Optimal. Leaf size=198 \[ \frac {16 b^2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac {8 b (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac {2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{21 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]
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Rubi [A] time = 0.12, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ \frac {16 b^2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac {8 b (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac {2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{21 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]
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)^{11/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {(b B d+2 A b e-3 a B e) \int \frac {\sqrt {a+b x}}{(d+e x)^{9/2}} \, dx}{3 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{21 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {(4 b (b B d+2 A b e-3 a B e)) \int \frac {\sqrt {a+b x}}{(d+e x)^{7/2}} \, dx}{21 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{21 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {8 b (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac {\left (8 b^2 (b B d+2 A b e-3 a B e)\right ) \int \frac {\sqrt {a+b x}}{(d+e x)^{5/2}} \, dx}{105 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{21 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {8 b (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{315 e (b d-a e)^4 (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 113, normalized size = 0.57 \[ \frac {2 (a+b x)^{3/2} \left (105 (B d-A e)-\frac {3 (d+e x) \left (4 b (d+e x) (-3 a e+5 b d+2 b e x)+15 (b d-a e)^2\right ) (-3 a B e+2 A b e+b B d)}{(b d-a e)^3}\right )}{945 e (d+e x)^{9/2} (a e-b d)} \]
Antiderivative was successfully verified.
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fricas [B] time = 108.25, size = 710, normalized size = 3.59 \[ -\frac {2 \, {\left (35 \, A a^{4} e^{3} - 8 \, {\left (B b^{4} d e^{2} - {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} e^{3}\right )} x^{4} + 21 \, {\left (2 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} d^{3} - 9 \, {\left (4 \, B a^{3} b - 21 \, A a^{2} b^{2}\right )} d^{2} e + 5 \, {\left (2 \, B a^{4} - 27 \, A a^{3} b\right )} d e^{2} - 4 \, {\left (9 \, B b^{4} d^{2} e - 2 \, {\left (14 \, B a b^{3} - 9 \, A b^{4}\right )} d e^{2} + {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} e^{3}\right )} x^{3} - 3 \, {\left (21 \, B b^{4} d^{3} - 3 \, {\left (23 \, B a b^{3} - 14 \, A b^{4}\right )} d^{2} e + {\left (19 \, B a^{2} b^{2} - 12 \, A a b^{3}\right )} d e^{2} - {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} e^{3}\right )} x^{2} - {\left (21 \, {\left (B a b^{3} + 5 \, A b^{4}\right )} d^{3} - 9 \, {\left (23 \, B a^{2} b^{2} + 7 \, A a b^{3}\right )} d^{2} e + {\left (167 \, B a^{3} b + 27 \, A a^{2} b^{2}\right )} d e^{2} - 5 \, {\left (9 \, B a^{4} + A a^{3} b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{315 \, {\left (b^{4} d^{9} - 4 \, a b^{3} d^{8} e + 6 \, a^{2} b^{2} d^{7} e^{2} - 4 \, a^{3} b d^{6} e^{3} + a^{4} d^{5} e^{4} + {\left (b^{4} d^{4} e^{5} - 4 \, a b^{3} d^{3} e^{6} + 6 \, a^{2} b^{2} d^{2} e^{7} - 4 \, a^{3} b d e^{8} + a^{4} e^{9}\right )} x^{5} + 5 \, {\left (b^{4} d^{5} e^{4} - 4 \, a b^{3} d^{4} e^{5} + 6 \, a^{2} b^{2} d^{3} e^{6} - 4 \, a^{3} b d^{2} e^{7} + a^{4} d e^{8}\right )} x^{4} + 10 \, {\left (b^{4} d^{6} e^{3} - 4 \, a b^{3} d^{5} e^{4} + 6 \, a^{2} b^{2} d^{4} e^{5} - 4 \, a^{3} b d^{3} e^{6} + a^{4} d^{2} e^{7}\right )} x^{3} + 10 \, {\left (b^{4} d^{7} e^{2} - 4 \, a b^{3} d^{6} e^{3} + 6 \, a^{2} b^{2} d^{5} e^{4} - 4 \, a^{3} b d^{4} e^{5} + a^{4} d^{3} e^{6}\right )} x^{2} + 5 \, {\left (b^{4} d^{8} e - 4 \, a b^{3} d^{7} e^{2} + 6 \, a^{2} b^{2} d^{6} e^{3} - 4 \, a^{3} b d^{5} e^{4} + a^{4} d^{4} e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.62, size = 579, normalized size = 2.92 \[ \frac {2 \, {\left ({\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (B b^{10} d {\left | b \right |} e^{6} - 3 \, B a b^{9} {\left | b \right |} e^{7} + 2 \, A b^{10} {\left | b \right |} e^{7}\right )} {\left (b x + a\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}} + \frac {9 \, {\left (B b^{11} d^{2} {\left | b \right |} e^{5} - 4 \, B a b^{10} d {\left | b \right |} e^{6} + 2 \, A b^{11} d {\left | b \right |} e^{6} + 3 \, B a^{2} b^{9} {\left | b \right |} e^{7} - 2 \, A a b^{10} {\left | b \right |} e^{7}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} + \frac {63 \, {\left (B b^{12} d^{3} {\left | b \right |} e^{4} - 5 \, B a b^{11} d^{2} {\left | b \right |} e^{5} + 2 \, A b^{12} d^{2} {\left | b \right |} e^{5} + 7 \, B a^{2} b^{10} d {\left | b \right |} e^{6} - 4 \, A a b^{11} d {\left | b \right |} e^{6} - 3 \, B a^{3} b^{9} {\left | b \right |} e^{7} + 2 \, A a^{2} b^{10} {\left | b \right |} e^{7}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} {\left (b x + a\right )} - \frac {105 \, {\left (B a b^{12} d^{3} {\left | b \right |} e^{4} - A b^{13} d^{3} {\left | b \right |} e^{4} - 3 \, B a^{2} b^{11} d^{2} {\left | b \right |} e^{5} + 3 \, A a b^{12} d^{2} {\left | b \right |} e^{5} + 3 \, B a^{3} b^{10} d {\left | b \right |} e^{6} - 3 \, A a^{2} b^{11} d {\left | b \right |} e^{6} - B a^{4} b^{9} {\left | b \right |} e^{7} + A a^{3} b^{10} {\left | b \right |} e^{7}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{315 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 322, normalized size = 1.63 \[ -\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-16 A \,b^{3} e^{3} x^{3}+24 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}-72 A \,b^{3} d \,e^{2} x^{2}-36 B \,a^{2} b \,e^{3} x^{2}+120 B a \,b^{2} d \,e^{2} x^{2}-36 B \,b^{3} d^{2} e \,x^{2}-30 A \,a^{2} b \,e^{3} x +108 A a \,b^{2} d \,e^{2} x -126 A \,b^{3} d^{2} e x +45 B \,a^{3} e^{3} x -177 B \,a^{2} b d \,e^{2} x +243 B a \,b^{2} d^{2} e x -63 B \,b^{3} d^{3} x +35 A \,a^{3} e^{3}-135 A \,a^{2} b d \,e^{2}+189 A a \,b^{2} d^{2} e -105 A \,b^{3} d^{3}+10 B \,a^{3} d \,e^{2}-36 B \,a^{2} b \,d^{2} e +42 B a \,b^{2} d^{3}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )} \]
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 = 2.34, size = 428, normalized size = 2.16 \[ \frac {\sqrt {d+e\,x}\,\left (\frac {x\,\sqrt {a+b\,x}\,\left (-90\,B\,a^4\,e^3+334\,B\,a^3\,b\,d\,e^2-10\,A\,a^3\,b\,e^3-414\,B\,a^2\,b^2\,d^2\,e+54\,A\,a^2\,b^2\,d\,e^2+42\,B\,a\,b^3\,d^3-126\,A\,a\,b^3\,d^2\,e+210\,A\,b^4\,d^3\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^4}-\frac {\sqrt {a+b\,x}\,\left (20\,B\,a^4\,d\,e^2+70\,A\,a^4\,e^3-72\,B\,a^3\,b\,d^2\,e-270\,A\,a^3\,b\,d\,e^2+84\,B\,a^2\,b^2\,d^3+378\,A\,a^2\,b^2\,d^2\,e-210\,A\,a\,b^3\,d^3\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{315\,e^3\,{\left (a\,e-b\,d\right )}^4}-\frac {8\,b^2\,x^3\,\left (a\,e-9\,b\,d\right )\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{315\,e^4\,{\left (a\,e-b\,d\right )}^4}+\frac {2\,b\,x^2\,\sqrt {a+b\,x}\,\left (a^2\,e^2-6\,a\,b\,d\,e+21\,b^2\,d^2\right )\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{105\,e^5\,{\left (a\,e-b\,d\right )}^4}\right )}{x^5+\frac {d^5}{e^5}+\frac {5\,d\,x^4}{e}+\frac {5\,d^4\,x}{e^4}+\frac {10\,d^2\,x^3}{e^2}+\frac {10\,d^3\,x^2}{e^3}} \]
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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