Optimal. Leaf size=218 \[ -\frac {7 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}}+\frac {7 d^4 \sqrt {c+d x}}{128 b (a+b x) (b c-a d)^4}-\frac {7 d^3 \sqrt {c+d x}}{192 b (a+b x)^2 (b c-a d)^3}+\frac {7 d^2 \sqrt {c+d x}}{240 b (a+b x)^3 (b c-a d)^2}-\frac {d \sqrt {c+d x}}{40 b (a+b x)^4 (b c-a d)}-\frac {\sqrt {c+d x}}{5 b (a+b x)^5} \]
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Rubi [A] time = 0.15, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {47, 51, 63, 208} \begin {gather*} -\frac {7 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}}+\frac {7 d^4 \sqrt {c+d x}}{128 b (a+b x) (b c-a d)^4}-\frac {7 d^3 \sqrt {c+d x}}{192 b (a+b x)^2 (b c-a d)^3}+\frac {7 d^2 \sqrt {c+d x}}{240 b (a+b x)^3 (b c-a d)^2}-\frac {d \sqrt {c+d x}}{40 b (a+b x)^4 (b c-a d)}-\frac {\sqrt {c+d x}}{5 b (a+b x)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x}}{(a+b x)^6} \, dx &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}+\frac {d \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx}{10 b}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}-\frac {\left (7 d^2\right ) \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx}{80 b (b c-a d)}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac {7 d^2 \sqrt {c+d x}}{240 b (b c-a d)^2 (a+b x)^3}+\frac {\left (7 d^3\right ) \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{96 b (b c-a d)^2}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac {7 d^2 \sqrt {c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac {7 d^3 \sqrt {c+d x}}{192 b (b c-a d)^3 (a+b x)^2}-\frac {\left (7 d^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{128 b (b c-a d)^3}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac {7 d^2 \sqrt {c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac {7 d^3 \sqrt {c+d x}}{192 b (b c-a d)^3 (a+b x)^2}+\frac {7 d^4 \sqrt {c+d x}}{128 b (b c-a d)^4 (a+b x)}+\frac {\left (7 d^5\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{256 b (b c-a d)^4}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac {7 d^2 \sqrt {c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac {7 d^3 \sqrt {c+d x}}{192 b (b c-a d)^3 (a+b x)^2}+\frac {7 d^4 \sqrt {c+d x}}{128 b (b c-a d)^4 (a+b x)}+\frac {\left (7 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{128 b (b c-a d)^4}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac {7 d^2 \sqrt {c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac {7 d^3 \sqrt {c+d x}}{192 b (b c-a d)^3 (a+b x)^2}+\frac {7 d^4 \sqrt {c+d x}}{128 b (b c-a d)^4 (a+b x)}-\frac {7 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 52, normalized size = 0.24 \begin {gather*} \frac {2 d^5 (c+d x)^{3/2} \, _2F_1\left (\frac {3}{2},6;\frac {5}{2};-\frac {b (c+d x)}{a d-b c}\right )}{3 (a d-b c)^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.52, size = 317, normalized size = 1.45 \begin {gather*} \frac {d^5 \sqrt {c+d x} \left (105 a^4 d^4-790 a^3 b d^3 (c+d x)-420 a^3 b c d^3+630 a^2 b^2 c^2 d^2-896 a^2 b^2 d^2 (c+d x)^2+2370 a^2 b^2 c d^2 (c+d x)-420 a b^3 c^3 d-2370 a b^3 c^2 d (c+d x)-490 a b^3 d (c+d x)^3+1792 a b^3 c d (c+d x)^2+105 b^4 c^4+790 b^4 c^3 (c+d x)-896 b^4 c^2 (c+d x)^2-105 b^4 (c+d x)^4+490 b^4 c (c+d x)^3\right )}{1920 b (b c-a d)^4 (-a d-b (c+d x)+b c)^5}-\frac {7 d^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x} \sqrt {a d-b c}}{b c-a d}\right )}{128 b^{3/2} (b c-a d)^4 \sqrt {a d-b c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.34, size = 1673, normalized size = 7.67
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.47, size = 432, normalized size = 1.98 \begin {gather*} \frac {7 \, d^{5} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{128 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \sqrt {-b^{2} c + a b d}} + \frac {105 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{4} d^{5} - 490 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{4} c d^{5} + 896 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4} c^{2} d^{5} - 790 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{4} c^{3} d^{5} - 105 \, \sqrt {d x + c} b^{4} c^{4} d^{5} + 490 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{3} d^{6} - 1792 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{3} c d^{6} + 2370 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} c^{2} d^{6} + 420 \, \sqrt {d x + c} a b^{3} c^{3} d^{6} + 896 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{2} d^{7} - 2370 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{2} c d^{7} - 630 \, \sqrt {d x + c} a^{2} b^{2} c^{2} d^{7} + 790 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b d^{8} + 420 \, \sqrt {d x + c} a^{3} b c d^{8} - 105 \, \sqrt {d x + c} a^{4} d^{9}}{1920 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 337, normalized size = 1.55 \begin {gather*} \frac {7 \left (d x +c \right )^{\frac {9}{2}} b^{3} d^{5}}{128 \left (b d x +a d \right )^{5} \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 )}+\frac {49 \left (d x +c \right )^{\frac {7}{2}} b^{2} d^{5}}{192 \left (b d x +a d \right )^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {7 \left (d x +c \right )^{\frac {5}{2}} b \,d^{5}}{15 \left (b d x +a d \right )^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {7 d^{5} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 \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 ) \sqrt {\left (a d -b c \right ) b}\, b}+\frac {79 \left (d x +c \right )^{\frac {3}{2}} d^{5}}{192 \left (b d x +a d \right )^{5} \left (a d -b c \right )}-\frac {7 \sqrt {d x +c}\, d^{5}}{128 \left (b d x +a d \right )^{5} b} \end {gather*}
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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mupad [B] time = 0.49, size = 401, normalized size = 1.84 \begin {gather*} \frac {\frac {79\,d^5\,{\left (c+d\,x\right )}^{3/2}}{192\,\left (a\,d-b\,c\right )}-\frac {7\,d^5\,\sqrt {c+d\,x}}{128\,b}+\frac {49\,b^2\,d^5\,{\left (c+d\,x\right )}^{7/2}}{192\,{\left (a\,d-b\,c\right )}^3}+\frac {7\,b^3\,d^5\,{\left (c+d\,x\right )}^{9/2}}{128\,{\left (a\,d-b\,c\right )}^4}+\frac {7\,b\,d^5\,{\left (c+d\,x\right )}^{5/2}}{15\,{\left (a\,d-b\,c\right )}^2}}{b^5\,{\left (c+d\,x\right )}^5-{\left (c+d\,x\right )}^2\,\left (-10\,a^3\,b^2\,d^3+30\,a^2\,b^3\,c\,d^2-30\,a\,b^4\,c^2\,d+10\,b^5\,c^3\right )-\left (5\,b^5\,c-5\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^4+a^5\,d^5-b^5\,c^5+{\left (c+d\,x\right )}^3\,\left (10\,a^2\,b^3\,d^2-20\,a\,b^4\,c\,d+10\,b^5\,c^2\right )+\left (c+d\,x\right )\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )-10\,a^2\,b^3\,c^3\,d^2+10\,a^3\,b^2\,c^2\,d^3+5\,a\,b^4\,c^4\,d-5\,a^4\,b\,c\,d^4}+\frac {7\,d^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{128\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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