Optimal. Leaf size=210 \[ -\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 x^5 (a+b x)}-\frac {3 a b \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^4 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{3 x^3 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)} \]
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Rubi [A] time = 0.08, antiderivative size = 210, 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^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 x^5 (a+b x)}-\frac {3 a b \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^4 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{3 x^3 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (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 )^{3/2}}{x^7} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3 (A+B x)}{x^7} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^3 A b^3}{x^7}+\frac {a^2 b^3 (3 A b+a B)}{x^6}+\frac {3 a b^4 (A b+a B)}{x^5}+\frac {b^5 (A b+3 a B)}{x^4}+\frac {b^6 B}{x^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {3 a b (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b^2 (A b+3 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 87, normalized size = 0.41 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (2 a^3 (5 A+6 B x)+9 a^2 b x (4 A+5 B x)+15 a b^2 x^2 (3 A+4 B x)+10 b^3 x^3 (2 A+3 B x)\right )}{60 x^6 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 2.34, size = 614, normalized size = 2.92 \begin {gather*} \frac {8 b^5 \sqrt {a^2+2 a b x+b^2 x^2} \left (-10 a^8 A b-12 a^8 b B x-86 a^7 A b^2 x-105 a^7 b^2 B x^2-325 a^6 A b^3 x^2-405 a^6 b^3 B x^3-705 a^5 A b^4 x^3-900 a^5 b^4 B x^4-960 a^4 A b^5 x^4-1260 a^4 b^5 B x^5-840 a^3 A b^6 x^5-1137 a^3 b^6 B x^6-461 a^2 A b^7 x^6-645 a^2 b^7 B x^7-145 a A b^8 x^7-210 a b^8 B x^8-20 A b^9 x^8-30 b^9 B x^9\right )+8 \sqrt {b^2} b^5 \left (10 a^9 A+12 a^9 B x+96 a^8 A b x+117 a^8 b B x^2+411 a^7 A b^2 x^2+510 a^7 b^2 B x^3+1030 a^6 A b^3 x^3+1305 a^6 b^3 B x^4+1665 a^5 A b^4 x^4+2160 a^5 b^4 B x^5+1800 a^4 A b^5 x^5+2397 a^4 b^5 B x^6+1301 a^3 A b^6 x^6+1782 a^3 b^6 B x^7+606 a^2 A b^7 x^7+855 a^2 b^7 B x^8+165 a A b^8 x^8+240 a b^8 B x^9+20 A b^9 x^9+30 b^9 B x^{10}\right )}{15 \sqrt {b^2} x^6 \sqrt {a^2+2 a b x+b^2 x^2} \left (-32 a^5 b^5-160 a^4 b^6 x-320 a^3 b^7 x^2-320 a^2 b^8 x^3-160 a b^9 x^4-32 b^{10} x^5\right )+15 x^6 \left (32 a^6 b^6+192 a^5 b^7 x+480 a^4 b^8 x^2+640 a^3 b^9 x^3+480 a^2 b^{10} x^4+192 a b^{11} x^5+32 b^{12} x^6\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 73, normalized size = 0.35 \begin {gather*} -\frac {30 \, B b^{3} x^{4} + 10 \, A a^{3} + 20 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 12 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{60 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 149, normalized size = 0.71 \begin {gather*} \frac {{\left (3 \, B a b^{5} - A b^{6}\right )} \mathrm {sgn}\left (b x + a\right )}{60 \, a^{3}} - \frac {30 \, B b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 60 \, B a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, A b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, B a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 45 \, A a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) + 36 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 10 \, A a^{3} \mathrm {sgn}\left (b x + a\right )}{60 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 92, normalized size = 0.44 \begin {gather*} -\frac {\left (30 B \,b^{3} x^{4}+20 A \,b^{3} x^{3}+60 B a \,b^{2} x^{3}+45 A a \,b^{2} x^{2}+45 B \,a^{2} b \,x^{2}+36 A \,a^{2} b x +12 B \,a^{3} x +10 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 \left (b x +a \right )^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 375, normalized size = 1.79 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{5}}{4 \, a^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{6}}{4 \, a^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{4}}{4 \, a^{4} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{5}}{4 \, a^{5} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{3}}{4 \, a^{5} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{4}}{4 \, a^{6} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{2}}{4 \, a^{4} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{3}}{4 \, a^{5} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b}{4 \, a^{3} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{2}}{4 \, a^{4} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B}{5 \, a^{2} x^{5}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b}{30 \, a^{3} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A}{6 \, a^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 195, normalized size = 0.93 \begin {gather*} -\frac {\left (\frac {B\,a^3}{5}+\frac {3\,A\,b\,a^2}{5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^3}{3}+B\,a\,b^2\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^3\,\left (a+b\,x\right )}-\frac {A\,a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\left (a+b\,x\right )}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^2\,\left (a+b\,x\right )}-\frac {3\,a\,b\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\left (a+b\,x\right )} \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 {3}{2}}}{x^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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