Optimal. Leaf size=171 \[ -\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {128 d^3 (c+d x)^{3/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}-\frac {256 d^4 (c+d x)^{3/2}}{3465 (b c-a d)^5 (a+b x)^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37}
\begin {gather*} -\frac {256 d^4 (c+d x)^{3/2}}{3465 (a+b x)^{3/2} (b c-a d)^5}+\frac {128 d^3 (c+d x)^{3/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac {32 d^2 (c+d x)^{3/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac {16 d (c+d x)^{3/2}}{99 (a+b x)^{9/2} (b c-a d)^2}-\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx &=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}-\frac {(8 d) \int \frac {\sqrt {c+d x}}{(a+b x)^{11/2}} \, dx}{11 (b c-a d)}\\ &=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}+\frac {\left (16 d^2\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^{9/2}} \, dx}{33 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}-\frac {\left (64 d^3\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx}{231 (b c-a d)^3}\\ &=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {128 d^3 (c+d x)^{3/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}+\frac {\left (128 d^4\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx}{1155 (b c-a d)^4}\\ &=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {128 d^3 (c+d x)^{3/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}-\frac {256 d^4 (c+d x)^{3/2}}{3465 (b c-a d)^5 (a+b x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 117, normalized size = 0.68 \begin {gather*} -\frac {2 (c+d x)^{3/2} \left (1155 d^4-\frac {2772 b d^3 (c+d x)}{a+b x}+\frac {2970 b^2 d^2 (c+d x)^2}{(a+b x)^2}-\frac {1540 b^3 d (c+d x)^3}{(a+b x)^3}+\frac {315 b^4 (c+d x)^4}{(a+b x)^4}\right )}{3465 (b c-a d)^5 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 248, normalized size = 1.45
method | result | size |
default | \(-\frac {\sqrt {d x +c}}{5 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{11 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {11}{2}}}-\frac {10 d \left (-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{11 \left (-a d +b c \right )}\right )}{10 b}\) | \(248\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (128 d^{4} x^{4} b^{4}+704 a \,b^{3} d^{4} x^{3}-192 b^{4} c \,d^{3} x^{3}+1584 a^{2} b^{2} d^{4} x^{2}-1056 a \,b^{3} c \,d^{3} x^{2}+240 b^{4} c^{2} d^{2} x^{2}+1848 a^{3} b \,d^{4} x -2376 a^{2} b^{2} c \,d^{3} x +1320 a \,b^{3} c^{2} d^{2} x -280 b^{4} c^{3} d x +1155 a^{4} d^{4}-2772 a^{3} b c \,d^{3}+2970 a^{2} b^{2} c^{2} d^{2}-1540 a \,b^{3} c^{3} d +315 b^{4} c^{4}\right )}{3465 \left (b x +a \right )^{\frac {11}{2}} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\) | \(256\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 781 vs.
\(2 (141) = 282\).
time = 7.14, size = 781, normalized size = 4.57 \begin {gather*} -\frac {2 \, {\left (128 \, b^{4} d^{5} x^{5} + 315 \, b^{4} c^{5} - 1540 \, a b^{3} c^{4} d + 2970 \, a^{2} b^{2} c^{3} d^{2} - 2772 \, a^{3} b c^{2} d^{3} + 1155 \, a^{4} c d^{4} - 64 \, {\left (b^{4} c d^{4} - 11 \, a b^{3} d^{5}\right )} x^{4} + 16 \, {\left (3 \, b^{4} c^{2} d^{3} - 22 \, a b^{3} c d^{4} + 99 \, a^{2} b^{2} d^{5}\right )} x^{3} - 8 \, {\left (5 \, b^{4} c^{3} d^{2} - 33 \, a b^{3} c^{2} d^{3} + 99 \, a^{2} b^{2} c d^{4} - 231 \, a^{3} b d^{5}\right )} x^{2} + {\left (35 \, b^{4} c^{4} d - 220 \, a b^{3} c^{3} d^{2} + 594 \, a^{2} b^{2} c^{2} d^{3} - 924 \, a^{3} b c d^{4} + 1155 \, a^{4} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3465 \, {\left (a^{6} b^{5} c^{5} - 5 \, a^{7} b^{4} c^{4} d + 10 \, a^{8} b^{3} c^{3} d^{2} - 10 \, a^{9} b^{2} c^{2} d^{3} + 5 \, a^{10} b c d^{4} - a^{11} d^{5} + {\left (b^{11} c^{5} - 5 \, a b^{10} c^{4} d + 10 \, a^{2} b^{9} c^{3} d^{2} - 10 \, a^{3} b^{8} c^{2} d^{3} + 5 \, a^{4} b^{7} c d^{4} - a^{5} b^{6} d^{5}\right )} x^{6} + 6 \, {\left (a b^{10} c^{5} - 5 \, a^{2} b^{9} c^{4} d + 10 \, a^{3} b^{8} c^{3} d^{2} - 10 \, a^{4} b^{7} c^{2} d^{3} + 5 \, a^{5} b^{6} c d^{4} - a^{6} b^{5} d^{5}\right )} x^{5} + 15 \, {\left (a^{2} b^{9} c^{5} - 5 \, a^{3} b^{8} c^{4} d + 10 \, a^{4} b^{7} c^{3} d^{2} - 10 \, a^{5} b^{6} c^{2} d^{3} + 5 \, a^{6} b^{5} c d^{4} - a^{7} b^{4} d^{5}\right )} x^{4} + 20 \, {\left (a^{3} b^{8} c^{5} - 5 \, a^{4} b^{7} c^{4} d + 10 \, a^{5} b^{6} c^{3} d^{2} - 10 \, a^{6} b^{5} c^{2} d^{3} + 5 \, a^{7} b^{4} c d^{4} - a^{8} b^{3} d^{5}\right )} x^{3} + 15 \, {\left (a^{4} b^{7} c^{5} - 5 \, a^{5} b^{6} c^{4} d + 10 \, a^{6} b^{5} c^{3} d^{2} - 10 \, a^{7} b^{4} c^{2} d^{3} + 5 \, a^{8} b^{3} c d^{4} - a^{9} b^{2} d^{5}\right )} x^{2} + 6 \, {\left (a^{5} b^{6} c^{5} - 5 \, a^{6} b^{5} c^{4} d + 10 \, a^{7} b^{4} c^{3} d^{2} - 10 \, a^{8} b^{3} c^{2} d^{3} + 5 \, a^{9} b^{2} c d^{4} - a^{10} b d^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1345 vs.
\(2 (141) = 282\).
time = 1.54, size = 1345, normalized size = 7.87 \begin {gather*} -\frac {512 \, {\left (\sqrt {b d} b^{16} c^{6} d^{5} - 6 \, \sqrt {b d} a b^{15} c^{5} d^{6} + 15 \, \sqrt {b d} a^{2} b^{14} c^{4} d^{7} - 20 \, \sqrt {b d} a^{3} b^{13} c^{3} d^{8} + 15 \, \sqrt {b d} a^{4} b^{12} c^{2} d^{9} - 6 \, \sqrt {b d} a^{5} b^{11} c d^{10} + \sqrt {b d} a^{6} b^{10} d^{11} - 11 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{14} c^{5} d^{5} + 55 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{13} c^{4} d^{6} - 110 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{12} c^{3} d^{7} + 110 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{11} c^{2} d^{8} - 55 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{10} c d^{9} + 11 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{5} b^{9} d^{10} + 55 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{12} c^{4} d^{5} - 220 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{11} c^{3} d^{6} + 330 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{10} c^{2} d^{7} - 220 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{9} c d^{8} + 55 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{4} b^{8} d^{9} - 165 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{10} c^{3} d^{5} + 495 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{9} c^{2} d^{6} - 495 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} b^{8} c d^{7} + 165 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{3} b^{7} d^{8} + 330 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} b^{8} c^{2} d^{5} - 660 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} a b^{7} c d^{6} + 330 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} a^{2} b^{6} d^{7} + 924 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{10} b^{6} c d^{5} - 924 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{10} a b^{5} d^{6} + 1386 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{12} b^{4} d^{5}\right )} {\left | b \right |}}{3465 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{11} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.43, size = 397, normalized size = 2.32 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {2310\,a^4\,c\,d^4-5544\,a^3\,b\,c^2\,d^3+5940\,a^2\,b^2\,c^3\,d^2-3080\,a\,b^3\,c^4\,d+630\,b^4\,c^5}{3465\,b^5\,{\left (a\,d-b\,c\right )}^5}+\frac {x\,\left (2310\,a^4\,d^5-1848\,a^3\,b\,c\,d^4+1188\,a^2\,b^2\,c^2\,d^3-440\,a\,b^3\,c^3\,d^2+70\,b^4\,c^4\,d\right )}{3465\,b^5\,{\left (a\,d-b\,c\right )}^5}+\frac {256\,d^5\,x^5}{3465\,b\,{\left (a\,d-b\,c\right )}^5}+\frac {16\,d^2\,x^2\,\left (231\,a^3\,d^3-99\,a^2\,b\,c\,d^2+33\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{3465\,b^4\,{\left (a\,d-b\,c\right )}^5}+\frac {128\,d^4\,x^4\,\left (11\,a\,d-b\,c\right )}{3465\,b^2\,{\left (a\,d-b\,c\right )}^5}+\frac {32\,d^3\,x^3\,\left (99\,a^2\,d^2-22\,a\,b\,c\,d+3\,b^2\,c^2\right )}{3465\,b^3\,{\left (a\,d-b\,c\right )}^5}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^5\,\sqrt {a+b\,x}}{b^5}+\frac {10\,a^2\,x^3\,\sqrt {a+b\,x}}{b^2}+\frac {10\,a^3\,x^2\,\sqrt {a+b\,x}}{b^3}+\frac {5\,a\,x^4\,\sqrt {a+b\,x}}{b}+\frac {5\,a^4\,x\,\sqrt {a+b\,x}}{b^4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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