Optimal. Leaf size=258 \[ \frac {x^{7/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {x^{5/2} (7 a B+A b)}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 \sqrt {x} (7 a B+A b)}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 x^{3/2} (7 a B+A b)}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (a+b x) (7 a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{3/2} b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.13, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {770, 78, 47, 63, 205} \begin {gather*} \frac {x^{7/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {x^{5/2} (7 a B+A b)}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 x^{3/2} (7 a B+A b)}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 \sqrt {x} (7 a B+A b)}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (a+b x) (7 a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{3/2} b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 205
Rule 770
Rubi steps
\begin {align*} \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {x^{5/2} (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) x^{7/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (A b+7 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{5/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 a \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) x^{7/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b+7 a B) x^{5/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 (A b+7 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{3/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 a \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) x^{7/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b+7 a B) x^{5/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 (A b+7 a B) x^{3/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 (A b+7 a B) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {x}}{\left (a b+b^2 x\right )^2} \, dx}{64 a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 (A b+7 a B) \sqrt {x}}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{7/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b+7 a B) x^{5/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 (A b+7 a B) x^{3/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 (A b+7 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{128 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 (A b+7 a B) \sqrt {x}}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{7/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b+7 a B) x^{5/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 (A b+7 a B) x^{3/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 (A b+7 a B) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 (A b+7 a B) \sqrt {x}}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{7/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b+7 a B) x^{5/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 (A b+7 a B) x^{3/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (A b+7 a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{3/2} b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 146, normalized size = 0.57 \begin {gather*} \frac {15 (a+b x)^4 (7 a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )-\sqrt {a} \sqrt {b} \sqrt {x} \left (105 a^4 B+5 a^3 b (3 A+77 B x)+a^2 b^2 x (55 A+511 B x)+a b^3 x^2 (73 A+279 B x)-15 A b^4 x^3\right )}{192 a^{3/2} b^{9/2} (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 23.55, size = 152, normalized size = 0.59 \begin {gather*} \frac {(a+b x) \left (\frac {5 (7 a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{3/2} b^{9/2}}-\frac {\sqrt {x} \left (105 a^4 B+15 a^3 A b+385 a^3 b B x+55 a^2 A b^2 x+511 a^2 b^2 B x^2+73 a A b^3 x^2+279 a b^3 B x^3-15 A b^4 x^3\right )}{192 a b^4 (a+b x)^4}\right )}{\sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 525, normalized size = 2.03 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B a^{5} + A a^{4} b + {\left (7 \, B a b^{4} + A b^{5}\right )} x^{4} + 4 \, {\left (7 \, B a^{2} b^{3} + A a b^{4}\right )} x^{3} + 6 \, {\left (7 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (7 \, B a^{4} b + A a^{3} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (105 \, B a^{5} b + 15 \, A a^{4} b^{2} + 3 \, {\left (93 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{3} + 73 \, {\left (7 \, B a^{3} b^{3} + A a^{2} b^{4}\right )} x^{2} + 55 \, {\left (7 \, B a^{4} b^{2} + A a^{3} b^{3}\right )} x\right )} \sqrt {x}}{384 \, {\left (a^{2} b^{9} x^{4} + 4 \, a^{3} b^{8} x^{3} + 6 \, a^{4} b^{7} x^{2} + 4 \, a^{5} b^{6} x + a^{6} b^{5}\right )}}, -\frac {15 \, {\left (7 \, B a^{5} + A a^{4} b + {\left (7 \, B a b^{4} + A b^{5}\right )} x^{4} + 4 \, {\left (7 \, B a^{2} b^{3} + A a b^{4}\right )} x^{3} + 6 \, {\left (7 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (7 \, B a^{4} b + A a^{3} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (105 \, B a^{5} b + 15 \, A a^{4} b^{2} + 3 \, {\left (93 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{3} + 73 \, {\left (7 \, B a^{3} b^{3} + A a^{2} b^{4}\right )} x^{2} + 55 \, {\left (7 \, B a^{4} b^{2} + A a^{3} b^{3}\right )} x\right )} \sqrt {x}}{192 \, {\left (a^{2} b^{9} x^{4} + 4 \, a^{3} b^{8} x^{3} + 6 \, a^{4} b^{7} x^{2} + 4 \, a^{5} b^{6} x + a^{6} b^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 147, normalized size = 0.57 \begin {gather*} \frac {5 \, {\left (7 \, B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a b^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {279 \, B a b^{3} x^{\frac {7}{2}} - 15 \, A b^{4} x^{\frac {7}{2}} + 511 \, B a^{2} b^{2} x^{\frac {5}{2}} + 73 \, A a b^{3} x^{\frac {5}{2}} + 385 \, B a^{3} b x^{\frac {3}{2}} + 55 \, A a^{2} b^{2} x^{\frac {3}{2}} + 105 \, B a^{4} \sqrt {x} + 15 \, A a^{3} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} a b^{4} \mathrm {sgn}\left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 357, normalized size = 1.38 \begin {gather*} \frac {\left (15 A \,b^{5} x^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+105 B a \,b^{4} x^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+60 A a \,b^{4} x^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+420 B \,a^{2} b^{3} x^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+90 A \,a^{2} b^{3} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+630 B \,a^{3} b^{2} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+15 \sqrt {a b}\, A \,b^{4} x^{\frac {7}{2}}-279 \sqrt {a b}\, B a \,b^{3} x^{\frac {7}{2}}+60 A \,a^{3} b^{2} x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+420 B \,a^{4} b x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-73 \sqrt {a b}\, A a \,b^{3} x^{\frac {5}{2}}-511 \sqrt {a b}\, B \,a^{2} b^{2} x^{\frac {5}{2}}+15 A \,a^{4} b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+105 B \,a^{5} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-55 \sqrt {a b}\, A \,a^{2} b^{2} x^{\frac {3}{2}}-385 \sqrt {a b}\, B \,a^{3} b \,x^{\frac {3}{2}}-15 \sqrt {a b}\, A \,a^{3} b \sqrt {x}-105 \sqrt {a b}\, B \,a^{4} \sqrt {x}\right ) \left (b x +a \right )}{192 \sqrt {a b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a \,b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.75, size = 376, normalized size = 1.46 \begin {gather*} -\frac {5 \, {\left (7 \, {\left (9 \, B a b^{5} + A b^{6}\right )} x^{2} - 3 \, {\left (7 \, B a^{2} b^{4} + 3 \, A a b^{5}\right )} x\right )} x^{\frac {9}{2}} + 10 \, {\left (7 \, {\left (9 \, B a^{2} b^{4} + A a b^{5}\right )} x^{2} - 9 \, {\left (7 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x\right )} x^{\frac {7}{2}} + 20 \, {\left (2 \, {\left (33 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{2} - {\left (13 \, B a^{4} b^{2} + 33 \, A a^{3} b^{3}\right )} x\right )} x^{\frac {5}{2}} + 2 \, {\left (45 \, {\left (9 \, B a^{4} b^{2} + A a^{3} b^{3}\right )} x^{2} - 11 \, {\left (7 \, B a^{5} b + 3 \, A a^{4} b^{2}\right )} x\right )} x^{\frac {3}{2}} + {\left (21 \, {\left (9 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} - 5 \, {\left (7 \, B a^{6} + 3 \, A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{3} b^{8} x^{5} + 5 \, a^{4} b^{7} x^{4} + 10 \, a^{5} b^{6} x^{3} + 10 \, a^{6} b^{5} x^{2} + 5 \, a^{7} b^{4} x + a^{8} b^{3}\right )}} + \frac {5 \, {\left (7 \, B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a b^{4}} + \frac {7 \, {\left (9 \, B a b + A b^{2}\right )} x^{\frac {3}{2}} - 30 \, {\left (7 \, B a^{2} + A a b\right )} \sqrt {x}}{384 \, a^{3} b^{4}} \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 = -0.00 \begin {gather*} \int \frac {x^{5/2}\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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