Optimal. Leaf size=309 \[ \frac{256 b^4 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{765765 e (d+e x)^{7/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{109395 e (d+e x)^{9/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{12155 e (d+e x)^{11/2} (b d-a e)^4}+\frac{16 b (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{3315 e (d+e x)^{13/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{255 e (d+e x)^{15/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)} \]
[Out]
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Rubi [A] time = 0.584694, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{256 b^4 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{765765 e (d+e x)^{7/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{109395 e (d+e x)^{9/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{12155 e (d+e x)^{11/2} (b d-a e)^4}+\frac{16 b (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{3315 e (d+e x)^{13/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{255 e (d+e x)^{15/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(19/2),x]
[Out]
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Rubi in Sympy [A] time = 67.1479, size = 301, normalized size = 0.97 \[ \frac{256 b^{4} \left (a + b x\right )^{\frac{7}{2}} \left (10 A b e - 17 B a e + 7 B b d\right )}{765765 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{6}} - \frac{128 b^{3} \left (a + b x\right )^{\frac{7}{2}} \left (10 A b e - 17 B a e + 7 B b d\right )}{109395 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{5}} + \frac{32 b^{2} \left (a + b x\right )^{\frac{7}{2}} \left (10 A b e - 17 B a e + 7 B b d\right )}{12155 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{4}} - \frac{16 b \left (a + b x\right )^{\frac{7}{2}} \left (10 A b e - 17 B a e + 7 B b d\right )}{3315 e \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (10 A b e - 17 B a e + 7 B b d\right )}{255 e \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{17 e \left (d + e x\right )^{\frac{17}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(19/2),x)
[Out]
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Mathematica [A] time = 0.877125, size = 331, normalized size = 1.07 \[ \frac{2 \sqrt{a+b x} \left (\frac{128 b^7 (d+e x)^8 (-17 a B e+10 A b e+7 b B d)}{(b d-a e)^6}+\frac{64 b^6 (d+e x)^7 (-17 a B e+10 A b e+7 b B d)}{(b d-a e)^5}+\frac{48 b^5 (d+e x)^6 (-17 a B e+10 A b e+7 b B d)}{(b d-a e)^4}+\frac{40 b^4 (d+e x)^5 (-17 a B e+10 A b e+7 b B d)}{(b d-a e)^3}+\frac{35 b^3 (d+e x)^4 (-17 a B e+10 A b e+7 b B d)}{(b d-a e)^2}-\frac{63 b^2 (d+e x)^3 (1207 a B e+5 A b e-1212 b B d)}{a e-b d}+231 b (d+e x)^2 (-527 a B e-275 A b e+802 b B d)-3003 (d+e x) (a e-b d) (17 a B e+35 A b e-52 b B d)+45045 (b d-a e)^2 (B d-A e)\right )}{765765 e^4 (d+e x)^{17/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(19/2),x]
[Out]
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Maple [B] time = 0.017, size = 722, normalized size = 2.3 \[ -{\frac{-2560\,A{b}^{5}{e}^{5}{x}^{5}+4352\,Ba{b}^{4}{e}^{5}{x}^{5}-1792\,B{b}^{5}d{e}^{4}{x}^{5}+8960\,Aa{b}^{4}{e}^{5}{x}^{4}-21760\,A{b}^{5}d{e}^{4}{x}^{4}-15232\,B{a}^{2}{b}^{3}{e}^{5}{x}^{4}+43264\,Ba{b}^{4}d{e}^{4}{x}^{4}-15232\,B{b}^{5}{d}^{2}{e}^{3}{x}^{4}-20160\,A{a}^{2}{b}^{3}{e}^{5}{x}^{3}+76160\,Aa{b}^{4}d{e}^{4}{x}^{3}-81600\,A{b}^{5}{d}^{2}{e}^{3}{x}^{3}+34272\,B{a}^{3}{b}^{2}{e}^{5}{x}^{3}-143584\,B{a}^{2}{b}^{3}d{e}^{4}{x}^{3}+192032\,Ba{b}^{4}{d}^{2}{e}^{3}{x}^{3}-57120\,B{b}^{5}{d}^{3}{e}^{2}{x}^{3}+36960\,A{a}^{3}{b}^{2}{e}^{5}{x}^{2}-171360\,A{a}^{2}{b}^{3}d{e}^{4}{x}^{2}+285600\,Aa{b}^{4}{d}^{2}{e}^{3}{x}^{2}-176800\,A{b}^{5}{d}^{3}{e}^{2}{x}^{2}-62832\,B{a}^{4}b{e}^{5}{x}^{2}+317184\,B{a}^{3}{b}^{2}d{e}^{4}{x}^{2}-605472\,B{a}^{2}{b}^{3}{d}^{2}{e}^{3}{x}^{2}+500480\,Ba{b}^{4}{d}^{3}{e}^{2}{x}^{2}-123760\,B{b}^{5}{d}^{4}e{x}^{2}-60060\,A{a}^{4}b{e}^{5}x+314160\,A{a}^{3}{b}^{2}d{e}^{4}x-642600\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}x+618800\,Aa{b}^{4}{d}^{3}{e}^{2}x-243100\,A{b}^{5}{d}^{4}ex+102102\,B{a}^{5}{e}^{5}x-576114\,B{a}^{4}bd{e}^{4}x+1312332\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}x-1501780\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}x+846430\,Ba{b}^{4}{d}^{4}ex-170170\,B{b}^{5}{d}^{5}x+90090\,A{a}^{5}{e}^{5}-510510\,A{a}^{4}bd{e}^{4}+1178100\,A{a}^{3}{b}^{2}{d}^{2}{e}^{3}-1392300\,A{a}^{2}{b}^{3}{d}^{3}{e}^{2}+850850\,Aa{b}^{4}{d}^{4}e-218790\,A{b}^{5}{d}^{5}+12012\,B{a}^{5}d{e}^{4}-62832\,B{a}^{4}b{d}^{2}{e}^{3}+128520\,B{a}^{3}{b}^{2}{d}^{3}{e}^{2}-123760\,B{a}^{2}{b}^{3}{d}^{4}e+48620\,Ba{b}^{4}{d}^{5}}{765765\,{a}^{6}{e}^{6}-4594590\,{a}^{5}bd{e}^{5}+11486475\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-15315300\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+11486475\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-4594590\,a{b}^{5}{d}^{5}e+765765\,{b}^{6}{d}^{6}} \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( ex+d \right ) ^{-{\frac{17}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(19/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(19/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 68.8735, size = 2603, normalized size = 8.42 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(19/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(19/2),x)
[Out]
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GIAC/XCAS [A] time = 1.01832, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(19/2),x, algorithm="giac")
[Out]