Optimal. Leaf size=95 \[ \frac {35 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}}-\frac {35 a \sqrt {x}}{4 b^4}-\frac {7 x^{5/2}}{4 b^2 (a+b x)}-\frac {x^{7/2}}{2 b (a+b x)^2}+\frac {35 x^{3/2}}{12 b^3} \]
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Rubi [A] time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {47, 50, 63, 205} \begin {gather*} \frac {35 a^{3/2} \tan ^{-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 205
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} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 27, normalized size = 0.28 \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 = 0.13, size = 89, normalized size = 0.94 \begin {gather*} \frac {35 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}}+\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 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 227, normalized size = 2.39 \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 = 77, normalized size = 0.81 \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 = 79, normalized size = 0.83 \begin {gather*} -\frac {13 a^{2} x^{\frac {3}{2}}}{4 \left (b x +a \right )^{2} b^{3}}-\frac {11 a^{3} \sqrt {x}}{4 \left (b x +a \right )^{2} b^{4}}+\frac {35 a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, b^{4}}+\frac {2 x^{\frac {3}{2}}}{3 b^{3}}-\frac {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 = 3.07, size = 86, normalized size = 0.91 \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} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \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 = 0.12, size = 81, normalized size = 0.85 \begin {gather*} \frac {2\,x^{3/2}}{3\,b^3}-\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 {6\,a\,\sqrt {x}}{b^4}+\frac {35\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,b^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 135.24, size = 906, normalized size = 9.54 \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 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}}}{24 i a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} + 48 i a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 i \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} - \frac {350 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}}{24 i a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} + 48 i a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 i \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} - \frac {112 i a^{\frac {3}{2}} b^{3} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}}{24 i a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} + 48 i a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 i \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} + \frac {16 i \sqrt {a} b^{4} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}}{24 i a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} + 48 i a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 i \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} + \frac {105 a^{4} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} + 48 i a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 i \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} - \frac {105 a^{4} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} + 48 i a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 i \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} + \frac {210 a^{3} b x \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} + 48 i a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 i \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} - \frac {210 a^{3} b x \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} + 48 i a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 i \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} + \frac {105 a^{2} b^{2} x^{2} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} + 48 i a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 i \sqrt {a} b^{7} x^{2} \sqrt {\frac {1}{b}}} - \frac {105 a^{2} b^{2} x^{2} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {5}{2}} b^{5} \sqrt {\frac {1}{b}} + 48 i a^{\frac {3}{2}} b^{6} x \sqrt {\frac {1}{b}} + 24 i \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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