Optimal. Leaf size=182 \[ \frac{2}{5} a^2 x^{5/2} (a B+3 A b)+\frac{2}{3} a^3 A x^{3/2}+\frac{2}{11} x^{11/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{6}{13} c x^{13/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{9} x^{9/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{6}{7} a x^{7/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{15} c^2 x^{15/2} (A c+3 b B)+\frac{2}{17} B c^3 x^{17/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11492, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {765} \[ \frac{2}{5} a^2 x^{5/2} (a B+3 A b)+\frac{2}{3} a^3 A x^{3/2}+\frac{2}{11} x^{11/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{6}{13} c x^{13/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{9} x^{9/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{6}{7} a x^{7/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{15} c^2 x^{15/2} (A c+3 b B)+\frac{2}{17} B c^3 x^{17/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin{align*} \int \sqrt{x} (A+B x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A \sqrt{x}+a^2 (3 A b+a B) x^{3/2}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^{5/2}+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{7/2}+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{9/2}+3 c \left (b^2 B+A b c+a B c\right ) x^{11/2}+c^2 (3 b B+A c) x^{13/2}+B c^3 x^{15/2}\right ) \, dx\\ &=\frac{2}{3} a^3 A x^{3/2}+\frac{2}{5} a^2 (3 A b+a B) x^{5/2}+\frac{6}{7} a \left (a b B+A \left (b^2+a c\right )\right ) x^{7/2}+\frac{2}{9} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{9/2}+\frac{2}{11} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{11/2}+\frac{6}{13} c \left (b^2 B+A b c+a B c\right ) x^{13/2}+\frac{2}{15} c^2 (3 b B+A c) x^{15/2}+\frac{2}{17} B c^3 x^{17/2}\\ \end{align*}
Mathematica [A] time = 0.20607, size = 178, normalized size = 0.98 \[ \frac{2 x^{3/2} \left (7293 a^2 x (9 A (7 b+5 c x)+5 B x (9 b+7 c x))+51051 a^3 (5 A+3 B x)+255 a x^2 \left (13 A \left (99 b^2+154 b c x+63 c^2 x^2\right )+7 B x \left (143 b^2+234 b c x+99 c^2 x^2\right )\right )+7 x^3 \left (17 A \left (1755 b^2 c x+715 b^3+1485 b c^2 x^2+429 c^3 x^3\right )+9 B x \left (2805 b^2 c x+1105 b^3+2431 b c^2 x^2+715 c^3 x^3\right )\right )\right )}{765765} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 192, normalized size = 1.1 \begin{align*}{\frac{90090\,B{c}^{3}{x}^{7}+102102\,A{c}^{3}{x}^{6}+306306\,B{x}^{6}b{c}^{2}+353430\,A{x}^{5}b{c}^{2}+353430\,aB{c}^{2}{x}^{5}+353430\,B{x}^{5}{b}^{2}c+417690\,aA{c}^{2}{x}^{4}+417690\,A{x}^{4}{b}^{2}c+835380\,B{x}^{4}abc+139230\,B{x}^{4}{b}^{3}+1021020\,A{x}^{3}abc+170170\,A{b}^{3}{x}^{3}+510510\,{a}^{2}Bc{x}^{3}+510510\,B{x}^{3}a{b}^{2}+656370\,{a}^{2}Ac{x}^{2}+656370\,A{x}^{2}a{b}^{2}+656370\,B{x}^{2}{a}^{2}b+918918\,A{a}^{2}bx+306306\,{a}^{3}Bx+510510\,A{a}^{3}}{765765}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.15005, size = 224, normalized size = 1.23 \begin{align*} \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{15} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{15}{2}} + \frac{6}{13} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{11}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} + \frac{2}{9} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac{9}{2}} + \frac{6}{7} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.00826, size = 424, normalized size = 2.33 \begin{align*} \frac{2}{765765} \,{\left (45045 \, B c^{3} x^{8} + 51051 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + 176715 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{6} + 69615 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{5} + 255255 \, A a^{3} x + 85085 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{4} + 328185 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} + 153153 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 8.20165, size = 216, normalized size = 1.19 \begin{align*} \frac{2 A a^{3} x^{\frac{3}{2}}}{3} + \frac{2 B c^{3} x^{\frac{17}{2}}}{17} + \frac{2 x^{\frac{15}{2}} \left (A c^{3} + 3 B b c^{2}\right )}{15} + \frac{2 x^{\frac{13}{2}} \left (3 A b c^{2} + 3 B a c^{2} + 3 B b^{2} c\right )}{13} + \frac{2 x^{\frac{11}{2}} \left (3 A a c^{2} + 3 A b^{2} c + 6 B a b c + B b^{3}\right )}{11} + \frac{2 x^{\frac{9}{2}} \left (6 A a b c + A b^{3} + 3 B a^{2} c + 3 B a b^{2}\right )}{9} + \frac{2 x^{\frac{7}{2}} \left (3 A a^{2} c + 3 A a b^{2} + 3 B a^{2} b\right )}{7} + \frac{2 x^{\frac{5}{2}} \left (3 A a^{2} b + B a^{3}\right )}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.29825, size = 261, normalized size = 1.43 \begin{align*} \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{5} \, B b c^{2} x^{\frac{15}{2}} + \frac{2}{15} \, A c^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B b^{2} c x^{\frac{13}{2}} + \frac{6}{13} \, B a c^{2} x^{\frac{13}{2}} + \frac{6}{13} \, A b c^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B b^{3} x^{\frac{11}{2}} + \frac{12}{11} \, B a b c x^{\frac{11}{2}} + \frac{6}{11} \, A b^{2} c x^{\frac{11}{2}} + \frac{6}{11} \, A a c^{2} x^{\frac{11}{2}} + \frac{2}{3} \, B a b^{2} x^{\frac{9}{2}} + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}} + \frac{2}{3} \, B a^{2} c x^{\frac{9}{2}} + \frac{4}{3} \, A a b c x^{\frac{9}{2}} + \frac{6}{7} \, B a^{2} b x^{\frac{7}{2}} + \frac{6}{7} \, A a b^{2} x^{\frac{7}{2}} + \frac{6}{7} \, A a^{2} c x^{\frac{7}{2}} + \frac{2}{5} \, B a^{3} x^{\frac{5}{2}} + \frac{6}{5} \, A a^{2} b x^{\frac{5}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]