Optimal. Leaf size=130 \[ \frac {a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}}-\frac {a \sqrt {x} (5 A b-7 a B)}{b^4}+\frac {x^{3/2} (5 A b-7 a B)}{3 b^3}-\frac {x^{5/2} (5 A b-7 a B)}{5 a b^2}+\frac {x^{7/2} (A b-a B)}{a b (a+b x)} \]
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Rubi [A] time = 0.07, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {27, 78, 50, 63, 205} \[ \frac {a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}}-\frac {x^{5/2} (5 A b-7 a B)}{5 a b^2}+\frac {x^{3/2} (5 A b-7 a B)}{3 b^3}-\frac {a \sqrt {x} (5 A b-7 a B)}{b^4}+\frac {x^{7/2} (A b-a B)}{a b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 50
Rule 63
Rule 78
Rule 205
Rubi steps
\begin {align*} \int \frac {x^{5/2} (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {x^{5/2} (A+B x)}{(a+b x)^2} \, dx\\ &=\frac {(A b-a B) x^{7/2}}{a b (a+b x)}-\frac {\left (\frac {5 A b}{2}-\frac {7 a B}{2}\right ) \int \frac {x^{5/2}}{a+b x} \, dx}{a b}\\ &=-\frac {(5 A b-7 a B) x^{5/2}}{5 a b^2}+\frac {(A b-a B) x^{7/2}}{a b (a+b x)}+\frac {(5 A b-7 a B) \int \frac {x^{3/2}}{a+b x} \, dx}{2 b^2}\\ &=\frac {(5 A b-7 a B) x^{3/2}}{3 b^3}-\frac {(5 A b-7 a B) x^{5/2}}{5 a b^2}+\frac {(A b-a B) x^{7/2}}{a b (a+b x)}-\frac {(a (5 A b-7 a B)) \int \frac {\sqrt {x}}{a+b x} \, dx}{2 b^3}\\ &=-\frac {a (5 A b-7 a B) \sqrt {x}}{b^4}+\frac {(5 A b-7 a B) x^{3/2}}{3 b^3}-\frac {(5 A b-7 a B) x^{5/2}}{5 a b^2}+\frac {(A b-a B) x^{7/2}}{a b (a+b x)}+\frac {\left (a^2 (5 A b-7 a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 b^4}\\ &=-\frac {a (5 A b-7 a B) \sqrt {x}}{b^4}+\frac {(5 A b-7 a B) x^{3/2}}{3 b^3}-\frac {(5 A b-7 a B) x^{5/2}}{5 a b^2}+\frac {(A b-a B) x^{7/2}}{a b (a+b x)}+\frac {\left (a^2 (5 A b-7 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^4}\\ &=-\frac {a (5 A b-7 a B) \sqrt {x}}{b^4}+\frac {(5 A b-7 a B) x^{3/2}}{3 b^3}-\frac {(5 A b-7 a B) x^{5/2}}{5 a b^2}+\frac {(A b-a B) x^{7/2}}{a b (a+b x)}+\frac {a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 110, normalized size = 0.85 \[ \frac {\sqrt {x} \left (105 a^3 B+a^2 (70 b B x-75 A b)-2 a b^2 x (25 A+7 B x)+2 b^3 x^2 (5 A+3 B x)\right )}{15 b^4 (a+b x)}-\frac {a^{3/2} (7 a B-5 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 290, normalized size = 2.23 \[ \left [-\frac {15 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b + {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (6 \, B b^{3} x^{3} + 105 \, B a^{3} - 75 \, A a^{2} b - 2 \, {\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 10 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt {x}}{30 \, {\left (b^{5} x + a b^{4}\right )}}, -\frac {15 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b + {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (6 \, B b^{3} x^{3} + 105 \, B a^{3} - 75 \, A a^{2} b - 2 \, {\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 10 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt {x}}{15 \, {\left (b^{5} x + a b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 122, normalized size = 0.94 \[ -\frac {{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {B a^{3} \sqrt {x} - A a^{2} b \sqrt {x}}{{\left (b x + a\right )} b^{4}} + \frac {2 \, {\left (3 \, B b^{8} x^{\frac {5}{2}} - 10 \, B a b^{7} x^{\frac {3}{2}} + 5 \, A b^{8} x^{\frac {3}{2}} + 45 \, B a^{2} b^{6} \sqrt {x} - 30 \, A a b^{7} \sqrt {x}\right )}}{15 \, b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 139, normalized size = 1.07 \[ \frac {2 B \,x^{\frac {5}{2}}}{5 b^{2}}+\frac {5 A \,a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}-\frac {7 B \,a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{4}}-\frac {A \,a^{2} \sqrt {x}}{\left (b x +a \right ) b^{3}}+\frac {2 A \,x^{\frac {3}{2}}}{3 b^{2}}+\frac {B \,a^{3} \sqrt {x}}{\left (b x +a \right ) b^{4}}-\frac {4 B a \,x^{\frac {3}{2}}}{3 b^{3}}-\frac {4 A a \sqrt {x}}{b^{3}}+\frac {6 B \,a^{2} \sqrt {x}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 115, normalized size = 0.88 \[ \frac {{\left (B a^{3} - A a^{2} b\right )} \sqrt {x}}{b^{5} x + a b^{4}} - \frac {{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {2 \, {\left (3 \, B b^{2} x^{\frac {5}{2}} - 5 \, {\left (2 \, B a b - A b^{2}\right )} x^{\frac {3}{2}} + 15 \, {\left (3 \, B a^{2} - 2 \, A a b\right )} \sqrt {x}\right )}}{15 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 146, normalized size = 1.12 \[ x^{3/2}\,\left (\frac {2\,A}{3\,b^2}-\frac {4\,B\,a}{3\,b^3}\right )-\sqrt {x}\,\left (\frac {2\,a\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{b}+\frac {2\,B\,a^2}{b^4}\right )+\frac {2\,B\,x^{5/2}}{5\,b^2}+\frac {\sqrt {x}\,\left (B\,a^3-A\,a^2\,b\right )}{x\,b^5+a\,b^4}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,\sqrt {x}\,\left (5\,A\,b-7\,B\,a\right )}{7\,B\,a^3-5\,A\,a^2\,b}\right )\,\left (5\,A\,b-7\,B\,a\right )}{b^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 52.86, size = 1068, normalized size = 8.22 \[ \begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {9}{2}}}{9}}{a^{2}} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{b^{2}} & \text {for}\: a = 0 \\- \frac {150 i A a^{\frac {5}{2}} b^{2} \sqrt {x} \sqrt {\frac {1}{b}}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} - \frac {100 i A a^{\frac {3}{2}} b^{3} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} + \frac {20 i A \sqrt {a} b^{4} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} + \frac {75 A a^{3} b \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} - \frac {75 A a^{3} b \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} + \frac {75 A a^{2} b^{2} x \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} - \frac {75 A a^{2} b^{2} x \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} + \frac {210 i B a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} + \frac {140 i B a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} - \frac {28 i B a^{\frac {3}{2}} b^{3} x^{\frac {5}{2}} \sqrt {\frac {1}{b}}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} + \frac {12 i B \sqrt {a} b^{4} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} - \frac {105 B a^{4} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} + \frac {105 B a^{4} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} - \frac {105 B a^{3} b x \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} + \frac {105 B a^{3} b x \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{30 i a^{\frac {3}{2}} b^{5} \sqrt {\frac {1}{b}} + 30 i \sqrt {a} b^{6} x \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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