Optimal. Leaf size=306 \[ -\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac {5 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac {5 a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 306, 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 = {784, 77}
\begin {gather*} -\frac {5 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^8 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{6 x^6 (a+b x)}-\frac {5 a b^3 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{7 x^7 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{10 x^{10} (a+b x)}-\frac {5 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 x^9 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rule 784
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{x^{12}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^5 A b^5}{x^{12}}+\frac {a^4 b^5 (5 A b+a B)}{x^{11}}+\frac {5 a^3 b^6 (2 A b+a B)}{x^{10}}+\frac {10 a^2 b^7 (A b+a B)}{x^9}+\frac {5 a b^8 (A b+2 a B)}{x^8}+\frac {b^9 (A b+5 a B)}{x^7}+\frac {b^{10} B}{x^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac {5 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac {5 a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 125, normalized size = 0.41 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (462 b^5 x^5 (5 A+6 B x)+1650 a b^4 x^4 (6 A+7 B x)+2475 a^2 b^3 x^3 (7 A+8 B x)+1925 a^3 b^2 x^2 (8 A+9 B x)+770 a^4 b x (9 A+10 B x)+126 a^5 (10 A+11 B x)\right )}{13860 x^{11} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.73, size = 140, normalized size = 0.46
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} B \,x^{6}}{5}+\left (-\frac {1}{6} b^{5} A -\frac {5}{6} a \,b^{4} B \right ) x^{5}+\left (-\frac {5}{7} a \,b^{4} A -\frac {10}{7} a^{2} b^{3} B \right ) x^{4}+\left (-\frac {5}{4} a^{2} b^{3} A -\frac {5}{4} a^{3} b^{2} B \right ) x^{3}+\left (-\frac {10}{9} a^{3} b^{2} A -\frac {5}{9} a^{4} b B \right ) x^{2}+\left (-\frac {1}{2} a^{4} b A -\frac {1}{10} a^{5} B \right ) x -\frac {a^{5} A}{11}\right )}{\left (b x +a \right ) x^{11}}\) | \(136\) |
gosper | \(-\frac {\left (2772 b^{5} B \,x^{6}+2310 A \,b^{5} x^{5}+11550 B a \,b^{4} x^{5}+9900 A a \,b^{4} x^{4}+19800 B \,a^{2} b^{3} x^{4}+17325 A \,a^{2} b^{3} x^{3}+17325 B \,a^{3} b^{2} x^{3}+15400 A \,a^{3} b^{2} x^{2}+7700 B \,a^{4} b \,x^{2}+6930 a^{4} b A x +1386 a^{5} B x +1260 a^{5} A \right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{13860 x^{11} \left (b x +a \right )^{5}}\) | \(140\) |
default | \(-\frac {\left (2772 b^{5} B \,x^{6}+2310 A \,b^{5} x^{5}+11550 B a \,b^{4} x^{5}+9900 A a \,b^{4} x^{4}+19800 B \,a^{2} b^{3} x^{4}+17325 A \,a^{2} b^{3} x^{3}+17325 B \,a^{3} b^{2} x^{3}+15400 A \,a^{3} b^{2} x^{2}+7700 B \,a^{4} b \,x^{2}+6930 a^{4} b A x +1386 a^{5} B x +1260 a^{5} A \right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{13860 x^{11} \left (b x +a \right )^{5}}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 675 vs.
\(2 (215) = 430\).
time = 0.29, size = 675, normalized size = 2.21 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{10}}{6 \, a^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{11}}{6 \, a^{11}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{9}}{6 \, a^{9} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{10}}{6 \, a^{10} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{8}}{6 \, a^{10} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{9}}{6 \, a^{11} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{7}}{6 \, a^{9} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{8}}{6 \, a^{10} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{6}}{6 \, a^{8} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{7}}{6 \, a^{9} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{6 \, a^{7} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{6 \, a^{8} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{6 \, a^{6} x^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{6 \, a^{7} x^{6}} + \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{1260 \, a^{5} x^{7}} - \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{2772 \, a^{6} x^{7}} - \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{180 \, a^{4} x^{8}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{396 \, a^{5} x^{8}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{90 \, a^{3} x^{9}} - \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{198 \, a^{4} x^{9}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{10 \, a^{2} x^{10}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{22 \, a^{3} x^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{11 \, a^{2} x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.37, size = 119, normalized size = 0.39 \begin {gather*} -\frac {2772 \, B b^{5} x^{6} + 1260 \, A a^{5} + 2310 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 9900 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 17325 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 7700 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 1386 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{13860 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{12}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.23, size = 221, normalized size = 0.72 \begin {gather*} -\frac {{\left (11 \, B a b^{10} - 5 \, A b^{11}\right )} \mathrm {sgn}\left (b x + a\right )}{13860 \, a^{6}} - \frac {2772 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 11550 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 2310 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 19800 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 9900 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 17325 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 17325 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 7700 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 15400 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1386 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 6930 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 1260 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{13860 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.39, size = 284, normalized size = 0.93 \begin {gather*} -\frac {\left (\frac {B\,a^5}{10}+\frac {A\,b\,a^4}{2}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^{10}\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{6}+\frac {5\,B\,a\,b^4}{6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^6\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{11\,x^{11}\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,x^5\,\left (a+b\,x\right )}-\frac {5\,a\,b^3\,\left (A\,b+2\,B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {5\,a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^8\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________