Optimal. Leaf size=297 \[ \frac {10 a^2 b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {b^4 x^3 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{3 (a+b x)}+\frac {5 a b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{2 (a+b x)}+\frac {b^5 B x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{x (a+b x)}+\frac {5 a^3 b \log (x) \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x} \]
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Rubi [A] time = 0.12, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 76} \begin {gather*} -\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{x (a+b x)}+\frac {10 a^2 b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {5 a b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{2 (a+b x)}+\frac {b^4 x^3 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{3 (a+b x)}+\frac {5 a^3 b \log (x) \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}+\frac {b^5 B x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^3} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{x^3} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (10 a^2 b^7 (A b+a B)+\frac {a^5 A b^5}{x^3}+\frac {a^4 b^5 (5 A b+a B)}{x^2}+\frac {5 a^3 b^6 (2 A b+a B)}{x}+5 a b^8 (A b+2 a B) x+b^9 (A b+5 a B) x^2+b^{10} B x^3\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {10 a^2 b^2 (A b+a B) x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^3 (A b+2 a B) x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^4 (A b+5 a B) x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b^5 B x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {5 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 126, normalized size = 0.42 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-6 a^5 (A+2 B x)-60 a^4 A b x+60 a^3 b x^2 \log (x) (a B+2 A b)+120 a^3 b^2 B x^3+60 a^2 b^3 x^3 (2 A+B x)+10 a b^4 x^4 (3 A+2 B x)+b^5 x^5 (4 A+3 B x)\right )}{12 x^2 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 2.20, size = 719, normalized size = 2.42 \begin {gather*} -\frac {5}{2} a^4 \sqrt {b^2} B \log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right )-\frac {5}{2} a^4 \sqrt {b^2} B \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )+5 a^4 b B \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 a b x+b^2 x^2}}{a}\right )-5 a^3 A b \sqrt {b^2} \log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right )-5 a^3 A b \sqrt {b^2} \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )+10 a^3 A b^2 \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 a b x+b^2 x^2}}{a}\right )+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-96 a^6 A b^2-192 a^6 b^2 B x-1056 a^5 A b^3 x-587 a^5 b^3 B x^2-2416 a^4 A b^4 x^2+1525 a^4 b^4 B x^3+464 a^3 A b^5 x^3+2880 a^3 b^5 B x^4+2400 a^2 A b^6 x^4+1280 a^2 b^6 B x^5+544 a A b^7 x^5+368 a b^7 B x^6+64 A b^8 x^6+48 b^8 B x^7\right )+\sqrt {b^2} \left (96 a^7 A b+192 a^7 b B x+1152 a^6 A b^2 x+779 a^6 b^2 B x^2+3472 a^5 A b^3 x^2-938 a^5 b^3 B x^3+1952 a^4 A b^4 x^3-4405 a^4 b^4 B x^4-2864 a^3 A b^5 x^4-4160 a^3 b^5 B x^5-2944 a^2 A b^6 x^5-1648 a^2 b^6 B x^6-608 a A b^7 x^6-416 a b^7 B x^7-64 A b^8 x^7-48 b^8 B x^8\right )}{96 \sqrt {b^2} x^2 \left (-2 a b-2 b^2 x\right ) \sqrt {a^2+2 a b x+b^2 x^2}+96 x^2 \left (2 a^2 b^2+4 a b^3 x+2 b^4 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 121, normalized size = 0.41 \begin {gather*} \frac {3 \, B b^{5} x^{6} - 6 \, A a^{5} + 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 60 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} \log \relax (x) - 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 191, normalized size = 0.64 \begin {gather*} \frac {1}{4} \, B b^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a b^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, A b^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, B a^{2} b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a b^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, B a^{3} b^{2} x \mathrm {sgn}\left (b x + a\right ) + 10 \, A a^{2} b^{3} x \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (B a^{4} b \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} b^{2} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right ) - \frac {A a^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, {\left (B a^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, A a^{4} b \mathrm {sgn}\left (b x + a\right )\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 144, normalized size = 0.48 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (3 B \,b^{5} x^{6}+4 A \,b^{5} x^{5}+20 B a \,b^{4} x^{5}+30 A a \,b^{4} x^{4}+60 B \,a^{2} b^{3} x^{4}+120 A \,a^{3} b^{2} x^{2} \ln \relax (x )+120 A \,a^{2} b^{3} x^{3}+60 B \,a^{4} b \,x^{2} \ln \relax (x )+120 B \,a^{3} b^{2} x^{3}-60 A \,a^{4} b x -12 B \,a^{5} x -6 A \,a^{5}\right )}{12 \left (b x +a \right )^{5} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 461, normalized size = 1.55 \begin {gather*} 5 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} B a^{4} b \log \left (2 \, b^{2} x + 2 \, a b\right ) + 10 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} A a^{3} b^{2} \log \left (2 \, b^{2} x + 2 \, a b\right ) - 5 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} B a^{4} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) - 10 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A a^{3} b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {5}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{2} b^{2} x + 5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a b^{3} x + \frac {15}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{3} b + 15 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a^{2} b^{2} + \frac {5}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{2} x + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{3} x}{2 \, a} + \frac {35}{12} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a b + \frac {35}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{2} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{2}}{2 \, a^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B}{x} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b}{2 \, a x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{2 \, a^{2} x^{2}} \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 (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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