Optimal. Leaf size=316 \[ \frac {20 a^2 b^2 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {2 b^4 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{5 (a+b x)}+\frac {10 a b^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{3 (a+b x)}+\frac {2 b^5 B x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}-\frac {2 a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac {2 a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 x^{3/2} (a+b x)}-\frac {10 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{\sqrt {x} (a+b x)} \]
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Rubi [A] time = 0.12, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 76} \begin {gather*} -\frac {2 a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 x^{3/2} (a+b x)}-\frac {10 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{\sqrt {x} (a+b x)}+\frac {20 a^2 b^2 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {10 a b^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{3 (a+b x)}+\frac {2 b^4 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{5 (a+b x)}-\frac {2 a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}+\frac {2 b^5 B x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (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^{7/2}} \, 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^{7/2}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^5 A b^5}{x^{7/2}}+\frac {a^4 b^5 (5 A b+a B)}{x^{5/2}}+\frac {5 a^3 b^6 (2 A b+a B)}{x^{3/2}}+\frac {10 a^2 b^7 (A b+a B)}{\sqrt {x}}+5 a b^8 (A b+2 a B) \sqrt {x}+b^9 (A b+5 a B) x^{3/2}+b^{10} B x^{5/2}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {2 a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac {2 a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac {10 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {20 a^2 b^2 (A b+a B) \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {10 a b^3 (A b+2 a B) x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b^4 (A b+5 a B) x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 b^5 B x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 122, normalized size = 0.39 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (7 a^5 (3 A+5 B x)+175 a^4 b x (A+3 B x)+1050 a^3 b^2 x^2 (A-B x)-350 a^2 b^3 x^3 (3 A+B x)-35 a b^4 x^4 (5 A+3 B x)-3 b^5 x^5 (7 A+5 B x)\right )}{105 x^{5/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 15.28, size = 145, normalized size = 0.46 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-21 a^5 A-35 a^5 B x-175 a^4 A b x-525 a^4 b B x^2-1050 a^3 A b^2 x^2+1050 a^3 b^2 B x^3+1050 a^2 A b^3 x^3+350 a^2 b^3 B x^4+175 a A b^4 x^4+105 a b^4 B x^5+21 A b^5 x^5+15 b^5 B x^6\right )}{105 x^{5/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 119, normalized size = 0.38 \begin {gather*} \frac {2 \, {\left (15 \, B b^{5} x^{6} - 21 \, A a^{5} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 175 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 1050 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 525 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 35 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x\right )}}{105 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 196, normalized size = 0.62 \begin {gather*} \frac {2}{7} \, B b^{5} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a b^{4} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, A b^{5} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, B a^{2} b^{3} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, A a b^{4} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 20 \, B a^{3} b^{2} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) + 20 \, A a^{2} b^{3} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) - \frac {2 \, {\left (75 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 150 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 25 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 3 \, A a^{5} \mathrm {sgn}\left (b x + a\right )\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 140, normalized size = 0.44 \begin {gather*} -\frac {2 \left (-15 B \,b^{5} x^{6}-21 A \,b^{5} x^{5}-105 B a \,b^{4} x^{5}-175 A a \,b^{4} x^{4}-350 B \,a^{2} b^{3} x^{4}-1050 A \,a^{2} b^{3} x^{3}-1050 B \,a^{3} b^{2} x^{3}+1050 A \,a^{3} b^{2} x^{2}+525 B \,a^{4} b \,x^{2}+175 A \,a^{4} b x +35 B \,a^{5} x +21 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{105 \left (b x +a \right )^{5} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 234, normalized size = 0.74 \begin {gather*} \frac {2}{15} \, {\left ({\left (3 \, b^{5} x^{2} + 5 \, a b^{4} x\right )} \sqrt {x} + \frac {20 \, {\left (a b^{4} x^{2} + 3 \, a^{2} b^{3} x\right )}}{\sqrt {x}} + \frac {90 \, {\left (a^{2} b^{3} x^{2} - a^{3} b^{2} x\right )}}{x^{\frac {3}{2}}} - \frac {20 \, {\left (3 \, a^{3} b^{2} x^{2} + a^{4} b x\right )}}{x^{\frac {5}{2}}} - \frac {5 \, a^{4} b x^{2} + 3 \, a^{5} x}{x^{\frac {7}{2}}}\right )} A + \frac {2}{105} \, {\left (3 \, {\left (5 \, b^{5} x^{2} + 7 \, a b^{4} x\right )} x^{\frac {3}{2}} + 28 \, {\left (3 \, a b^{4} x^{2} + 5 \, a^{2} b^{3} x\right )} \sqrt {x} + \frac {210 \, {\left (a^{2} b^{3} x^{2} + 3 \, a^{3} b^{2} x\right )}}{\sqrt {x}} + \frac {420 \, {\left (a^{3} b^{2} x^{2} - a^{4} b x\right )}}{x^{\frac {3}{2}}} - \frac {35 \, {\left (3 \, a^{4} b x^{2} + a^{5} x\right )}}{x^{\frac {5}{2}}}\right )} B \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^{7/2}} \,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^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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