Optimal. Leaf size=147 \[ \frac {4 b (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac {2 (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]
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Rubi [A] time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ \frac {4 b (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac {2 (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {(3 b B d+4 A b e-7 a B e) \int \frac {\sqrt {a+b x}}{(d+e x)^{7/2}} \, dx}{7 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {2 (3 b B d+4 A b e-7 a B e) (a+b x)^{3/2}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac {(2 b (3 b B d+4 A b e-7 a B e)) \int \frac {\sqrt {a+b x}}{(d+e x)^{5/2}} \, dx}{35 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {2 (3 b B d+4 A b e-7 a B e) (a+b x)^{3/2}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac {4 b (3 b B d+4 A b e-7 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 135, normalized size = 0.92 \[ \frac {2 (a+b x)^{3/2} \left (A \left (15 a^2 e^2-6 a b e (7 d+2 e x)+b^2 \left (35 d^2+28 d e x+8 e^2 x^2\right )\right )+B \left (3 a^2 e (2 d+7 e x)-2 a b \left (7 d^2+29 d e x+7 e^2 x^2\right )+3 b^2 d x (7 d+2 e x)\right )\right )}{105 (d+e x)^{7/2} (b d-a e)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 25.26, size = 440, normalized size = 2.99 \[ \frac {2 \, {\left (15 \, A a^{3} e^{2} + 2 \, {\left (3 \, B b^{3} d e - {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x^{3} - 7 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} d^{2} + 6 \, {\left (B a^{3} - 7 \, A a^{2} b\right )} d e + {\left (21 \, B b^{3} d^{2} - 4 \, {\left (13 \, B a b^{2} - 7 \, A b^{3}\right )} d e + {\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x^{2} + {\left (7 \, {\left (B a b^{2} + 5 \, A b^{3}\right )} d^{2} - 2 \, {\left (26 \, B a^{2} b + 7 \, A a b^{2}\right )} d e + 3 \, {\left (7 \, B a^{3} + A a^{2} b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{105 \, {\left (b^{3} d^{7} - 3 \, a b^{2} d^{6} e + 3 \, a^{2} b d^{5} e^{2} - a^{3} d^{4} e^{3} + {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{4} + 4 \, {\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{3} + 6 \, {\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x^{2} + 4 \, {\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} e^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.66, size = 355, normalized size = 2.41 \[ \frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (3 \, B b^{8} d {\left | b \right |} e^{4} - 7 \, B a b^{7} {\left | b \right |} e^{5} + 4 \, A b^{8} {\left | b \right |} e^{5}\right )} {\left (b x + a\right )}}{b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}} + \frac {7 \, {\left (3 \, B b^{9} d^{2} {\left | b \right |} e^{3} - 10 \, B a b^{8} d {\left | b \right |} e^{4} + 4 \, A b^{9} d {\left | b \right |} e^{4} + 7 \, B a^{2} b^{7} {\left | b \right |} e^{5} - 4 \, A a b^{8} {\left | b \right |} e^{5}\right )}}{b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}}\right )} - \frac {35 \, {\left (B a b^{9} d^{2} {\left | b \right |} e^{3} - A b^{10} d^{2} {\left | b \right |} e^{3} - 2 \, B a^{2} b^{8} d {\left | b \right |} e^{4} + 2 \, A a b^{9} d {\left | b \right |} e^{4} + B a^{3} b^{7} {\left | b \right |} e^{5} - A a^{2} b^{8} {\left | b \right |} e^{5}\right )}}{b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}}\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{105 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 177, normalized size = 1.20 \[ -\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (8 A \,b^{2} e^{2} x^{2}-14 B a b \,e^{2} x^{2}+6 B \,b^{2} d e \,x^{2}-12 A a b \,e^{2} x +28 A \,b^{2} d e x +21 B \,a^{2} e^{2} x -58 B a b d e x +21 B \,b^{2} d^{2} x +15 A \,a^{2} e^{2}-42 A a b d e +35 A \,b^{2} d^{2}+6 B \,a^{2} d e -14 B a b \,d^{2}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.10, size = 295, normalized size = 2.01 \[ -\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (12\,B\,a^3\,d\,e+30\,A\,a^3\,e^2-28\,B\,a^2\,b\,d^2-84\,A\,a^2\,b\,d\,e+70\,A\,a\,b^2\,d^2\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\sqrt {a+b\,x}\,\left (42\,B\,a^3\,e^2-104\,B\,a^2\,b\,d\,e+6\,A\,a^2\,b\,e^2+14\,B\,a\,b^2\,d^2-28\,A\,a\,b^2\,d\,e+70\,A\,b^3\,d^2\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^3}+\frac {4\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-7\,B\,a\,e+3\,B\,b\,d\right )}{105\,e^3\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,b\,x^2\,\left (a\,e-7\,b\,d\right )\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-7\,B\,a\,e+3\,B\,b\,d\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^3}\right )}{x^4+\frac {d^4}{e^4}+\frac {4\,d\,x^3}{e}+\frac {4\,d^3\,x}{e^3}+\frac {6\,d^2\,x^2}{e^2}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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