Optimal. Leaf size=180 \[ -\frac {35 d^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 \sqrt {b} (b c-a d)^{9/2}}+\frac {35 d^3 \sqrt {c+d x}}{64 (a+b x) (b c-a d)^4}-\frac {35 d^2 \sqrt {c+d x}}{96 (a+b x)^2 (b c-a d)^3}+\frac {7 d \sqrt {c+d x}}{24 (a+b x)^3 (b c-a d)^2}-\frac {\sqrt {c+d x}}{4 (a+b x)^4 (b c-a d)} \]
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Rubi [A] time = 0.06, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {51, 63, 208} \[ \frac {35 d^3 \sqrt {c+d x}}{64 (a+b x) (b c-a d)^4}-\frac {35 d^2 \sqrt {c+d x}}{96 (a+b x)^2 (b c-a d)^3}-\frac {35 d^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 \sqrt {b} (b c-a d)^{9/2}}+\frac {7 d \sqrt {c+d x}}{24 (a+b x)^3 (b c-a d)^2}-\frac {\sqrt {c+d x}}{4 (a+b x)^4 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx &=-\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}-\frac {(7 d) \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx}{8 (b c-a d)}\\ &=-\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}+\frac {\left (35 d^2\right ) \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{48 (b c-a d)^2}\\ &=-\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}-\frac {35 d^2 \sqrt {c+d x}}{96 (b c-a d)^3 (a+b x)^2}-\frac {\left (35 d^3\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{64 (b c-a d)^3}\\ &=-\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}-\frac {35 d^2 \sqrt {c+d x}}{96 (b c-a d)^3 (a+b x)^2}+\frac {35 d^3 \sqrt {c+d x}}{64 (b c-a d)^4 (a+b x)}+\frac {\left (35 d^4\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{128 (b c-a d)^4}\\ &=-\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}-\frac {35 d^2 \sqrt {c+d x}}{96 (b c-a d)^3 (a+b x)^2}+\frac {35 d^3 \sqrt {c+d x}}{64 (b c-a d)^4 (a+b x)}+\frac {\left (35 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{64 (b c-a d)^4}\\ &=-\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}-\frac {35 d^2 \sqrt {c+d x}}{96 (b c-a d)^3 (a+b x)^2}+\frac {35 d^3 \sqrt {c+d x}}{64 (b c-a d)^4 (a+b x)}-\frac {35 d^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 \sqrt {b} (b c-a d)^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.28 \[ \frac {2 d^4 \sqrt {c+d x} \, _2F_1\left (\frac {1}{2},5;\frac {3}{2};-\frac {b (c+d x)}{a d-b c}\right )}{(a d-b c)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 1325, normalized size = 7.36 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.13, size = 331, normalized size = 1.84 \[ \frac {35 \, d^{4} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{64 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b^{2} c + a b d}} + \frac {105 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d^{4} - 385 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c d^{4} + 511 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{2} d^{4} - 279 \, \sqrt {d x + c} b^{3} c^{3} d^{4} + 385 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} d^{5} - 1022 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c d^{5} + 837 \, \sqrt {d x + c} a b^{2} c^{2} d^{5} + 511 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{6} - 837 \, \sqrt {d x + c} a^{2} b c d^{6} + 279 \, \sqrt {d x + c} a^{3} d^{7}}{192 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 179, normalized size = 0.99 \[ \frac {35 d^{4} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{64 \left (a d -b c \right )^{4} \sqrt {\left (a d -b c \right ) b}}+\frac {\sqrt {d x +c}\, d^{4}}{4 \left (a d -b c \right ) \left (b d x +a d \right )^{4}}+\frac {7 \sqrt {d x +c}\, d^{4}}{24 \left (a d -b c \right )^{2} \left (b d x +a d \right )^{3}}+\frac {35 \sqrt {d x +c}\, d^{4}}{96 \left (a d -b c \right )^{3} \left (b d x +a d \right )^{2}}+\frac {35 \sqrt {d x +c}\, d^{4}}{64 \left (a d -b c \right )^{4} \left (b d x +a d \right )} \]
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 = 0.46, size = 307, normalized size = 1.71 \[ \frac {\frac {93\,d^4\,\sqrt {c+d\,x}}{64\,\left (a\,d-b\,c\right )}+\frac {385\,b^2\,d^4\,{\left (c+d\,x\right )}^{5/2}}{192\,{\left (a\,d-b\,c\right )}^3}+\frac {35\,b^3\,d^4\,{\left (c+d\,x\right )}^{7/2}}{64\,{\left (a\,d-b\,c\right )}^4}+\frac {511\,b\,d^4\,{\left (c+d\,x\right )}^{3/2}}{192\,{\left (a\,d-b\,c\right )}^2}}{b^4\,{\left (c+d\,x\right )}^4-\left (4\,b^4\,c-4\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^3-\left (c+d\,x\right )\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )+a^4\,d^4+b^4\,c^4+{\left (c+d\,x\right )}^2\,\left (6\,a^2\,b^2\,d^2-12\,a\,b^3\,c\,d+6\,b^4\,c^2\right )+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d-4\,a^3\,b\,c\,d^3}+\frac {35\,d^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{64\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{9/2}} \]
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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