Optimal. Leaf size=303 \[ \frac {a^2 b^2 x^{10} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {b^4 x^{12} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{12 (a+b x)}+\frac {5 a b^3 x^{11} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{11 (a+b x)}+\frac {b^5 B x^{13} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac {a^5 A x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {a^4 x^8 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{8 (a+b x)}+\frac {5 a^3 b x^9 \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 76} \[ \frac {b^4 x^{12} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{12 (a+b x)}+\frac {5 a b^3 x^{11} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{11 (a+b x)}+\frac {a^2 b^2 x^{10} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {5 a^3 b x^9 \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 (a+b x)}+\frac {a^4 x^8 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{8 (a+b x)}+\frac {a^5 A x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {b^5 B x^{13} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 76
Rule 770
Rubi steps
\begin {align*} \int x^6 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^6 \left (a b+b^2 x\right )^5 (A+B x) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a^5 A b^5 x^6+a^4 b^5 (5 A b+a B) x^7+5 a^3 b^6 (2 A b+a B) x^8+10 a^2 b^7 (A b+a B) x^9+5 a b^8 (A b+2 a B) x^{10}+b^9 (A b+5 a B) x^{11}+b^{10} B x^{12}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {a^5 A x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {a^4 (5 A b+a B) x^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac {5 a^3 b (2 A b+a B) x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {a^2 b^2 (A b+a B) x^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^3 (A b+2 a B) x^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {b^4 (A b+5 a B) x^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{12 (a+b x)}+\frac {b^5 B x^{13} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 125, normalized size = 0.41 \[ \frac {x^7 \sqrt {(a+b x)^2} \left (1287 a^5 (8 A+7 B x)+5005 a^4 b x (9 A+8 B x)+8008 a^3 b^2 x^2 (10 A+9 B x)+6552 a^2 b^3 x^3 (11 A+10 B x)+2730 a b^4 x^4 (12 A+11 B x)+462 b^5 x^5 (13 A+12 B x)\right )}{72072 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 118, normalized size = 0.39 \[ \frac {1}{13} \, B b^{5} x^{13} + \frac {1}{7} \, A a^{5} x^{7} + \frac {1}{12} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{12} + \frac {5}{11} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{11} + {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{10} + \frac {5}{9} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 220, normalized size = 0.73 \[ \frac {1}{13} \, B b^{5} x^{13} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{12} \, B a b^{4} x^{12} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{12} \, A b^{5} x^{12} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{11} \, B a^{2} b^{3} x^{11} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{11} \, A a b^{4} x^{11} \mathrm {sgn}\left (b x + a\right ) + B a^{3} b^{2} x^{10} \mathrm {sgn}\left (b x + a\right ) + A a^{2} b^{3} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{9} \, B a^{4} b x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, A a^{3} b^{2} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{8} \, B a^{5} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, A a^{4} b x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, A a^{5} x^{7} \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (7 \, B a^{13} - 13 \, A a^{12} b\right )} \mathrm {sgn}\left (b x + a\right )}{72072 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 140, normalized size = 0.46 \[ \frac {\left (5544 B \,b^{5} x^{6}+6006 x^{5} A \,b^{5}+30030 x^{5} B a \,b^{4}+32760 x^{4} A a \,b^{4}+65520 x^{4} B \,a^{2} b^{3}+72072 A \,a^{2} b^{3} x^{3}+72072 B \,a^{3} b^{2} x^{3}+80080 x^{2} A \,a^{3} b^{2}+40040 x^{2} B \,a^{4} b +45045 x A \,a^{4} b +9009 x B \,a^{5}+10296 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x^{7}}{72072 \left (b x +a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.66, size = 481, normalized size = 1.59 \[ \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B x^{6}}{13 \, b^{2}} - \frac {19 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a x^{5}}{156 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A x^{5}}{12 \, b^{2}} + \frac {251 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{2} x^{4}}{1716 \, b^{4}} - \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a x^{4}}{132 \, b^{3}} - \frac {68 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{3} x^{3}}{429 \, b^{5}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a^{2} x^{3}}{33 \, b^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{7} x}{6 \, b^{7}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a^{6} x}{6 \, b^{6}} + \frac {211 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{4} x^{2}}{1287 \, b^{6}} - \frac {16 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a^{3} x^{2}}{99 \, b^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{8}}{6 \, b^{8}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a^{7}}{6 \, b^{7}} - \frac {1709 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{5} x}{10296 \, b^{7}} + \frac {131 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a^{4} x}{792 \, b^{6}} + \frac {1715 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{6}}{10296 \, b^{8}} - \frac {923 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a^{5}}{5544 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^6\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{6} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________