Optimal. Leaf size=151 \[ \frac {2 a^4 (a+b x)^{7/2} (A b-a B)}{7 b^6}-\frac {2 a^3 (a+b x)^{9/2} (4 A b-5 a B)}{9 b^6}+\frac {4 a^2 (a+b x)^{11/2} (3 A b-5 a B)}{11 b^6}+\frac {2 (a+b x)^{15/2} (A b-5 a B)}{15 b^6}-\frac {4 a (a+b x)^{13/2} (2 A b-5 a B)}{13 b^6}+\frac {2 B (a+b x)^{17/2}}{17 b^6} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \begin {gather*} \frac {4 a^2 (a+b x)^{11/2} (3 A b-5 a B)}{11 b^6}-\frac {2 a^3 (a+b x)^{9/2} (4 A b-5 a B)}{9 b^6}+\frac {2 a^4 (a+b x)^{7/2} (A b-a B)}{7 b^6}+\frac {2 (a+b x)^{15/2} (A b-5 a B)}{15 b^6}-\frac {4 a (a+b x)^{13/2} (2 A b-5 a B)}{13 b^6}+\frac {2 B (a+b x)^{17/2}}{17 b^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin {align*} \int x^4 (a+b x)^{5/2} (A+B x) \, dx &=\int \left (-\frac {a^4 (-A b+a B) (a+b x)^{5/2}}{b^5}+\frac {a^3 (-4 A b+5 a B) (a+b x)^{7/2}}{b^5}-\frac {2 a^2 (-3 A b+5 a B) (a+b x)^{9/2}}{b^5}+\frac {2 a (-2 A b+5 a B) (a+b x)^{11/2}}{b^5}+\frac {(A b-5 a B) (a+b x)^{13/2}}{b^5}+\frac {B (a+b x)^{15/2}}{b^5}\right ) \, dx\\ &=\frac {2 a^4 (A b-a B) (a+b x)^{7/2}}{7 b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{9/2}}{9 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{11/2}}{11 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{13/2}}{13 b^6}+\frac {2 (A b-5 a B) (a+b x)^{15/2}}{15 b^6}+\frac {2 B (a+b x)^{17/2}}{17 b^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 106, normalized size = 0.70 \begin {gather*} \frac {2 (a+b x)^{7/2} \left (-1280 a^5 B+128 a^4 b (17 A+35 B x)-224 a^3 b^2 x (34 A+45 B x)+336 a^2 b^3 x^2 (51 A+55 B x)-462 a b^4 x^3 (68 A+65 B x)+3003 b^5 x^4 (17 A+15 B x)\right )}{765765 b^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.07, size = 137, normalized size = 0.91 \begin {gather*} \frac {2 (a+b x)^{7/2} \left (-109395 a^5 B+109395 a^4 A b+425425 a^4 B (a+b x)-340340 a^3 A b (a+b x)-696150 a^3 B (a+b x)^2+417690 a^2 A b (a+b x)^2+589050 a^2 B (a+b x)^3-235620 a A b (a+b x)^3+51051 A b (a+b x)^4-255255 a B (a+b x)^4+45045 B (a+b x)^5\right )}{765765 b^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.31, size = 192, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (45045 \, B b^{8} x^{8} - 1280 \, B a^{8} + 2176 \, A a^{7} b + 3003 \, {\left (35 \, B a b^{7} + 17 \, A b^{8}\right )} x^{7} + 231 \, {\left (275 \, B a^{2} b^{6} + 527 \, A a b^{7}\right )} x^{6} + 63 \, {\left (5 \, B a^{3} b^{5} + 1207 \, A a^{2} b^{6}\right )} x^{5} - 35 \, {\left (10 \, B a^{4} b^{4} - 17 \, A a^{3} b^{5}\right )} x^{4} + 40 \, {\left (10 \, B a^{5} b^{3} - 17 \, A a^{4} b^{4}\right )} x^{3} - 48 \, {\left (10 \, B a^{6} b^{2} - 17 \, A a^{5} b^{3}\right )} x^{2} + 64 \, {\left (10 \, B a^{7} b - 17 \, A a^{6} b^{2}\right )} x\right )} \sqrt {b x + a}}{765765 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.38, size = 708, normalized size = 4.69
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 119, normalized size = 0.79 \begin {gather*} \frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (45045 B \,b^{5} x^{5}+51051 A \,b^{5} x^{4}-30030 B a \,b^{4} x^{4}-31416 A a \,b^{4} x^{3}+18480 B \,a^{2} b^{3} x^{3}+17136 A \,a^{2} b^{3} x^{2}-10080 B \,a^{3} b^{2} x^{2}-7616 A \,a^{3} b^{2} x +4480 B \,a^{4} b x +2176 A \,a^{4} b -1280 B \,a^{5}\right )}{765765 b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.90, size = 123, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (45045 \, {\left (b x + a\right )}^{\frac {17}{2}} B - 51051 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {15}{2}} + 117810 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {13}{2}} - 139230 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 109395 \, {\left (B a^{5} - A a^{4} b\right )} {\left (b x + a\right )}^{\frac {7}{2}}\right )}}{765765 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.37, size = 137, normalized size = 0.91 \begin {gather*} \frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{13/2}}{13\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{17/2}}{17\,b^6}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{15/2}}{15\,b^6}-\frac {\left (2\,B\,a^5-2\,A\,a^4\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6}+\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 20.85, size = 586, normalized size = 3.88 \begin {gather*} \frac {2 A a^{2} \left (\frac {a^{4} \left (a + b x\right )^{\frac {3}{2}}}{3} - \frac {4 a^{3} \left (a + b x\right )^{\frac {5}{2}}}{5} + \frac {6 a^{2} \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {4 a \left (a + b x\right )^{\frac {9}{2}}}{9} + \frac {\left (a + b x\right )^{\frac {11}{2}}}{11}\right )}{b^{5}} + \frac {4 A a \left (- \frac {a^{5} \left (a + b x\right )^{\frac {3}{2}}}{3} + a^{4} \left (a + b x\right )^{\frac {5}{2}} - \frac {10 a^{3} \left (a + b x\right )^{\frac {7}{2}}}{7} + \frac {10 a^{2} \left (a + b x\right )^{\frac {9}{2}}}{9} - \frac {5 a \left (a + b x\right )^{\frac {11}{2}}}{11} + \frac {\left (a + b x\right )^{\frac {13}{2}}}{13}\right )}{b^{5}} + \frac {2 A \left (\frac {a^{6} \left (a + b x\right )^{\frac {3}{2}}}{3} - \frac {6 a^{5} \left (a + b x\right )^{\frac {5}{2}}}{5} + \frac {15 a^{4} \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {20 a^{3} \left (a + b x\right )^{\frac {9}{2}}}{9} + \frac {15 a^{2} \left (a + b x\right )^{\frac {11}{2}}}{11} - \frac {6 a \left (a + b x\right )^{\frac {13}{2}}}{13} + \frac {\left (a + b x\right )^{\frac {15}{2}}}{15}\right )}{b^{5}} + \frac {2 B a^{2} \left (- \frac {a^{5} \left (a + b x\right )^{\frac {3}{2}}}{3} + a^{4} \left (a + b x\right )^{\frac {5}{2}} - \frac {10 a^{3} \left (a + b x\right )^{\frac {7}{2}}}{7} + \frac {10 a^{2} \left (a + b x\right )^{\frac {9}{2}}}{9} - \frac {5 a \left (a + b x\right )^{\frac {11}{2}}}{11} + \frac {\left (a + b x\right )^{\frac {13}{2}}}{13}\right )}{b^{6}} + \frac {4 B a \left (\frac {a^{6} \left (a + b x\right )^{\frac {3}{2}}}{3} - \frac {6 a^{5} \left (a + b x\right )^{\frac {5}{2}}}{5} + \frac {15 a^{4} \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {20 a^{3} \left (a + b x\right )^{\frac {9}{2}}}{9} + \frac {15 a^{2} \left (a + b x\right )^{\frac {11}{2}}}{11} - \frac {6 a \left (a + b x\right )^{\frac {13}{2}}}{13} + \frac {\left (a + b x\right )^{\frac {15}{2}}}{15}\right )}{b^{6}} + \frac {2 B \left (- \frac {a^{7} \left (a + b x\right )^{\frac {3}{2}}}{3} + \frac {7 a^{6} \left (a + b x\right )^{\frac {5}{2}}}{5} - 3 a^{5} \left (a + b x\right )^{\frac {7}{2}} + \frac {35 a^{4} \left (a + b x\right )^{\frac {9}{2}}}{9} - \frac {35 a^{3} \left (a + b x\right )^{\frac {11}{2}}}{11} + \frac {21 a^{2} \left (a + b x\right )^{\frac {13}{2}}}{13} - \frac {7 a \left (a + b x\right )^{\frac {15}{2}}}{15} + \frac {\left (a + b x\right )^{\frac {17}{2}}}{17}\right )}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________