Optimal. Leaf size=170 \[ \frac {32 b^3 \sqrt {b x+c x^2} (8 b B-9 A c)}{315 c^5 \sqrt {x}}-\frac {16 b^2 \sqrt {x} \sqrt {b x+c x^2} (8 b B-9 A c)}{315 c^4}+\frac {4 b x^{3/2} \sqrt {b x+c x^2} (8 b B-9 A c)}{105 c^3}-\frac {2 x^{5/2} \sqrt {b x+c x^2} (8 b B-9 A c)}{63 c^2}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c} \]
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Rubi [A] time = 0.15, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {794, 656, 648} \begin {gather*} -\frac {16 b^2 \sqrt {x} \sqrt {b x+c x^2} (8 b B-9 A c)}{315 c^4}+\frac {32 b^3 \sqrt {b x+c x^2} (8 b B-9 A c)}{315 c^5 \sqrt {x}}-\frac {2 x^{5/2} \sqrt {b x+c x^2} (8 b B-9 A c)}{63 c^2}+\frac {4 b x^{3/2} \sqrt {b x+c x^2} (8 b B-9 A c)}{105 c^3}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int \frac {x^{7/2} (A+B x)}{\sqrt {b x+c x^2}} \, dx &=\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c}+\frac {\left (2 \left (\frac {7}{2} (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {x^{7/2}}{\sqrt {b x+c x^2}} \, dx}{9 c}\\ &=-\frac {2 (8 b B-9 A c) x^{5/2} \sqrt {b x+c x^2}}{63 c^2}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c}+\frac {(2 b (8 b B-9 A c)) \int \frac {x^{5/2}}{\sqrt {b x+c x^2}} \, dx}{21 c^2}\\ &=\frac {4 b (8 b B-9 A c) x^{3/2} \sqrt {b x+c x^2}}{105 c^3}-\frac {2 (8 b B-9 A c) x^{5/2} \sqrt {b x+c x^2}}{63 c^2}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c}-\frac {\left (8 b^2 (8 b B-9 A c)\right ) \int \frac {x^{3/2}}{\sqrt {b x+c x^2}} \, dx}{105 c^3}\\ &=-\frac {16 b^2 (8 b B-9 A c) \sqrt {x} \sqrt {b x+c x^2}}{315 c^4}+\frac {4 b (8 b B-9 A c) x^{3/2} \sqrt {b x+c x^2}}{105 c^3}-\frac {2 (8 b B-9 A c) x^{5/2} \sqrt {b x+c x^2}}{63 c^2}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c}+\frac {\left (16 b^3 (8 b B-9 A c)\right ) \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx}{315 c^4}\\ &=\frac {32 b^3 (8 b B-9 A c) \sqrt {b x+c x^2}}{315 c^5 \sqrt {x}}-\frac {16 b^2 (8 b B-9 A c) \sqrt {x} \sqrt {b x+c x^2}}{315 c^4}+\frac {4 b (8 b B-9 A c) x^{3/2} \sqrt {b x+c x^2}}{105 c^3}-\frac {2 (8 b B-9 A c) x^{5/2} \sqrt {b x+c x^2}}{63 c^2}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 94, normalized size = 0.55 \begin {gather*} \frac {2 \sqrt {x (b+c x)} \left (-16 b^3 c (9 A+4 B x)+24 b^2 c^2 x (3 A+2 B x)-2 b c^3 x^2 (27 A+20 B x)+5 c^4 x^3 (9 A+7 B x)+128 b^4 B\right )}{315 c^5 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 107, normalized size = 0.63 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-144 A b^3 c+72 A b^2 c^2 x-54 A b c^3 x^2+45 A c^4 x^3+128 b^4 B-64 b^3 B c x+48 b^2 B c^2 x^2-40 b B c^3 x^3+35 B c^4 x^4\right )}{315 c^5 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 103, normalized size = 0.61 \begin {gather*} \frac {2 \, {\left (35 \, B c^{4} x^{4} + 128 \, B b^{4} - 144 \, A b^{3} c - 5 \, {\left (8 \, B b c^{3} - 9 \, A c^{4}\right )} x^{3} + 6 \, {\left (8 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{2} - 8 \, {\left (8 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{315 \, c^{5} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 135, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (B b^{4} - A b^{3} c\right )} \sqrt {c x + b}}{c^{5}} + \frac {2 \, {\left (35 \, {\left (c x + b\right )}^{\frac {9}{2}} B - 180 \, {\left (c x + b\right )}^{\frac {7}{2}} B b + 378 \, {\left (c x + b\right )}^{\frac {5}{2}} B b^{2} - 420 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{3} + 45 \, {\left (c x + b\right )}^{\frac {7}{2}} A c - 189 \, {\left (c x + b\right )}^{\frac {5}{2}} A b c + 315 \, {\left (c x + b\right )}^{\frac {3}{2}} A b^{2} c\right )}}{315 \, c^{5}} - \frac {32 \, {\left (8 \, B b^{\frac {9}{2}} - 9 \, A b^{\frac {7}{2}} c\right )}}{315 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 107, normalized size = 0.63 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-35 B \,x^{4} c^{4}-45 A \,c^{4} x^{3}+40 B b \,c^{3} x^{3}+54 A b \,c^{3} x^{2}-48 B \,b^{2} c^{2} x^{2}-72 A \,b^{2} c^{2} x +64 B \,b^{3} c x +144 A \,b^{3} c -128 b^{4} B \right ) \sqrt {x}}{315 \sqrt {c \,x^{2}+b x}\, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 120, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (5 \, c^{4} x^{4} - b c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} - 8 \, b^{3} c x - 16 \, b^{4}\right )} A}{35 \, \sqrt {c x + b} c^{4}} + \frac {2 \, {\left (35 \, c^{5} x^{5} - 5 \, b c^{4} x^{4} + 8 \, b^{2} c^{3} x^{3} - 16 \, b^{3} c^{2} x^{2} + 64 \, b^{4} c x + 128 \, b^{5}\right )} B}{315 \, \sqrt {c x + b} c^{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 = -0.01 \begin {gather*} \int \frac {x^{7/2}\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {7}{2}} \left (A + B x\right )}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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