Optimal. Leaf size=82 \[ \frac {15 \sqrt {x}}{4 b^3}-\frac {x^{5/2}}{2 b (a+b x)^2}-\frac {5 x^{3/2}}{4 b^2 (a+b x)}-\frac {15 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {43, 52, 65, 211}
\begin {gather*} -\frac {15 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}}-\frac {5 x^{3/2}}{4 b^2 (a+b x)}-\frac {x^{5/2}}{2 b (a+b x)^2}+\frac {15 \sqrt {x}}{4 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 211
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{(a+b x)^3} \, dx &=-\frac {x^{5/2}}{2 b (a+b x)^2}+\frac {5 \int \frac {x^{3/2}}{(a+b x)^2} \, dx}{4 b}\\ &=-\frac {x^{5/2}}{2 b (a+b x)^2}-\frac {5 x^{3/2}}{4 b^2 (a+b x)}+\frac {15 \int \frac {\sqrt {x}}{a+b x} \, dx}{8 b^2}\\ &=\frac {15 \sqrt {x}}{4 b^3}-\frac {x^{5/2}}{2 b (a+b x)^2}-\frac {5 x^{3/2}}{4 b^2 (a+b x)}-\frac {(15 a) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 b^3}\\ &=\frac {15 \sqrt {x}}{4 b^3}-\frac {x^{5/2}}{2 b (a+b x)^2}-\frac {5 x^{3/2}}{4 b^2 (a+b x)}-\frac {(15 a) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^3}\\ &=\frac {15 \sqrt {x}}{4 b^3}-\frac {x^{5/2}}{2 b (a+b x)^2}-\frac {5 x^{3/2}}{4 b^2 (a+b x)}-\frac {15 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 70, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x} \left (15 a^2+25 a b x+8 b^2 x^2\right )}{4 b^3 (a+b x)^2}-\frac {15 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 34.87, size = 665, normalized size = 8.11 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\sqrt {x}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {2 x^{\frac {7}{2}}}{7 a^3},b\text {==}0\right \},\left \{\frac {2 \sqrt {x}}{b^3},a\text {==}0\right \}\right \},\frac {-15 a^3 \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {-\frac {a}{b}}+16 a b^5 x \sqrt {-\frac {a}{b}}+8 b^6 x^2 \sqrt {-\frac {a}{b}}}+\frac {15 a^3 \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {-\frac {a}{b}}+16 a b^5 x \sqrt {-\frac {a}{b}}+8 b^6 x^2 \sqrt {-\frac {a}{b}}}+\frac {30 a^2 b \sqrt {x} \sqrt {-\frac {a}{b}}}{8 a^2 b^4 \sqrt {-\frac {a}{b}}+16 a b^5 x \sqrt {-\frac {a}{b}}+8 b^6 x^2 \sqrt {-\frac {a}{b}}}-\frac {30 a^2 b x \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {-\frac {a}{b}}+16 a b^5 x \sqrt {-\frac {a}{b}}+8 b^6 x^2 \sqrt {-\frac {a}{b}}}+\frac {30 a^2 b x \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {-\frac {a}{b}}+16 a b^5 x \sqrt {-\frac {a}{b}}+8 b^6 x^2 \sqrt {-\frac {a}{b}}}+\frac {50 a b^2 x^{\frac {3}{2}} \sqrt {-\frac {a}{b}}}{8 a^2 b^4 \sqrt {-\frac {a}{b}}+16 a b^5 x \sqrt {-\frac {a}{b}}+8 b^6 x^2 \sqrt {-\frac {a}{b}}}-\frac {15 a b^2 x^2 \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {-\frac {a}{b}}+16 a b^5 x \sqrt {-\frac {a}{b}}+8 b^6 x^2 \sqrt {-\frac {a}{b}}}+\frac {15 a b^2 x^2 \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {-\frac {a}{b}}+16 a b^5 x \sqrt {-\frac {a}{b}}+8 b^6 x^2 \sqrt {-\frac {a}{b}}}+\frac {16 b^3 x^{\frac {5}{2}} \sqrt {-\frac {a}{b}}}{8 a^2 b^4 \sqrt {-\frac {a}{b}}+16 a b^5 x \sqrt {-\frac {a}{b}}+8 b^6 x^2 \sqrt {-\frac {a}{b}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.12, size = 56, normalized size = 0.68
method | result | size |
derivativedivides | \(\frac {2 \sqrt {x}}{b^{3}}-\frac {2 a \left (\frac {-\frac {9 b \,x^{\frac {3}{2}}}{8}-\frac {7 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {15 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3}}\) | \(56\) |
default | \(\frac {2 \sqrt {x}}{b^{3}}-\frac {2 a \left (\frac {-\frac {9 b \,x^{\frac {3}{2}}}{8}-\frac {7 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {15 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3}}\) | \(56\) |
risch | \(\frac {2 \sqrt {x}}{b^{3}}+\frac {9 a \,x^{\frac {3}{2}}}{4 b^{2} \left (b x +a \right )^{2}}+\frac {7 a^{2} \sqrt {x}}{4 b^{3} \left (b x +a \right )^{2}}-\frac {15 a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b^{3} \sqrt {a b}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 73, normalized size = 0.89 \begin {gather*} \frac {9 \, a b x^{\frac {3}{2}} + 7 \, a^{2} \sqrt {x}}{4 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} - \frac {15 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{3}} + \frac {2 \, \sqrt {x}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 200, normalized size = 2.44 \begin {gather*} \left [\frac {15 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\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^{2} x^{2} + 25 \, a b x + 15 \, a^{2}\right )} \sqrt {x}}{8 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac {15 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (8 \, b^{2} x^{2} + 25 \, a b x + 15 \, a^{2}\right )} \sqrt {x}}{4 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 34.88, size = 683, normalized size = 8.33 \begin {gather*} \begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{b^{3}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a^{3}} & \text {for}\: b = 0 \\- \frac {15 a^{3} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {15 a^{3} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {30 a^{2} b \sqrt {x} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {30 a^{2} b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {30 a^{2} b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {50 a b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {15 a b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {15 a b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {16 b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 85, normalized size = 1.04 \begin {gather*} 2 \left (\frac {\sqrt {x}}{b^{3}}+\frac {9 \sqrt {x} x b a+7 \sqrt {x} a^{2}}{8 b^{3} \left (x b+a\right )^{2}}-\frac {15 a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b^{3}\cdot 2 \sqrt {a b}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 69, normalized size = 0.84 \begin {gather*} \frac {\frac {7\,a^2\,\sqrt {x}}{4}+\frac {9\,a\,b\,x^{3/2}}{4}}{a^2\,b^3+2\,a\,b^4\,x+b^5\,x^2}+\frac {2\,\sqrt {x}}{b^3}-\frac {15\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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