Optimal. Leaf size=346 \[ \frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{40 a^2 x^4}+\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{11/2} c^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{240 a^3 x^3}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 a^3 d^3+481 a^2 b c d^2-749 a b^2 c^2 d+315 b^3 c^3\right )}{960 a^4 c x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4-90 a^3 b c d^3+1564 a^2 b^2 c^2 d^2-2310 a b^3 c^3 d+945 b^4 c^4\right )}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5} \]
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Rubi [A] time = 0.40, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {98, 149, 151, 12, 93, 208} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (481 a^2 b c d^2-15 a^3 d^3-749 a b^2 c^2 d+315 b^3 c^3\right )}{960 a^4 c x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{240 a^3 x^3}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4-2310 a b^3 c^3 d+945 b^4 c^4\right )}{1920 a^5 c^2 x}+\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{11/2} c^{5/2}}+\frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{40 a^2 x^4}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 149
Rule 151
Rule 208
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx &=-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (9 b c-13 a d)+d (3 b c-5 a d) x\right )}{x^5 \sqrt {a+b x}} \, dx}{5 a}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}-\frac {\int \frac {-\frac {1}{4} c \left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right )-\frac {1}{2} d \left (27 b^2 c^2-63 a b c d+40 a^2 d^2\right ) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{20 a^2}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}+\frac {\int \frac {-\frac {1}{8} c \left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right )-\frac {1}{2} b c d \left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{60 a^3 c}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}-\frac {\int \frac {-\frac {1}{16} c \left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right )-\frac {1}{8} b c d \left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^4 c^2}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {\left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}+\frac {\int -\frac {15 c (b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^5 c^3}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {\left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^5 c^2}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {\left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 a^5 c^2}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {\left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}+\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{11/2} c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 232, normalized size = 0.67 \begin {gather*} \frac {\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt {c}}-8 \sqrt {a+b x} (c+d x)^{5/2}\right )}{24 a x^3}-\frac {16 a c \sqrt {a+b x} (c+d x)^{7/2}}{x^5}+\frac {6 \sqrt {a+b x} (c+d x)^{7/2} (a d+3 b c)}{x^4}}{80 a^2 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.61, size = 444, normalized size = 1.28 \begin {gather*} \frac {\sqrt {a+b x} (a d-b c)^3 \left (45 a^6 d^2-\frac {210 a^5 c d^2 (a+b x)}{c+d x}+210 a^5 b c d-2895 a^4 b^2 c^2-\frac {384 a^4 c^2 d^2 (a+b x)^2}{(c+d x)^2}+\frac {1580 a^4 b c^2 d (a+b x)}{c+d x}+\frac {7110 a^3 b^2 c^3 (a+b x)}{c+d x}+\frac {210 a^3 c^3 d^2 (a+b x)^3}{(c+d x)^3}-\frac {1792 a^3 b c^3 d (a+b x)^2}{(c+d x)^2}-\frac {8064 a^2 b^2 c^4 (a+b x)^2}{(c+d x)^2}-\frac {45 a^2 c^4 d^2 (a+b x)^4}{(c+d x)^4}+\frac {980 a^2 b c^4 d (a+b x)^3}{(c+d x)^3}-\frac {945 b^2 c^6 (a+b x)^4}{(c+d x)^4}+\frac {4410 a b^2 c^5 (a+b x)^3}{(c+d x)^3}-\frac {210 a b c^5 d (a+b x)^4}{(c+d x)^4}\right )}{1920 a^5 c^2 \sqrt {c+d x} \left (a-\frac {c (a+b x)}{c+d x}\right )^5}-\frac {(a d-b c)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{11/2} c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 15.61, size = 732, normalized size = 2.12 \begin {gather*} \left [-\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {a c} x^{5} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (384 \, a^{5} c^{5} + {\left (945 \, a b^{4} c^{5} - 2310 \, a^{2} b^{3} c^{4} d + 1564 \, a^{3} b^{2} c^{3} d^{2} - 90 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (315 \, a^{2} b^{3} c^{5} - 749 \, a^{3} b^{2} c^{4} d + 481 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (63 \, a^{3} b^{2} c^{5} - 148 \, a^{4} b c^{4} d + 93 \, a^{5} c^{3} d^{2}\right )} x^{2} - 144 \, {\left (3 \, a^{4} b c^{5} - 7 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{6} c^{3} x^{5}}, -\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (384 \, a^{5} c^{5} + {\left (945 \, a b^{4} c^{5} - 2310 \, a^{2} b^{3} c^{4} d + 1564 \, a^{3} b^{2} c^{3} d^{2} - 90 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (315 \, a^{2} b^{3} c^{5} - 749 \, a^{3} b^{2} c^{4} d + 481 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (63 \, a^{3} b^{2} c^{5} - 148 \, a^{4} b c^{4} d + 93 \, a^{5} c^{3} d^{2}\right )} x^{2} - 144 \, {\left (3 \, a^{4} b c^{5} - 7 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{6} c^{3} x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 0.03, size = 813, normalized size = 2.35 \begin {gather*} -\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (45 a^{5} d^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+75 a^{4} b c \,d^{4} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+450 a^{3} b^{2} c^{2} d^{3} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-2250 a^{2} b^{3} c^{3} d^{2} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+2625 a \,b^{4} c^{4} d \,x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-945 b^{5} c^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-90 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} d^{4} x^{4}-180 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b c \,d^{3} x^{4}+3128 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}-4620 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{3} d \,x^{4}+1890 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{4} c^{4} x^{4}+60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c \,d^{3} x^{3}-1924 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{2} d^{2} x^{3}+2996 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{3} d \,x^{3}-1260 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{4} x^{3}+1488 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c^{2} d^{2} x^{2}-2368 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{3} d \,x^{2}+1008 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{4} x^{2}+2016 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c^{3} d x -864 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{4} x +768 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c^{4}\right )}{3840 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{5} c^{2} x^{5}} \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 [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{x^6\,\sqrt {a+b\,x}} \,d x \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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