Optimal. Leaf size=326 \[ \frac {2}{35} \sqrt {x^2-x+1} \sqrt {x+1} \left (7 a x+5 b x^2\right )+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} \left (7 a-5 \left (1-\sqrt {3}\right ) b\right ) F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{35 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}+\frac {6 b \sqrt {x^2-x+1} \sqrt {x+1}}{7 \left (x+\sqrt {3}+1\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{7 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {2}{35} \sqrt {x^2-x+1} \sqrt {x+1} \left (7 a x+5 b x^2\right )+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} \left (7 a-5 \left (1-\sqrt {3}\right ) b\right ) F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{35 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}+\frac {6 b \sqrt {x^2-x+1} \sqrt {x+1}}{7 \left (x+\sqrt {3}+1\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{7 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 218
Rule 809
Rule 1853
Rule 1877
Rule 1878
Rubi steps
\begin {align*} \int \sqrt {1+x} (a+b x) \sqrt {1-x+x^2} \, dx &=\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \int (a+b x) \sqrt {1+x^3} \, dx}{\sqrt {1+x^3}}\\ &=\frac {2}{35} \sqrt {1+x} \sqrt {1-x+x^2} \left (7 a x+5 b x^2\right )+\frac {\left (3 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {\frac {2 a}{5}+\frac {2 b x}{7}}{\sqrt {1+x^3}} \, dx}{2 \sqrt {1+x^3}}\\ &=\frac {2}{35} \sqrt {1+x} \sqrt {1-x+x^2} \left (7 a x+5 b x^2\right )+\frac {\left (3 b \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx}{7 \sqrt {1+x^3}}+\frac {\left (3 \left (7 a-5 \left (1-\sqrt {3}\right ) b\right ) \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{35 \sqrt {1+x^3}}\\ &=\frac {6 b \sqrt {1+x} \sqrt {1-x+x^2}}{7 \left (1+\sqrt {3}+x\right )}+\frac {2}{35} \sqrt {1+x} \sqrt {1-x+x^2} \left (7 a x+5 b x^2\right )-\frac {3 \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 )}{7 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} \left (7 a-5 \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 )}{35 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.60, size = 423, normalized size = 1.30 \[ \frac {2}{35} x \sqrt {x+1} \sqrt {x^2-x+1} (7 a+5 b x)-\frac {(x+1)^{3/2} \left (\frac {\sqrt {2} \sqrt {\frac {-\frac {6 i}{x+1}+\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt {\frac {\frac {6 i}{x+1}+\sqrt {3}-3 i}{\sqrt {3}-3 i}} \left (5 \left (3-i \sqrt {3}\right ) b-14 i \sqrt {3} a\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {x+1}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {x+1}}-\frac {60 \sqrt {-\frac {i}{\sqrt {3}+3 i}} b \left (x^2-x+1\right )}{(x+1)^2}+\frac {15 i \sqrt {2} \left (\sqrt {3}+i\right ) b \sqrt {\frac {-\frac {6 i}{x+1}+\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt {\frac {\frac {6 i}{x+1}+\sqrt {3}-3 i}{\sqrt {3}-3 i}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {x+1}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {x+1}}\right )}{70 \sqrt {-\frac {i}{\sqrt {3}+3 i}} \sqrt {x^2-x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.96, size = 596, normalized size = 1.83 \[ -\frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (-10 b \,x^{5}-14 a \,x^{4}-10 b \,x^{2}-14 a x +21 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{i \sqrt {3}-3}}\, a \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )-63 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{i \sqrt {3}-3}}\, a \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )+90 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{i \sqrt {3}-3}}\, b \EllipticE \left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )-15 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{i \sqrt {3}-3}}\, b \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )-45 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{i \sqrt {3}-3}}\, b \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )\right )}{35 \left (x^{3}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {x+1}\,\left (a+b\,x\right )\,\sqrt {x^2-x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b x\right ) \sqrt {x + 1} \sqrt {x^{2} - x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________