Optimal. Leaf size=306 \[ \frac {x^{9/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^{7/2} (A b-9 a B)}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 (a+b x) (A b-9 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 \sqrt {a} b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 \sqrt {x} (a+b x) (A b-9 a B)}{64 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 x^{3/2} (A b-9 a B)}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 x^{5/2} (A b-9 a B)}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.15, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {770, 78, 47, 50, 63, 205} \begin {gather*} \frac {x^{9/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^{7/2} (A b-9 a B)}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 x^{5/2} (A b-9 a B)}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 x^{3/2} (A b-9 a B)}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 \sqrt {x} (a+b x) (A b-9 a B)}{64 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 (a+b x) (A b-9 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 \sqrt {a} b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 205
Rule 770
Rubi steps
\begin {align*} \int \frac {x^{7/2} (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^{7/2} (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^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (b^2 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{7/2}}{\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^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{5/2}}{\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^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{192 a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {x}}{a b+b^2 x} \, dx}{128 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (A b-9 a B) \sqrt {x} (a+b x)}{64 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{128 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (A b-9 a B) \sqrt {x} (a+b x)}{64 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 (A b-9 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 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (A b-9 a B) \sqrt {x} (a+b x)}{64 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 (A b-9 a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 \sqrt {a} b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 79, normalized size = 0.26 \begin {gather*} \frac {x^{9/2} \left (9 a^4 (A b-a B)-(a+b x)^4 (A b-9 a B) \, _2F_1\left (4,\frac {9}{2};\frac {11}{2};-\frac {b x}{a}\right )\right )}{36 a^5 b (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 32.09, size = 159, normalized size = 0.52 \begin {gather*} \frac {(a+b x) \left (\frac {\sqrt {x} \left (945 a^4 B-105 a^3 A b+3465 a^3 b B x-385 a^2 A b^2 x+4599 a^2 b^2 B x^2-511 a A b^3 x^2+2511 a b^3 B x^3-279 A b^4 x^3+384 b^4 B x^4\right )}{192 b^5 (a+b x)^4}-\frac {35 (9 a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 \sqrt {a} b^{11/2}}\right )}{\sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 555, normalized size = 1.81 \begin {gather*} \left [\frac {105 \, {\left (9 \, B a^{5} - A a^{4} b + {\left (9 \, B a b^{4} - A b^{5}\right )} x^{4} + 4 \, {\left (9 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 6 \, {\left (9 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (9 \, 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 (384 \, B a b^{5} x^{4} + 945 \, B a^{5} b - 105 \, A a^{4} b^{2} + 279 \, {\left (9 \, B a^{2} b^{4} - A a b^{5}\right )} x^{3} + 511 \, {\left (9 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{2} + 385 \, {\left (9 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x\right )} \sqrt {x}}{384 \, {\left (a b^{10} x^{4} + 4 \, a^{2} b^{9} x^{3} + 6 \, a^{3} b^{8} x^{2} + 4 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}, \frac {105 \, {\left (9 \, B a^{5} - A a^{4} b + {\left (9 \, B a b^{4} - A b^{5}\right )} x^{4} + 4 \, {\left (9 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 6 \, {\left (9 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (9 \, 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 (384 \, B a b^{5} x^{4} + 945 \, B a^{5} b - 105 \, A a^{4} b^{2} + 279 \, {\left (9 \, B a^{2} b^{4} - A a b^{5}\right )} x^{3} + 511 \, {\left (9 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{2} + 385 \, {\left (9 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x\right )} \sqrt {x}}{192 \, {\left (a b^{10} x^{4} + 4 \, a^{2} b^{9} x^{3} + 6 \, a^{3} b^{8} x^{2} + 4 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 159, normalized size = 0.52 \begin {gather*} \frac {2 \, B \sqrt {x}}{b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {35 \, {\left (9 \, B a - A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {975 \, B a b^{3} x^{\frac {7}{2}} - 279 \, A b^{4} x^{\frac {7}{2}} + 2295 \, B a^{2} b^{2} x^{\frac {5}{2}} - 511 \, A a b^{3} x^{\frac {5}{2}} + 1929 \, B a^{3} b x^{\frac {3}{2}} - 385 \, A a^{2} b^{2} x^{\frac {3}{2}} + 561 \, B a^{4} \sqrt {x} - 105 \, A a^{3} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} b^{5} \mathrm {sgn}\left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 368, normalized size = 1.20 \begin {gather*} -\frac {\left (-105 A \,b^{5} x^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+945 B a \,b^{4} x^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-420 A a \,b^{4} x^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+3780 B \,a^{2} b^{3} x^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-384 \sqrt {a b}\, B \,b^{4} x^{\frac {9}{2}}-630 A \,a^{2} b^{3} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+5670 B \,a^{3} b^{2} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+279 \sqrt {a b}\, A \,b^{4} x^{\frac {7}{2}}-2511 \sqrt {a b}\, B a \,b^{3} x^{\frac {7}{2}}-420 A \,a^{3} b^{2} x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+3780 B \,a^{4} b x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+511 \sqrt {a b}\, A a \,b^{3} x^{\frac {5}{2}}-4599 \sqrt {a b}\, B \,a^{2} b^{2} x^{\frac {5}{2}}-105 A \,a^{4} b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+945 B \,a^{5} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+385 \sqrt {a b}\, A \,a^{2} b^{2} x^{\frac {3}{2}}-3465 \sqrt {a b}\, B \,a^{3} b \,x^{\frac {3}{2}}+105 \sqrt {a b}\, A \,a^{3} b \sqrt {x}-945 \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}} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.62, size = 374, normalized size = 1.22 \begin {gather*} \frac {105 \, {\left (3 \, {\left (11 \, B a b^{5} - A b^{6}\right )} x^{2} + {\left (9 \, B a^{2} b^{4} + A a b^{5}\right )} x\right )} x^{\frac {9}{2}} + 30 \, {\left ({\left (359 \, B a^{2} b^{4} - 21 \, A a b^{5}\right )} x^{2} + {\left (61 \, B a^{3} b^{3} + 21 \, A a^{2} b^{4}\right )} x\right )} x^{\frac {7}{2}} + 20 \, {\left (66 \, {\left (11 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{2} + 13 \, {\left (9 \, B a^{4} b^{2} + A a^{3} b^{3}\right )} x\right )} x^{\frac {5}{2}} + 2 \, {\left (405 \, {\left (11 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 77 \, {\left (9 \, B a^{5} b + A a^{4} b^{2}\right )} x\right )} x^{\frac {3}{2}} + 7 \, {\left (27 \, {\left (11 \, B a^{5} b - A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{2} b^{9} x^{5} + 5 \, a^{3} b^{8} x^{4} + 10 \, a^{4} b^{7} x^{3} + 10 \, a^{5} b^{6} x^{2} + 5 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} - \frac {35 \, {\left (9 \, B a - A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} b^{5}} - \frac {7 \, {\left (3 \, {\left (11 \, B a b - A b^{2}\right )} x^{\frac {3}{2}} - 10 \, {\left (9 \, B a^{2} - A a b\right )} \sqrt {x}\right )}}{128 \, a^{2} b^{5}} \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^{7/2}\,\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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