Optimal. Leaf size=309 \[ \frac {256 b^4 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{765765 e (d+e x)^{7/2} (b d-a e)^6}+\frac {128 b^3 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{109395 e (d+e x)^{9/2} (b d-a e)^5}+\frac {32 b^2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{12155 e (d+e x)^{11/2} (b d-a e)^4}+\frac {16 b (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{3315 e (d+e x)^{13/2} (b d-a e)^3}+\frac {2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{255 e (d+e x)^{15/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)} \]
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Rubi [A] time = 0.20, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \begin {gather*} \frac {256 b^4 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{765765 e (d+e x)^{7/2} (b d-a e)^6}+\frac {128 b^3 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{109395 e (d+e x)^{9/2} (b d-a e)^5}+\frac {32 b^2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{12155 e (d+e x)^{11/2} (b d-a e)^4}+\frac {16 b (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{3315 e (d+e x)^{13/2} (b d-a e)^3}+\frac {2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{255 e (d+e x)^{15/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {(7 b B d+10 A b e-17 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{17/2}} \, dx}{17 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {(8 b (7 b B d+10 A b e-17 a B e)) \int \frac {(a+b x)^{5/2}}{(d+e x)^{15/2}} \, dx}{255 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac {\left (16 b^2 (7 b B d+10 A b e-17 a B e)\right ) \int \frac {(a+b x)^{5/2}}{(d+e x)^{13/2}} \, dx}{1105 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac {32 b^2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{12155 e (b d-a e)^4 (d+e x)^{11/2}}+\frac {\left (64 b^3 (7 b B d+10 A b e-17 a B e)\right ) \int \frac {(a+b x)^{5/2}}{(d+e x)^{11/2}} \, dx}{12155 e (b d-a e)^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac {32 b^2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{12155 e (b d-a e)^4 (d+e x)^{11/2}}+\frac {128 b^3 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{109395 e (b d-a e)^5 (d+e x)^{9/2}}+\frac {\left (128 b^4 (7 b B d+10 A b e-17 a B e)\right ) \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{109395 e (b d-a e)^5}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac {32 b^2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{12155 e (b d-a e)^4 (d+e x)^{11/2}}+\frac {128 b^3 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{109395 e (b d-a e)^5 (d+e x)^{9/2}}+\frac {256 b^4 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{765765 e (b d-a e)^6 (d+e x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 160, normalized size = 0.52 \begin {gather*} \frac {2 (a+b x)^{7/2} \left (45045 (B d-A e)-\frac {2 (d+e x) \left (8 b (d+e x) \left (2 b (d+e x) \left (4 b (d+e x) (-7 a e+9 b d+2 b e x)+63 (b d-a e)^2\right )+231 (b d-a e)^3\right )+3003 (b d-a e)^4\right ) \left (-\frac {17 a B e}{2}+5 A b e+\frac {7 b B d}{2}\right )}{(b d-a e)^5}\right )}{765765 e (d+e x)^{17/2} (a e-b d)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.65, size = 347, normalized size = 1.12 \begin {gather*} -\frac {2 (a+b x)^{17/2} \left (-\frac {109395 A b^5 (d+e x)^5}{(a+b x)^5}+\frac {425425 A b^4 e (d+e x)^4}{(a+b x)^4}-\frac {696150 A b^3 e^2 (d+e x)^3}{(a+b x)^3}+\frac {589050 A b^2 e^3 (d+e x)^2}{(a+b x)^2}-\frac {255255 A b e^4 (d+e x)}{a+b x}+\frac {109395 a b^4 B (d+e x)^5}{(a+b x)^5}-\frac {85085 b^4 B d (d+e x)^4}{(a+b x)^4}-\frac {340340 a b^3 B e (d+e x)^4}{(a+b x)^4}+\frac {278460 b^3 B d e (d+e x)^3}{(a+b x)^3}+\frac {417690 a b^2 B e^2 (d+e x)^3}{(a+b x)^3}-\frac {353430 b^2 B d e^2 (d+e x)^2}{(a+b x)^2}+\frac {51051 a B e^4 (d+e x)}{a+b x}-\frac {235620 a b B e^3 (d+e x)^2}{(a+b x)^2}+\frac {204204 b B d e^3 (d+e x)}{a+b x}+45045 A e^5-45045 B d e^4\right )}{765765 (d+e x)^{17/2} (b d-a e)^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [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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giac [B] time = 12.92, size = 1752, normalized size = 5.67
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 722, normalized size = 2.34 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-1280 A \,b^{5} e^{5} x^{5}+2176 B a \,b^{4} e^{5} x^{5}-896 B \,b^{5} d \,e^{4} x^{5}+4480 A a \,b^{4} e^{5} x^{4}-10880 A \,b^{5} d \,e^{4} x^{4}-7616 B \,a^{2} b^{3} e^{5} x^{4}+21632 B a \,b^{4} d \,e^{4} x^{4}-7616 B \,b^{5} d^{2} e^{3} x^{4}-10080 A \,a^{2} b^{3} e^{5} x^{3}+38080 A a \,b^{4} d \,e^{4} x^{3}-40800 A \,b^{5} d^{2} e^{3} x^{3}+17136 B \,a^{3} b^{2} e^{5} x^{3}-71792 B \,a^{2} b^{3} d \,e^{4} x^{3}+96016 B a \,b^{4} d^{2} e^{3} x^{3}-28560 B \,b^{5} d^{3} e^{2} x^{3}+18480 A \,a^{3} b^{2} e^{5} x^{2}-85680 A \,a^{2} b^{3} d \,e^{4} x^{2}+142800 A a \,b^{4} d^{2} e^{3} x^{2}-88400 A \,b^{5} d^{3} e^{2} x^{2}-31416 B \,a^{4} b \,e^{5} x^{2}+158592 B \,a^{3} b^{2} d \,e^{4} x^{2}-302736 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+250240 B a \,b^{4} d^{3} e^{2} x^{2}-61880 B \,b^{5} d^{4} e \,x^{2}-30030 A \,a^{4} b \,e^{5} x +157080 A \,a^{3} b^{2} d \,e^{4} x -321300 A \,a^{2} b^{3} d^{2} e^{3} x +309400 A a \,b^{4} d^{3} e^{2} x -121550 A \,b^{5} d^{4} e x +51051 B \,a^{5} e^{5} x -288057 B \,a^{4} b d \,e^{4} x +656166 B \,a^{3} b^{2} d^{2} e^{3} x -750890 B \,a^{2} b^{3} d^{3} e^{2} x +423215 B a \,b^{4} d^{4} e x -85085 B \,b^{5} d^{5} x +45045 A \,a^{5} e^{5}-255255 A \,a^{4} b d \,e^{4}+589050 A \,a^{3} b^{2} d^{2} e^{3}-696150 A \,a^{2} b^{3} d^{3} e^{2}+425425 A a \,b^{4} d^{4} e -109395 A \,b^{5} d^{5}+6006 B \,a^{5} d \,e^{4}-31416 B \,a^{4} b \,d^{2} e^{3}+64260 B \,a^{3} b^{2} d^{3} e^{2}-61880 B \,a^{2} b^{3} d^{4} e +24310 B a \,b^{4} d^{5}\right )}{765765 \left (e x +d \right )^{\frac {17}{2}} \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )} \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.46, size = 1144, normalized size = 3.70 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {x^2\,\sqrt {a+b\,x}\,\left (-243474\,B\,a^7\,b\,e^5+1375122\,B\,a^6\,b^2\,d\,e^4-127050\,A\,a^6\,b^2\,e^5-3143028\,B\,a^5\,b^3\,d^2\,e^3+760410\,A\,a^5\,b^3\,d\,e^4+3619300\,B\,a^4\,b^4\,d^3\,e^2-1892100\,A\,a^4\,b^4\,d^2\,e^3-2044250\,B\,a^3\,b^5\,d^4\,e+2497300\,A\,a^3\,b^5\,d^3\,e^2+364650\,B\,a^2\,b^6\,d^5-1823250\,A\,a^2\,b^6\,d^4\,e+656370\,A\,a\,b^7\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}-\frac {x\,\sqrt {a+b\,x}\,\left (102102\,B\,a^8\,e^5-540078\,B\,a^7\,b\,d\,e^4+210210\,A\,a^7\,b\,e^5+1123836\,B\,a^6\,b^2\,d^2\,e^3-1217370\,A\,a^6\,b^2\,d\,e^4-1116220\,B\,a^5\,b^3\,d^3\,e^2+2891700\,A\,a^5\,b^3\,d^2\,e^3+475150\,B\,a^4\,b^4\,d^4\,e-3558100\,A\,a^4\,b^4\,d^3\,e^2-24310\,B\,a^3\,b^5\,d^5+2309450\,A\,a^3\,b^5\,d^4\,e-656370\,A\,a^2\,b^6\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}-\frac {\sqrt {a+b\,x}\,\left (12012\,B\,a^8\,d\,e^4+90090\,A\,a^8\,e^5-62832\,B\,a^7\,b\,d^2\,e^3-510510\,A\,a^7\,b\,d\,e^4+128520\,B\,a^6\,b^2\,d^3\,e^2+1178100\,A\,a^6\,b^2\,d^2\,e^3-123760\,B\,a^5\,b^3\,d^4\,e-1392300\,A\,a^5\,b^3\,d^3\,e^2+48620\,B\,a^4\,b^4\,d^5+850850\,A\,a^4\,b^4\,d^4\,e-218790\,A\,a^3\,b^5\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}+\frac {x^3\,\sqrt {a+b\,x}\,\left (-152082\,B\,a^6\,b^2\,e^5+908362\,B\,a^5\,b^3\,d\,e^4-630\,A\,a^5\,b^3\,e^5-2249780\,B\,a^4\,b^4\,d^2\,e^3+5950\,A\,a^4\,b^4\,d\,e^4+2932500\,B\,a^3\,b^5\,d^3\,e^2-25500\,A\,a^3\,b^5\,d^2\,e^3-2044250\,B\,a^2\,b^6\,d^4\,e+66300\,A\,a^2\,b^6\,d^3\,e^2+461890\,B\,a\,b^7\,d^5-121550\,A\,a\,b^7\,d^4\,e+218790\,A\,b^8\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}+\frac {256\,b^7\,x^8\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )}{765765\,e^5\,{\left (a\,e-b\,d\right )}^6}-\frac {16\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )\,\left (5\,a^3\,e^3-51\,a^2\,b\,d\,e^2+255\,a\,b^2\,d^2\,e-1105\,b^3\,d^3\right )}{765765\,e^8\,{\left (a\,e-b\,d\right )}^6}-\frac {128\,b^6\,x^7\,\left (a\,e-17\,b\,d\right )\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )}{765765\,e^6\,{\left (a\,e-b\,d\right )}^6}+\frac {2\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )\,\left (7\,a^4\,e^4-68\,a^3\,b\,d\,e^3+306\,a^2\,b^2\,d^2\,e^2-884\,a\,b^3\,d^3\,e+2431\,b^4\,d^4\right )}{153153\,e^9\,{\left (a\,e-b\,d\right )}^6}+\frac {32\,b^5\,x^6\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-34\,a\,b\,d\,e+255\,b^2\,d^2\right )\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )}{765765\,e^7\,{\left (a\,e-b\,d\right )}^6}\right )}{x^9+\frac {d^9}{e^9}+\frac {9\,d\,x^8}{e}+\frac {9\,d^8\,x}{e^8}+\frac {36\,d^2\,x^7}{e^2}+\frac {84\,d^3\,x^6}{e^3}+\frac {126\,d^4\,x^5}{e^4}+\frac {126\,d^5\,x^4}{e^5}+\frac {84\,d^6\,x^3}{e^6}+\frac {36\,d^7\,x^2}{e^7}} \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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