Optimal. Leaf size=164 \[ -\frac {5 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}}+\frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2} \]
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Rubi [A] time = 0.06, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {50, 63, 217, 206} \begin {gather*} -\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int x^{5/2} (a+b x)^{5/2} \, dx &=\frac {1}{6} x^{7/2} (a+b x)^{5/2}+\frac {1}{12} (5 a) \int x^{5/2} (a+b x)^{3/2} \, dx\\ &=\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}+\frac {1}{8} a^2 \int x^{5/2} \sqrt {a+b x} \, dx\\ &=\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}+\frac {1}{64} a^3 \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx\\ &=\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {\left (5 a^4\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{384 b}\\ &=-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}+\frac {\left (5 a^5\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{512 b^2}\\ &=\frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {\left (5 a^6\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{1024 b^3}\\ &=\frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {\left (5 a^6\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{512 b^3}\\ &=\frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {\left (5 a^6\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^3}\\ &=\frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {5 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 118, normalized size = 0.72 \begin {gather*} \frac {\sqrt {a+b x} \left (\sqrt {b} \sqrt {x} \left (15 a^5-10 a^4 b x+8 a^3 b^2 x^2+432 a^2 b^3 x^3+640 a b^4 x^4+256 b^5 x^5\right )-\frac {15 a^{11/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {\frac {b x}{a}+1}}\right )}{1536 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 121, normalized size = 0.74 \begin {gather*} \frac {5 a^6 \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{512 b^{7/2}}+\frac {\sqrt {a+b x} \left (15 a^5 \sqrt {x}-10 a^4 b x^{3/2}+8 a^3 b^2 x^{5/2}+432 a^2 b^3 x^{7/2}+640 a b^4 x^{9/2}+256 b^5 x^{11/2}\right )}{1536 b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 206, normalized size = 1.26 \begin {gather*} \left [\frac {15 \, a^{6} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt {b x + a} \sqrt {x}}{3072 \, b^{4}}, \frac {15 \, a^{6} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt {b x + a} \sqrt {x}}{1536 \, b^{4}}\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 [A] time = 0.00, size = 156, normalized size = 0.95 \begin {gather*} -\frac {5 \sqrt {\left (b x +a \right ) x}\, a^{6} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{1024 \sqrt {b x +a}\, b^{\frac {7}{2}} \sqrt {x}}-\frac {5 \sqrt {b x +a}\, a^{5} \sqrt {x}}{512 b^{3}}-\frac {5 \left (b x +a \right )^{\frac {3}{2}} a^{4} \sqrt {x}}{768 b^{3}}+\frac {\left (b x +a \right )^{\frac {7}{2}} x^{\frac {5}{2}}}{6 b}-\frac {\left (b x +a \right )^{\frac {5}{2}} a^{3} \sqrt {x}}{192 b^{3}}-\frac {\left (b x +a \right )^{\frac {7}{2}} a \,x^{\frac {3}{2}}}{12 b^{2}}+\frac {\left (b x +a \right )^{\frac {7}{2}} a^{2} \sqrt {x}}{32 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.99, size = 244, normalized size = 1.49 \begin {gather*} \frac {5 \, a^{6} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{1024 \, b^{\frac {7}{2}}} + \frac {\frac {15 \, \sqrt {b x + a} a^{6} b^{5}}{\sqrt {x}} - \frac {85 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} b^{4}}{x^{\frac {3}{2}}} + \frac {198 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{6} b^{3}}{x^{\frac {5}{2}}} + \frac {198 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{6} b^{2}}{x^{\frac {7}{2}}} - \frac {85 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{6} b}{x^{\frac {9}{2}}} + \frac {15 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{6}}{x^{\frac {11}{2}}}}{1536 \, {\left (b^{9} - \frac {6 \, {\left (b x + a\right )} b^{8}}{x} + \frac {15 \, {\left (b x + a\right )}^{2} b^{7}}{x^{2}} - \frac {20 \, {\left (b x + a\right )}^{3} b^{6}}{x^{3}} + \frac {15 \, {\left (b x + a\right )}^{4} b^{5}}{x^{4}} - \frac {6 \, {\left (b x + a\right )}^{5} b^{4}}{x^{5}} + \frac {{\left (b x + a\right )}^{6} b^{3}}{x^{6}}\right )}} \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 = -0.01 \begin {gather*} \int x^{5/2}\,{\left (a+b\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 25.94, size = 207, normalized size = 1.26 \begin {gather*} \frac {5 a^{\frac {11}{2}} \sqrt {x}}{512 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {5 a^{\frac {9}{2}} x^{\frac {3}{2}}}{1536 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {a^{\frac {7}{2}} x^{\frac {5}{2}}}{768 b \sqrt {1 + \frac {b x}{a}}} + \frac {55 a^{\frac {5}{2}} x^{\frac {7}{2}}}{192 \sqrt {1 + \frac {b x}{a}}} + \frac {67 a^{\frac {3}{2}} b x^{\frac {9}{2}}}{96 \sqrt {1 + \frac {b x}{a}}} + \frac {7 \sqrt {a} b^{2} x^{\frac {11}{2}}}{12 \sqrt {1 + \frac {b x}{a}}} - \frac {5 a^{6} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{512 b^{\frac {7}{2}}} + \frac {b^{3} x^{\frac {13}{2}}}{6 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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