Optimal. Leaf size=129 \[ -\frac {8 b^3 (c+d x)^{9/2} (b c-a d)}{9 d^5}+\frac {12 b^2 (c+d x)^{7/2} (b c-a d)^2}{7 d^5}-\frac {8 b (c+d x)^{5/2} (b c-a d)^3}{5 d^5}+\frac {2 (c+d x)^{3/2} (b c-a d)^4}{3 d^5}+\frac {2 b^4 (c+d x)^{11/2}}{11 d^5} \]
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Rubi [A] time = 0.05, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \[ -\frac {8 b^3 (c+d x)^{9/2} (b c-a d)}{9 d^5}+\frac {12 b^2 (c+d x)^{7/2} (b c-a d)^2}{7 d^5}-\frac {8 b (c+d x)^{5/2} (b c-a d)^3}{5 d^5}+\frac {2 (c+d x)^{3/2} (b c-a d)^4}{3 d^5}+\frac {2 b^4 (c+d x)^{11/2}}{11 d^5} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x)^4 \sqrt {c+d x} \, dx &=\int \left (\frac {(-b c+a d)^4 \sqrt {c+d x}}{d^4}-\frac {4 b (b c-a d)^3 (c+d x)^{3/2}}{d^4}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{5/2}}{d^4}-\frac {4 b^3 (b c-a d) (c+d x)^{7/2}}{d^4}+\frac {b^4 (c+d x)^{9/2}}{d^4}\right ) \, dx\\ &=\frac {2 (b c-a d)^4 (c+d x)^{3/2}}{3 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{5/2}}{5 d^5}+\frac {12 b^2 (b c-a d)^2 (c+d x)^{7/2}}{7 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{9/2}}{9 d^5}+\frac {2 b^4 (c+d x)^{11/2}}{11 d^5}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 101, normalized size = 0.78 \[ \frac {2 (c+d x)^{3/2} \left (-1540 b^3 (c+d x)^3 (b c-a d)+2970 b^2 (c+d x)^2 (b c-a d)^2-2772 b (c+d x) (b c-a d)^3+1155 (b c-a d)^4+315 b^4 (c+d x)^4\right )}{3465 d^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 245, normalized size = 1.90 \[ \frac {2 \, {\left (315 \, b^{4} d^{5} x^{5} + 128 \, b^{4} c^{5} - 704 \, a b^{3} c^{4} d + 1584 \, a^{2} b^{2} c^{3} d^{2} - 1848 \, a^{3} b c^{2} d^{3} + 1155 \, a^{4} c d^{4} + 35 \, {\left (b^{4} c d^{4} + 44 \, a b^{3} d^{5}\right )} x^{4} - 10 \, {\left (4 \, b^{4} c^{2} d^{3} - 22 \, a b^{3} c d^{4} - 297 \, a^{2} b^{2} d^{5}\right )} x^{3} + 6 \, {\left (8 \, b^{4} c^{3} d^{2} - 44 \, a b^{3} c^{2} d^{3} + 99 \, a^{2} b^{2} c d^{4} + 462 \, a^{3} b d^{5}\right )} x^{2} - {\left (64 \, b^{4} c^{4} d - 352 \, a b^{3} c^{3} d^{2} + 792 \, a^{2} b^{2} c^{2} d^{3} - 924 \, a^{3} b c d^{4} - 1155 \, a^{4} d^{5}\right )} x\right )} \sqrt {d x + c}}{3465 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.25, size = 470, normalized size = 3.64 \[ \frac {2 \, {\left (3465 \, \sqrt {d x + c} a^{4} c + 1155 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{4} + \frac {4620 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{3} b c}{d} + \frac {1386 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a^{2} b^{2} c}{d^{2}} + \frac {924 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a^{3} b}{d} + \frac {396 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a b^{3} c}{d^{3}} + \frac {594 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a^{2} b^{2}}{d^{2}} + \frac {11 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} b^{4} c}{d^{4}} + \frac {44 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} a b^{3}}{d^{3}} + \frac {5 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} b^{4}}{d^{4}}\right )}}{3465 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 186, normalized size = 1.44 \[ \frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (315 b^{4} x^{4} d^{4}+1540 a \,b^{3} d^{4} x^{3}-280 b^{4} c \,d^{3} x^{3}+2970 a^{2} b^{2} d^{4} x^{2}-1320 a \,b^{3} c \,d^{3} x^{2}+240 b^{4} c^{2} d^{2} x^{2}+2772 a^{3} b \,d^{4} x -2376 a^{2} b^{2} c \,d^{3} x +1056 a \,b^{3} c^{2} d^{2} x -192 b^{4} c^{3} d x +1155 a^{4} d^{4}-1848 a^{3} b c \,d^{3}+1584 a^{2} b^{2} c^{2} d^{2}-704 a \,b^{3} c^{3} d +128 b^{4} c^{4}\right )}{3465 d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 181, normalized size = 1.40 \[ \frac {2 \, {\left (315 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{4} - 1540 \, {\left (b^{4} c - a b^{3} d\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 2970 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 2772 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{3465 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 112, normalized size = 0.87 \[ \frac {2\,b^4\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}-\frac {\left (8\,b^4\,c-8\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5}+\frac {2\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{3/2}}{3\,d^5}+\frac {12\,b^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}+\frac {8\,b\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{5/2}}{5\,d^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.19, size = 223, normalized size = 1.73 \[ \frac {2 \left (\frac {b^{4} \left (c + d x\right )^{\frac {11}{2}}}{11 d^{4}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (4 a b^{3} d - 4 b^{4} c\right )}{9 d^{4}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (6 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 6 b^{4} c^{2}\right )}{7 d^{4}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (4 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 12 a b^{3} c^{2} d - 4 b^{4} c^{3}\right )}{5 d^{4}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{3 d^{4}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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