Optimal. Leaf size=365 \[ \frac {18 \sqrt {x^2-x+1} \sqrt {x+1} \left (91 a x+55 b x^2\right )}{5005}+\frac {2}{143} \sqrt {x^2-x+1} \left (x^3+1\right ) \sqrt {x+1} \left (13 a x+11 b x^2\right )+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} \left (91 a-55 \left (1-\sqrt {3}\right ) b\right ) F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{5005 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}+\frac {54 b \sqrt {x^2-x+1} \sqrt {x+1}}{91 \left (x+\sqrt {3}+1\right )}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{91 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )} \]
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Rubi [A] time = 0.21, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {809, 1853, 1878, 218, 1877} \[ \frac {18 \sqrt {x^2-x+1} \sqrt {x+1} \left (91 a x+55 b x^2\right )}{5005}+\frac {2}{143} \sqrt {x^2-x+1} \left (x^3+1\right ) \sqrt {x+1} \left (13 a x+11 b x^2\right )+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} \left (91 a-55 \left (1-\sqrt {3}\right ) b\right ) F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{5005 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}+\frac {54 b \sqrt {x^2-x+1} \sqrt {x+1}}{91 \left (x+\sqrt {3}+1\right )}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{91 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )} \]
Antiderivative was successfully verified.
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Rule 218
Rule 809
Rule 1853
Rule 1877
Rule 1878
Rubi steps
\begin {align*} \int (1+x)^{3/2} (a+b x) \left (1-x+x^2\right )^{3/2} \, dx &=\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \int (a+b x) \left (1+x^3\right )^{3/2} \, dx}{\sqrt {1+x^3}}\\ &=\frac {2}{143} \sqrt {1+x} \sqrt {1-x+x^2} \left (13 a x+11 b x^2\right ) \left (1+x^3\right )+\frac {\left (9 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \left (\frac {2 a}{11}+\frac {2 b x}{13}\right ) \sqrt {1+x^3} \, dx}{2 \sqrt {1+x^3}}\\ &=\frac {18 \sqrt {1+x} \sqrt {1-x+x^2} \left (91 a x+55 b x^2\right )}{5005}+\frac {2}{143} \sqrt {1+x} \sqrt {1-x+x^2} \left (13 a x+11 b x^2\right ) \left (1+x^3\right )+\frac {\left (27 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {\frac {4 a}{55}+\frac {4 b x}{91}}{\sqrt {1+x^3}} \, dx}{4 \sqrt {1+x^3}}\\ &=\frac {18 \sqrt {1+x} \sqrt {1-x+x^2} \left (91 a x+55 b x^2\right )}{5005}+\frac {2}{143} \sqrt {1+x} \sqrt {1-x+x^2} \left (13 a x+11 b x^2\right ) \left (1+x^3\right )+\frac {\left (27 b \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx}{91 \sqrt {1+x^3}}+\frac {\left (27 \left (91 a-55 \left (1-\sqrt {3}\right ) b\right ) \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{5005 \sqrt {1+x^3}}\\ &=\frac {54 b \sqrt {1+x} \sqrt {1-x+x^2}}{91 \left (1+\sqrt {3}+x\right )}+\frac {18 \sqrt {1+x} \sqrt {1-x+x^2} \left (91 a x+55 b x^2\right )}{5005}+\frac {2}{143} \sqrt {1+x} \sqrt {1-x+x^2} \left (13 a x+11 b x^2\right ) \left (1+x^3\right )-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{91 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} \left (91 a-55 \left (1-\sqrt {3}\right ) b\right ) (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{5005 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )}\\ \end {align*}
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Mathematica [C] time = 1.83, size = 437, normalized size = 1.20 \[ \frac {2 x \sqrt {x+1} \sqrt {x^2-x+1} \left (91 a \left (5 x^3+14\right )+55 b x \left (7 x^3+16\right )\right )}{5005}-\frac {9 (x+1)^{3/2} \left (\frac {\sqrt {2} \sqrt {\frac {-\frac {6 i}{x+1}+\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt {\frac {\frac {6 i}{x+1}+\sqrt {3}-3 i}{\sqrt {3}-3 i}} \left (55 \left (3-i \sqrt {3}\right ) b-182 i \sqrt {3} a\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {x+1}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {x+1}}-\frac {660 \sqrt {-\frac {i}{\sqrt {3}+3 i}} b \left (x^2-x+1\right )}{(x+1)^2}+\frac {165 i \sqrt {2} \left (\sqrt {3}+i\right ) b \sqrt {\frac {-\frac {6 i}{x+1}+\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt {\frac {\frac {6 i}{x+1}+\sqrt {3}-3 i}{\sqrt {3}-3 i}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {x+1}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {x+1}}\right )}{10010 \sqrt {-\frac {i}{\sqrt {3}+3 i}} \sqrt {x^2-x+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{4} + a x^{3} + b x + a\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}, x\right ) \]
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 \[ \int {\left (b x + a\right )} {\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.81, size = 608, normalized size = 1.67 \[ -\frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (-770 b \,x^{8}-910 a \,x^{7}-2530 b \,x^{5}-3458 a \,x^{4}-1760 b \,x^{2}-2548 a x +2457 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{i \sqrt {3}-3}}\, a \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )-7371 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{i \sqrt {3}-3}}\, a \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )+8910 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{i \sqrt {3}-3}}\, b \EllipticE \left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )-1485 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{i \sqrt {3}-3}}\, b \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )-4455 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{i \sqrt {3}-3}}\, b \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )\right )}{5005 \left (x^{3}+1\right )} \]
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 \[ \int {\left (b x + a\right )} {\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (x+1\right )}^{3/2}\,\left (a+b\,x\right )\,{\left (x^2-x+1\right )}^{3/2} \,d x \]
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 \[ \int \left (a + b x\right ) \left (x + 1\right )^{\frac {3}{2}} \left (x^{2} - x + 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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