Optimal. Leaf size=198 \[ -\frac {3 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{7/2} (b c-a d)^{5/2}}+\frac {3 d^4 \sqrt {c+d x}}{128 b^3 (a+b x) (b c-a d)^2}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (a+b x)^2 (b c-a d)}-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5} \]
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Rubi [A] time = 0.09, antiderivative size = 198, 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} \[ \frac {3 d^4 \sqrt {c+d x}}{128 b^3 (a+b x) (b c-a d)^2}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (a+b x)^2 (b c-a d)}-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {3 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{7/2} (b c-a d)^{5/2}}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{(a+b x)^6} \, dx &=-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}+\frac {d \int \frac {(c+d x)^{3/2}}{(a+b x)^5} \, dx}{2 b}\\ &=-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}+\frac {\left (3 d^2\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx}{16 b^2}\\ &=-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}+\frac {d^3 \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{32 b^3}\\ &=-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (b c-a d) (a+b x)^2}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}-\frac {\left (3 d^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{128 b^3 (b c-a d)}\\ &=-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (b c-a d) (a+b x)^2}+\frac {3 d^4 \sqrt {c+d x}}{128 b^3 (b c-a d)^2 (a+b x)}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}+\frac {\left (3 d^5\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{256 b^3 (b c-a d)^2}\\ &=-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (b c-a d) (a+b x)^2}+\frac {3 d^4 \sqrt {c+d x}}{128 b^3 (b c-a d)^2 (a+b x)}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}+\frac {\left (3 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^3 (b c-a d)^2}\\ &=-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (b c-a d) (a+b x)^2}+\frac {3 d^4 \sqrt {c+d x}}{128 b^3 (b c-a d)^2 (a+b x)}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}-\frac {3 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{7/2} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 52, normalized size = 0.26 \[ \frac {2 d^5 (c+d x)^{7/2} \, _2F_1\left (\frac {7}{2},6;\frac {9}{2};-\frac {b (c+d x)}{a d-b c}\right )}{7 (a d-b c)^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 1337, normalized size = 6.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.25, size = 380, normalized size = 1.92 \[ \frac {3 \, d^{5} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{128 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {15 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{4} d^{5} - 70 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{4} c d^{5} - 128 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4} c^{2} d^{5} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{4} c^{3} d^{5} - 15 \, \sqrt {d x + c} b^{4} c^{4} d^{5} + 70 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{3} d^{6} + 256 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{3} c d^{6} - 210 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} c^{2} d^{6} + 60 \, \sqrt {d x + c} a b^{3} c^{3} d^{6} - 128 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{2} d^{7} + 210 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{2} c d^{7} - 90 \, \sqrt {d x + c} a^{2} b^{2} c^{2} d^{7} - 70 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b d^{8} + 60 \, \sqrt {d x + c} a^{3} b c d^{8} - 15 \, \sqrt {d x + c} a^{4} d^{9}}{640 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 305, normalized size = 1.54 \[ -\frac {3 \sqrt {d x +c}\, a^{2} d^{7}}{128 \left (b d x +a d \right )^{5} b^{3}}+\frac {3 \sqrt {d x +c}\, a c \,d^{6}}{64 \left (b d x +a d \right )^{5} b^{2}}+\frac {3 \left (d x +c \right )^{\frac {9}{2}} b \,d^{5}}{128 \left (b d x +a d \right )^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {3 \sqrt {d x +c}\, c^{2} d^{5}}{128 \left (b d x +a d \right )^{5} b}-\frac {7 \left (d x +c \right )^{\frac {3}{2}} a \,d^{6}}{64 \left (b d x +a d \right )^{5} b^{2}}+\frac {7 \left (d x +c \right )^{\frac {3}{2}} c \,d^{5}}{64 \left (b d x +a d \right )^{5} b}+\frac {7 \left (d x +c \right )^{\frac {7}{2}} d^{5}}{64 \left (b d x +a d \right )^{5} \left (a d -b c \right )}-\frac {\left (d x +c \right )^{\frac {5}{2}} d^{5}}{5 \left (b d x +a d \right )^{5} b}+\frac {3 d^{5} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {\left (a d -b c \right ) b}\, b^{3}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 411, normalized size = 2.08 \[ \frac {3\,d^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{128\,b^{7/2}\,{\left (a\,d-b\,c\right )}^{5/2}}-\frac {\frac {d^5\,{\left (c+d\,x\right )}^{5/2}}{5\,b}-\frac {7\,d^5\,{\left (c+d\,x\right )}^{7/2}}{64\,\left (a\,d-b\,c\right )}+\frac {3\,d^5\,\sqrt {c+d\,x}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{128\,b^3}+\frac {7\,d^5\,\left (a\,d-b\,c\right )\,{\left (c+d\,x\right )}^{3/2}}{64\,b^2}-\frac {3\,b\,d^5\,{\left (c+d\,x\right )}^{9/2}}{128\,{\left (a\,d-b\,c\right )}^2}}{b^5\,{\left (c+d\,x\right )}^5-{\left (c+d\,x\right )}^2\,\left (-10\,a^3\,b^2\,d^3+30\,a^2\,b^3\,c\,d^2-30\,a\,b^4\,c^2\,d+10\,b^5\,c^3\right )-\left (5\,b^5\,c-5\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^4+a^5\,d^5-b^5\,c^5+{\left (c+d\,x\right )}^3\,\left (10\,a^2\,b^3\,d^2-20\,a\,b^4\,c\,d+10\,b^5\,c^2\right )+\left (c+d\,x\right )\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )-10\,a^2\,b^3\,c^3\,d^2+10\,a^3\,b^2\,c^2\,d^3+5\,a\,b^4\,c^4\,d-5\,a^4\,b\,c\,d^4} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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