Optimal. Leaf size=152 \[ \frac {5}{4} b (4 A b+3 a B) \sqrt {x} \sqrt {a+b x}+\frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {5}{4} a \sqrt {b} (4 A b+3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {79, 49, 52, 65,
223, 212} \begin {gather*} -\frac {2 (a+b x)^{5/2} (3 a B+4 A b)}{3 a \sqrt {x}}+\frac {5 b \sqrt {x} (a+b x)^{3/2} (3 a B+4 A b)}{6 a}+\frac {5}{4} b \sqrt {x} \sqrt {a+b x} (3 a B+4 A b)+\frac {5}{4} a \sqrt {b} (3 a B+4 A b) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{x^{5/2}} \, dx &=-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {\left (2 \left (2 A b+\frac {3 a B}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{x^{3/2}} \, dx}{3 a}\\ &=-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {(5 b (4 A b+3 a B)) \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx}{3 a}\\ &=\frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {1}{4} (5 b (4 A b+3 a B)) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx\\ &=\frac {5}{4} b (4 A b+3 a B) \sqrt {x} \sqrt {a+b x}+\frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {1}{8} (5 a b (4 A b+3 a B)) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=\frac {5}{4} b (4 A b+3 a B) \sqrt {x} \sqrt {a+b x}+\frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {1}{4} (5 a b (4 A b+3 a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5}{4} b (4 A b+3 a B) \sqrt {x} \sqrt {a+b x}+\frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {1}{4} (5 a b (4 A b+3 a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=\frac {5}{4} b (4 A b+3 a B) \sqrt {x} \sqrt {a+b x}+\frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {5}{4} a \sqrt {b} (4 A b+3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 100, normalized size = 0.66 \begin {gather*} \frac {\sqrt {a+b x} \left (6 b^2 x^2 (2 A+B x)-8 a^2 (A+3 B x)+a b x (-56 A+27 B x)\right )}{12 x^{3/2}}-\frac {5}{4} a \sqrt {b} (4 A b+3 a B) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 207, normalized size = 1.36
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-6 b^{2} B \,x^{3}-12 A \,b^{2} x^{2}-27 B a b \,x^{2}+56 a A b x +24 a^{2} B x +8 a^{2} A \right )}{12 x^{\frac {3}{2}}}+\frac {\left (\frac {5 A \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) b^{\frac {3}{2}} a}{2}+\frac {15 B \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {b}\, a^{2}}{8}\right ) \sqrt {\left (b x +a \right ) x}}{\sqrt {b x +a}\, \sqrt {x}}\) | \(145\) |
default | \(\frac {\sqrt {b x +a}\, \left (12 B \,b^{\frac {5}{2}} \sqrt {\left (b x +a \right ) x}\, x^{3}+60 A \,b^{2} \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a \,x^{2}+24 A \,b^{\frac {5}{2}} \sqrt {\left (b x +a \right ) x}\, x^{2}+45 B b \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} x^{2}+54 B \,b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}\, a \,x^{2}-112 A a \,b^{\frac {3}{2}} x \sqrt {\left (b x +a \right ) x}-48 B \,a^{2} x \sqrt {b}\, \sqrt {\left (b x +a \right ) x}-16 A \,a^{2} \sqrt {b}\, \sqrt {\left (b x +a \right ) x}\right )}{24 x^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}\) | \(207\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 191, normalized size = 1.26 \begin {gather*} \frac {15}{8} \, B a^{2} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + \frac {5}{2} \, A a b^{\frac {3}{2}} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - \frac {15 \, \sqrt {b x^{2} + a x} B a^{2}}{4 \, x} - \frac {35 \, \sqrt {b x^{2} + a x} A a b}{6 \, x} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{4 \, x^{2}} - \frac {5 \, \sqrt {b x^{2} + a x} A a^{2}}{6 \, x^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{2 \, x^{3}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{6 \, x^{3}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.14, size = 217, normalized size = 1.43 \begin {gather*} \left [\frac {15 \, {\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt {b} x^{2} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (6 \, B b^{2} x^{3} - 8 \, A a^{2} + 3 \, {\left (9 \, B a b + 4 \, A b^{2}\right )} x^{2} - 8 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, x^{2}}, -\frac {15 \, {\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (6 \, B b^{2} x^{3} - 8 \, A a^{2} + 3 \, {\left (9 \, B a b + 4 \, A b^{2}\right )} x^{2} - 8 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{12 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 30.42, size = 230, normalized size = 1.51 \begin {gather*} A \left (- \frac {2 a^{2} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {14 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3} - \frac {5 a b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )}}{2} + 5 a b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{b x} + 1} + 1 \right )} + b^{\frac {5}{2}} x \sqrt {\frac {a}{b x} + 1}\right ) + B \left (- \frac {2 a^{\frac {5}{2}}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} + \frac {a^{\frac {3}{2}} b \sqrt {x}}{4 \sqrt {1 + \frac {b x}{a}}} + \frac {11 \sqrt {a} b^{2} x^{\frac {3}{2}}}{4 \sqrt {1 + \frac {b x}{a}}} + \frac {15 a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4} + \frac {b^{3} x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1 + \frac {b x}{a}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \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 \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{x^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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