Optimal. Leaf size=97 \[ -\frac {35 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}}+\frac {35 a \sqrt {x}}{4 b^4}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {35 x^{3/2}}{12 b^3} \]
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Rubi [A] time = 0.03, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {47, 50, 63, 208} \begin {gather*} -\frac {35 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}+\frac {35 a \sqrt {x}}{4 b^4}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {35 x^{3/2}}{12 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {x^{7/2}}{(-a+b x)^3} \, dx &=-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 \int \frac {x^{5/2}}{(-a+b x)^2} \, dx}{4 b}\\ &=-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}+\frac {35 \int \frac {x^{3/2}}{-a+b x} \, dx}{8 b^2}\\ &=\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}+\frac {(35 a) \int \frac {\sqrt {x}}{-a+b x} \, dx}{8 b^3}\\ &=\frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}+\frac {\left (35 a^2\right ) \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{8 b^4}\\ &=\frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}+\frac {\left (35 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^4}\\ &=\frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a-b x)^2}+\frac {7 x^{5/2}}{4 b^2 (a-b x)}-\frac {35 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.27 \begin {gather*} -\frac {2 x^{9/2} \, _2F_1\left (3,\frac {9}{2};\frac {11}{2};\frac {b x}{a}\right )}{9 a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 91, normalized size = 0.94 \begin {gather*} \frac {105 a^3 \sqrt {x}-175 a^2 b x^{3/2}+56 a b^2 x^{5/2}+8 b^3 x^{7/2}}{12 b^4 (b x-a)^2}-\frac {35 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 227, normalized size = 2.34 \begin {gather*} \left [\frac {105 \, {\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt {\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {\frac {a}{b}} + a}{b x - a}\right ) + 2 \, {\left (8 \, b^{3} x^{3} + 56 \, a b^{2} x^{2} - 175 \, a^{2} b x + 105 \, a^{3}\right )} \sqrt {x}}{24 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {105 \, {\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt {-\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {-\frac {a}{b}}}{a}\right ) + {\left (8 \, b^{3} x^{3} + 56 \, a b^{2} x^{2} - 175 \, a^{2} b x + 105 \, a^{3}\right )} \sqrt {x}}{12 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 81, normalized size = 0.84 \begin {gather*} \frac {35 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 \, \sqrt {-a b} b^{4}} - \frac {13 \, a^{2} b x^{\frac {3}{2}} - 11 \, a^{3} \sqrt {x}}{4 \, {\left (b x - a\right )}^{2} b^{4}} + \frac {2 \, {\left (b^{6} x^{\frac {3}{2}} + 9 \, a b^{5} \sqrt {x}\right )}}{3 \, b^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 70, normalized size = 0.72 \begin {gather*} \frac {2 \left (-\frac {35 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}+\frac {-\frac {13 b \,x^{\frac {3}{2}}}{8}+\frac {11 a \sqrt {x}}{8}}{\left (b x -a \right )^{2}}\right ) a^{2}}{b^{4}}+\frac {\frac {2 b \,x^{\frac {3}{2}}}{3}+6 a \sqrt {x}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 103, normalized size = 1.06 \begin {gather*} -\frac {13 \, a^{2} b x^{\frac {3}{2}} - 11 \, a^{3} \sqrt {x}}{4 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {35 \, a^{2} \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{4}} + \frac {2 \, {\left (b x^{\frac {3}{2}} + 9 \, a \sqrt {x}\right )}}{3 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 83, normalized size = 0.86 \begin {gather*} \frac {\frac {11\,a^3\,\sqrt {x}}{4}-\frac {13\,a^2\,b\,x^{3/2}}{4}}{a^2\,b^4-2\,a\,b^5\,x+b^6\,x^2}+\frac {2\,x^{3/2}}{3\,b^3}+\frac {6\,a\,\sqrt {x}}{b^4}+\frac {a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,35{}\mathrm {i}}{4\,b^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 136.15, size = 840, normalized size = 8.66 \begin {gather*} \begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 x^{\frac {9}{2}}}{9 a^{3}} & \text {for}\: b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 b^{3}} & \text {for}\: a = 0 \\\frac {210 a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}}}{24 a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} - 48 a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} - \frac {350 a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}}{24 a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} - 48 a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} + \frac {112 a^{\frac {3}{2}} b^{3} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}}{24 a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} - 48 a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} + \frac {16 \sqrt {a} b^{4} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}}{24 a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} - 48 a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} + \frac {105 a^{4} \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} - 48 a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} - \frac {105 a^{4} \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} - 48 a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} - \frac {210 a^{3} b x \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} - 48 a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} + \frac {210 a^{3} b x \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} - 48 a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} + \frac {105 a^{2} b^{2} x^{2} \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} - 48 a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} - \frac {105 a^{2} b^{2} x^{2} \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} - 48 a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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