Optimal. Leaf size=82 \[ -\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2}}-\frac {15}{4 a^3 \sqrt {x}}+\frac {5}{4 a^2 \sqrt {x} (a+b x)}+\frac {1}{2 a \sqrt {x} (a+b x)^2} \]
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Rubi [A] time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {51, 63, 205} \begin {gather*} \frac {5}{4 a^2 \sqrt {x} (a+b x)}-\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2}}-\frac {15}{4 a^3 \sqrt {x}}+\frac {1}{2 a \sqrt {x} (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} (a+b x)^3} \, dx &=\frac {1}{2 a \sqrt {x} (a+b x)^2}+\frac {5 \int \frac {1}{x^{3/2} (a+b x)^2} \, dx}{4 a}\\ &=\frac {1}{2 a \sqrt {x} (a+b x)^2}+\frac {5}{4 a^2 \sqrt {x} (a+b x)}+\frac {15 \int \frac {1}{x^{3/2} (a+b x)} \, dx}{8 a^2}\\ &=-\frac {15}{4 a^3 \sqrt {x}}+\frac {1}{2 a \sqrt {x} (a+b x)^2}+\frac {5}{4 a^2 \sqrt {x} (a+b x)}-\frac {(15 b) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a^3}\\ &=-\frac {15}{4 a^3 \sqrt {x}}+\frac {1}{2 a \sqrt {x} (a+b x)^2}+\frac {5}{4 a^2 \sqrt {x} (a+b x)}-\frac {(15 b) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^3}\\ &=-\frac {15}{4 a^3 \sqrt {x}}+\frac {1}{2 a \sqrt {x} (a+b x)^2}+\frac {5}{4 a^2 \sqrt {x} (a+b x)}-\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 25, normalized size = 0.30 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};-\frac {b x}{a}\right )}{a^3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 70, normalized size = 0.85 \begin {gather*} \frac {-8 a^2-25 a b x-15 b^2 x^2}{4 a^3 \sqrt {x} (a+b x)^2}-\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 214, normalized size = 2.61 \begin {gather*} \left [\frac {15 \, {\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) - 2 \, {\left (15 \, b^{2} x^{2} + 25 \, a b x + 8 \, a^{2}\right )} \sqrt {x}}{8 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}, \frac {15 \, {\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (15 \, b^{2} x^{2} + 25 \, a b x + 8 \, a^{2}\right )} \sqrt {x}}{4 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 59, normalized size = 0.72 \begin {gather*} -\frac {15 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{3}} - \frac {2}{a^{3} \sqrt {x}} - \frac {7 \, b^{2} x^{\frac {3}{2}} + 9 \, a b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 0.80 \begin {gather*} -\frac {7 b^{2} x^{\frac {3}{2}}}{4 \left (b x +a \right )^{2} a^{3}}-\frac {9 b \sqrt {x}}{4 \left (b x +a \right )^{2} a^{2}}-\frac {15 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a^{3}}-\frac {2}{a^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 73, normalized size = 0.89 \begin {gather*} -\frac {15 \, b^{2} x^{2} + 25 \, a b x + 8 \, a^{2}}{4 \, {\left (a^{3} b^{2} x^{\frac {5}{2}} + 2 \, a^{4} b x^{\frac {3}{2}} + a^{5} \sqrt {x}\right )}} - \frac {15 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 70, normalized size = 0.85 \begin {gather*} -\frac {\frac {2}{a}+\frac {15\,b^2\,x^2}{4\,a^3}+\frac {25\,b\,x}{4\,a^2}}{a^2\,\sqrt {x}+b^2\,x^{5/2}+2\,a\,b\,x^{3/2}}-\frac {15\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 54.35, size = 865, normalized size = 10.55 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{a^{3} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {2}{7 b^{3} x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {16 i a^{\frac {5}{2}} \sqrt {\frac {1}{b}}}{8 i a^{\frac {11}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 16 i a^{\frac {9}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 8 i a^{\frac {7}{2}} b^{2} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} - \frac {50 i a^{\frac {3}{2}} b x \sqrt {\frac {1}{b}}}{8 i a^{\frac {11}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 16 i a^{\frac {9}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 8 i a^{\frac {7}{2}} b^{2} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} - \frac {30 i \sqrt {a} b^{2} x^{2} \sqrt {\frac {1}{b}}}{8 i a^{\frac {11}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 16 i a^{\frac {9}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 8 i a^{\frac {7}{2}} b^{2} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} - \frac {15 a^{2} \sqrt {x} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {11}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 16 i a^{\frac {9}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 8 i a^{\frac {7}{2}} b^{2} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} + \frac {15 a^{2} \sqrt {x} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {11}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 16 i a^{\frac {9}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 8 i a^{\frac {7}{2}} b^{2} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} - \frac {30 a b x^{\frac {3}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {11}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 16 i a^{\frac {9}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 8 i a^{\frac {7}{2}} b^{2} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} + \frac {30 a b x^{\frac {3}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {11}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 16 i a^{\frac {9}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 8 i a^{\frac {7}{2}} b^{2} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} - \frac {15 b^{2} x^{\frac {5}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {11}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 16 i a^{\frac {9}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 8 i a^{\frac {7}{2}} b^{2} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} + \frac {15 b^{2} x^{\frac {5}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {11}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 16 i a^{\frac {9}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 8 i a^{\frac {7}{2}} b^{2} x^{\frac {5}{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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