a b c-3 b^3\right ) \left (2 \sqrt {c} \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4} \left (4 c \left (5 a+2 c x^4\right )-3 b^2+8 b c x^2\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )\right )}{2048 c^{9/2}}+x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 535, normalized size = 2.40 \[ \left [-\frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (5120 \, c^{7} x^{12} + 6400 \, b c^{6} x^{10} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{8} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{6} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{4} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{286720 \, c^{6}}, \frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, {\left (5120 \, c^{7} x^{12} + 6400 \, b c^{6} x^{10} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{8} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{6} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{4} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{143360 \, c^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 669, normalized size = 3.00 \[ \frac {1}{7680} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {7 \, b^{2} c^{2} - 16 \, a c^{3}}{c^{4}}\right )} x^{2} + \frac {35 \, b^{3} c - 116 \, a b c^{2}}{c^{4}}\right )} x^{2} - \frac {105 \, b^{4} - 460 \, a b^{2} c + 256 \, a^{2} c^{2}}{c^{4}}\right )} - \frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {9}{2}}}\right )} a + \frac {1}{30720} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {9 \, b^{2} c^{3} - 20 \, a c^{4}}{c^{5}}\right )} x^{2} + \frac {21 \, b^{3} c^{2} - 68 \, a b c^{3}}{c^{5}}\right )} x^{2} - \frac {105 \, b^{4} c - 448 \, a b^{2} c^{2} + 240 \, a^{2} c^{3}}{c^{5}}\right )} x^{2} + \frac {315 \, b^{5} - 1680 \, a b^{3} c + 1808 \, a^{2} b c^{2}}{c^{5}}\right )} + \frac {15 \, {\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {11}{2}}}\right )} b + \frac {1}{430080} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {11 \, b^{2} c^{4} - 24 \, a c^{5}}{c^{6}}\right )} x^{2} + \frac {99 \, b^{3} c^{3} - 316 \, a b c^{4}}{c^{6}}\right )} x^{2} - \frac {231 \, b^{4} c^{2} - 972 \, a b^{2} c^{3} + 512 \, a^{2} c^{4}}{c^{6}}\right )} x^{2} + \frac {1155 \, b^{5} c - 6048 \, a b^{3} c^{2} + 6352 \, a^{2} b c^{3}}{c^{6}}\right )} x^{2} - \frac {3465 \, b^{6} - 21840 \, a b^{4} c + 34608 \, a^{2} b^{2} c^{2} - 8192 \, a^{3} c^{3}}{c^{6}}\right )} - \frac {105 \, {\left (33 \, b^{7} - 252 \, a b^{5} c + 560 \, a^{2} b^{3} c^{2} - 320 \, a^{3} b c^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {13}{2}}}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 534, normalized size = 2.39 \[ \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, c \,x^{12}}{14}+\frac {5 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,x^{10}}{56}+\frac {4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,x^{8}}{35}+\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} x^{8}}{560 c}+\frac {11 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b \,x^{6}}{1120 c}-\frac {9 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3} x^{6}}{4480 c^{2}}+\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} x^{4}}{70 c}-\frac {31 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,b^{2} x^{4}}{2240 c^{2}}+\frac {3 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{4} x^{4}}{1280 c^{3}}-\frac {73 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} b \,x^{2}}{2240 c^{2}}+\frac {13 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,b^{3} x^{2}}{640 c^{3}}-\frac {3 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{5} x^{2}}{1024 c^{4}}+\frac {3 a^{3} b \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{64 c^{\frac {5}{2}}}-\frac {15 a^{2} b^{3} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{256 c^{\frac {7}{2}}}+\frac {21 a \,b^{5} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{1024 c^{\frac {9}{2}}}-\frac {9 b^{7} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4096 c^{\frac {11}{2}}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{3}}{35 c^{2}}+\frac {49 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} b^{2}}{640 c^{3}}-\frac {9 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,b^{4}}{256 c^{4}}+\frac {9 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{6}}{2048 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int x^7\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \]
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 \[ \int x^{7} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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