3.2193 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=309 \[ \frac{256 b^4 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{45045 e (d+e x)^{3/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac{16 b (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(3*b*B
*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2)
) + (16*b*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(1287*e*(b*d - a*e)^3
*(d + e*x)^(9/2)) + (32*b^2*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(30
03*e*(b*d - a*e)^4*(d + e*x)^(7/2)) + (128*b^3*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(
a + b*x)^(3/2))/(15015*e*(b*d - a*e)^5*(d + e*x)^(5/2)) + (256*b^4*(3*b*B*d + 10
*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(45045*e*(b*d - a*e)^6*(d + e*x)^(3/2))

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Rubi [A]  time = 0.577748, 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 \[ \frac{256 b^4 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{45045 e (d+e x)^{3/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac{16 b (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(3*b*B
*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2)
) + (16*b*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(1287*e*(b*d - a*e)^3
*(d + e*x)^(9/2)) + (32*b^2*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(30
03*e*(b*d - a*e)^4*(d + e*x)^(7/2)) + (128*b^3*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(
a + b*x)^(3/2))/(15015*e*(b*d - a*e)^5*(d + e*x)^(5/2)) + (256*b^4*(3*b*B*d + 10
*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(45045*e*(b*d - a*e)^6*(d + e*x)^(3/2))

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Rubi in Sympy [A]  time = 67.6272, size = 301, normalized size = 0.97 \[ \frac{256 b^{4} \left (a + b x\right )^{\frac{3}{2}} \left (10 A b e - 13 B a e + 3 B b d\right )}{45045 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{6}} - \frac{128 b^{3} \left (a + b x\right )^{\frac{3}{2}} \left (10 A b e - 13 B a e + 3 B b d\right )}{15015 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{5}} + \frac{32 b^{2} \left (a + b x\right )^{\frac{3}{2}} \left (10 A b e - 13 B a e + 3 B b d\right )}{3003 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4}} - \frac{16 b \left (a + b x\right )^{\frac{3}{2}} \left (10 A b e - 13 B a e + 3 B b d\right )}{1287 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (10 A b e - 13 B a e + 3 B b d\right )}{143 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (A e - B d\right )}{13 e \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(15/2),x)

[Out]

256*b**4*(a + b*x)**(3/2)*(10*A*b*e - 13*B*a*e + 3*B*b*d)/(45045*e*(d + e*x)**(3
/2)*(a*e - b*d)**6) - 128*b**3*(a + b*x)**(3/2)*(10*A*b*e - 13*B*a*e + 3*B*b*d)/
(15015*e*(d + e*x)**(5/2)*(a*e - b*d)**5) + 32*b**2*(a + b*x)**(3/2)*(10*A*b*e -
13*B*a*e + 3*B*b*d)/(3003*e*(d + e*x)**(7/2)*(a*e - b*d)**4) - 16*b*(a + b*x)**
(3/2)*(10*A*b*e - 13*B*a*e + 3*B*b*d)/(1287*e*(d + e*x)**(9/2)*(a*e - b*d)**3) +
2*(a + b*x)**(3/2)*(10*A*b*e - 13*B*a*e + 3*B*b*d)/(143*e*(d + e*x)**(11/2)*(a*
e - b*d)**2) - 2*(a + b*x)**(3/2)*(A*e - B*d)/(13*e*(d + e*x)**(13/2)*(a*e - b*d
))

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Mathematica [A]  time = 0.528408, size = 256, normalized size = 0.83 \[ \frac{2 \sqrt{a+b x} \left (\frac{128 b^5 (d+e x)^6 (-13 a B e+10 A b e+3 b B d)}{(b d-a e)^6}+\frac{64 b^4 (d+e x)^5 (-13 a B e+10 A b e+3 b B d)}{(b d-a e)^5}+\frac{48 b^3 (d+e x)^4 (-13 a B e+10 A b e+3 b B d)}{(b d-a e)^4}+\frac{40 b^2 (d+e x)^3 (-13 a B e+10 A b e+3 b B d)}{(b d-a e)^3}+\frac{35 b (d+e x)^2 (-13 a B e+10 A b e+3 b B d)}{(b d-a e)^2}-\frac{315 (d+e x) (13 a B e+A b e-14 b B d)}{a e-b d}+3465 (B d-A e)\right )}{45045 e^2 (d+e x)^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(2*Sqrt[a + b*x]*(3465*(B*d - A*e) - (315*(-14*b*B*d + A*b*e + 13*a*B*e)*(d + e*
x))/(-(b*d) + a*e) + (35*b*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(d + e*x)^2)/(b*d - a
*e)^2 + (40*b^2*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(d + e*x)^3)/(b*d - a*e)^3 + (48
*b^3*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(d + e*x)^4)/(b*d - a*e)^4 + (64*b^4*(3*b*B
*d + 10*A*b*e - 13*a*B*e)*(d + e*x)^5)/(b*d - a*e)^5 + (128*b^5*(3*b*B*d + 10*A*
b*e - 13*a*B*e)*(d + e*x)^6)/(b*d - a*e)^6))/(45045*e^2*(d + e*x)^(13/2))

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Maple [B]  time = 0.019, size = 722, normalized size = 2.3 \[ -{\frac{-2560\,A{b}^{5}{e}^{5}{x}^{5}+3328\,Ba{b}^{4}{e}^{5}{x}^{5}-768\,B{b}^{5}d{e}^{4}{x}^{5}+3840\,Aa{b}^{4}{e}^{5}{x}^{4}-16640\,A{b}^{5}d{e}^{4}{x}^{4}-4992\,B{a}^{2}{b}^{3}{e}^{5}{x}^{4}+22784\,Ba{b}^{4}d{e}^{4}{x}^{4}-4992\,B{b}^{5}{d}^{2}{e}^{3}{x}^{4}-4800\,A{a}^{2}{b}^{3}{e}^{5}{x}^{3}+24960\,Aa{b}^{4}d{e}^{4}{x}^{3}-45760\,A{b}^{5}{d}^{2}{e}^{3}{x}^{3}+6240\,B{a}^{3}{b}^{2}{e}^{5}{x}^{3}-33888\,B{a}^{2}{b}^{3}d{e}^{4}{x}^{3}+66976\,Ba{b}^{4}{d}^{2}{e}^{3}{x}^{3}-13728\,B{b}^{5}{d}^{3}{e}^{2}{x}^{3}+5600\,A{a}^{3}{b}^{2}{e}^{5}{x}^{2}-31200\,A{a}^{2}{b}^{3}d{e}^{4}{x}^{2}+68640\,Aa{b}^{4}{d}^{2}{e}^{3}{x}^{2}-68640\,A{b}^{5}{d}^{3}{e}^{2}{x}^{2}-7280\,B{a}^{4}b{e}^{5}{x}^{2}+42240\,B{a}^{3}{b}^{2}d{e}^{4}{x}^{2}-98592\,B{a}^{2}{b}^{3}{d}^{2}{e}^{3}{x}^{2}+109824\,Ba{b}^{4}{d}^{3}{e}^{2}{x}^{2}-20592\,B{b}^{5}{d}^{4}e{x}^{2}-6300\,A{a}^{4}b{e}^{5}x+36400\,A{a}^{3}{b}^{2}d{e}^{4}x-85800\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}x+102960\,Aa{b}^{4}{d}^{3}{e}^{2}x-60060\,A{b}^{5}{d}^{4}ex+8190\,B{a}^{5}{e}^{5}x-49210\,B{a}^{4}bd{e}^{4}x+122460\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}x-159588\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}x+108966\,Ba{b}^{4}{d}^{4}ex-18018\,B{b}^{5}{d}^{5}x+6930\,A{a}^{5}{e}^{5}-40950\,A{a}^{4}bd{e}^{4}+100100\,A{a}^{3}{b}^{2}{d}^{2}{e}^{3}-128700\,A{a}^{2}{b}^{3}{d}^{3}{e}^{2}+90090\,Aa{b}^{4}{d}^{4}e-30030\,A{b}^{5}{d}^{5}+1260\,B{a}^{5}d{e}^{4}-7280\,B{a}^{4}b{d}^{2}{e}^{3}+17160\,B{a}^{3}{b}^{2}{d}^{3}{e}^{2}-20592\,B{a}^{2}{b}^{3}{d}^{4}e+12012\,Ba{b}^{4}{d}^{5}}{45045\,{a}^{6}{e}^{6}-270270\,{a}^{5}bd{e}^{5}+675675\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-900900\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+675675\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-270270\,a{b}^{5}{d}^{5}e+45045\,{b}^{6}{d}^{6}} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(15/2),x)

[Out]

-2/45045*(b*x+a)^(3/2)*(-1280*A*b^5*e^5*x^5+1664*B*a*b^4*e^5*x^5-384*B*b^5*d*e^4
*x^5+1920*A*a*b^4*e^5*x^4-8320*A*b^5*d*e^4*x^4-2496*B*a^2*b^3*e^5*x^4+11392*B*a*
b^4*d*e^4*x^4-2496*B*b^5*d^2*e^3*x^4-2400*A*a^2*b^3*e^5*x^3+12480*A*a*b^4*d*e^4*
x^3-22880*A*b^5*d^2*e^3*x^3+3120*B*a^3*b^2*e^5*x^3-16944*B*a^2*b^3*d*e^4*x^3+334
88*B*a*b^4*d^2*e^3*x^3-6864*B*b^5*d^3*e^2*x^3+2800*A*a^3*b^2*e^5*x^2-15600*A*a^2
*b^3*d*e^4*x^2+34320*A*a*b^4*d^2*e^3*x^2-34320*A*b^5*d^3*e^2*x^2-3640*B*a^4*b*e^
5*x^2+21120*B*a^3*b^2*d*e^4*x^2-49296*B*a^2*b^3*d^2*e^3*x^2+54912*B*a*b^4*d^3*e^
2*x^2-10296*B*b^5*d^4*e*x^2-3150*A*a^4*b*e^5*x+18200*A*a^3*b^2*d*e^4*x-42900*A*a
^2*b^3*d^2*e^3*x+51480*A*a*b^4*d^3*e^2*x-30030*A*b^5*d^4*e*x+4095*B*a^5*e^5*x-24
605*B*a^4*b*d*e^4*x+61230*B*a^3*b^2*d^2*e^3*x-79794*B*a^2*b^3*d^3*e^2*x+54483*B*
a*b^4*d^4*e*x-9009*B*b^5*d^5*x+3465*A*a^5*e^5-20475*A*a^4*b*d*e^4+50050*A*a^3*b^
2*d^2*e^3-64350*A*a^2*b^3*d^3*e^2+45045*A*a*b^4*d^4*e-15015*A*b^5*d^5+630*B*a^5*
d*e^4-3640*B*a^4*b*d^2*e^3+8580*B*a^3*b^2*d^3*e^2-10296*B*a^2*b^3*d^4*e+6006*B*a
*b^4*d^5)/(e*x+d)^(13/2)/(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)

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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)*sqrt(b*x + a)/(e*x + d)^(15/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 14.5667, size = 1926, normalized size = 6.23 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(15/2),x, algorithm="fricas")

[Out]

-2/45045*(3465*A*a^6*e^5 - 128*(3*B*b^6*d*e^4 - (13*B*a*b^5 - 10*A*b^6)*e^5)*x^6
+ 3003*(2*B*a^2*b^4 - 5*A*a*b^5)*d^5 - 1287*(8*B*a^3*b^3 - 35*A*a^2*b^4)*d^4*e
+ 4290*(2*B*a^4*b^2 - 15*A*a^3*b^3)*d^3*e^2 - 910*(4*B*a^5*b - 55*A*a^4*b^2)*d^2
*e^3 + 315*(2*B*a^6 - 65*A*a^5*b)*d*e^4 - 64*(39*B*b^6*d^2*e^3 - 2*(86*B*a*b^5 -
65*A*b^6)*d*e^4 + (13*B*a^2*b^4 - 10*A*a*b^5)*e^5)*x^5 - 16*(429*B*b^6*d^3*e^2
- 13*(149*B*a*b^5 - 110*A*b^6)*d^2*e^3 + (347*B*a^2*b^4 - 260*A*a*b^5)*d*e^4 - 3
*(13*B*a^3*b^3 - 10*A*a^2*b^4)*e^5)*x^4 - 8*(1287*B*b^6*d^4*e - 858*(7*B*a*b^5 -
5*A*b^6)*d^3*e^2 + 26*(76*B*a^2*b^4 - 55*A*a*b^5)*d^2*e^3 - 6*(87*B*a^3*b^3 - 6
5*A*a^2*b^4)*d*e^4 + 5*(13*B*a^4*b^2 - 10*A*a^3*b^3)*e^5)*x^3 - (9009*B*b^6*d^5
- 429*(103*B*a*b^5 - 70*A*b^6)*d^4*e + 858*(29*B*a^2*b^4 - 20*A*a*b^5)*d^3*e^2 -
78*(153*B*a^3*b^3 - 110*A*a^2*b^4)*d^2*e^3 + 5*(697*B*a^4*b^2 - 520*A*a^3*b^3)*
d*e^4 - 35*(13*B*a^5*b - 10*A*a^4*b^2)*e^5)*x^2 - (3003*(B*a*b^5 + 5*A*b^6)*d^5
- 429*(103*B*a^2*b^4 + 35*A*a*b^5)*d^4*e + 858*(83*B*a^3*b^3 + 15*A*a^2*b^4)*d^3
*e^2 - 130*(443*B*a^4*b^2 + 55*A*a^3*b^3)*d^2*e^3 + 175*(137*B*a^5*b + 13*A*a^4*
b^2)*d*e^4 - 315*(13*B*a^6 + A*a^5*b)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^6*d
^13 - 6*a*b^5*d^12*e + 15*a^2*b^4*d^11*e^2 - 20*a^3*b^3*d^10*e^3 + 15*a^4*b^2*d^
9*e^4 - 6*a^5*b*d^8*e^5 + a^6*d^7*e^6 + (b^6*d^6*e^7 - 6*a*b^5*d^5*e^8 + 15*a^2*
b^4*d^4*e^9 - 20*a^3*b^3*d^3*e^10 + 15*a^4*b^2*d^2*e^11 - 6*a^5*b*d*e^12 + a^6*e
^13)*x^7 + 7*(b^6*d^7*e^6 - 6*a*b^5*d^6*e^7 + 15*a^2*b^4*d^5*e^8 - 20*a^3*b^3*d^
4*e^9 + 15*a^4*b^2*d^3*e^10 - 6*a^5*b*d^2*e^11 + a^6*d*e^12)*x^6 + 21*(b^6*d^8*e
^5 - 6*a*b^5*d^7*e^6 + 15*a^2*b^4*d^6*e^7 - 20*a^3*b^3*d^5*e^8 + 15*a^4*b^2*d^4*
e^9 - 6*a^5*b*d^3*e^10 + a^6*d^2*e^11)*x^5 + 35*(b^6*d^9*e^4 - 6*a*b^5*d^8*e^5 +
15*a^2*b^4*d^7*e^6 - 20*a^3*b^3*d^6*e^7 + 15*a^4*b^2*d^5*e^8 - 6*a^5*b*d^4*e^9
+ a^6*d^3*e^10)*x^4 + 35*(b^6*d^10*e^3 - 6*a*b^5*d^9*e^4 + 15*a^2*b^4*d^8*e^5 -
20*a^3*b^3*d^7*e^6 + 15*a^4*b^2*d^6*e^7 - 6*a^5*b*d^5*e^8 + a^6*d^4*e^9)*x^3 + 2
1*(b^6*d^11*e^2 - 6*a*b^5*d^10*e^3 + 15*a^2*b^4*d^9*e^4 - 20*a^3*b^3*d^8*e^5 + 1
5*a^4*b^2*d^7*e^6 - 6*a^5*b*d^6*e^7 + a^6*d^5*e^8)*x^2 + 7*(b^6*d^12*e - 6*a*b^5
*d^11*e^2 + 15*a^2*b^4*d^10*e^3 - 20*a^3*b^3*d^9*e^4 + 15*a^4*b^2*d^8*e^5 - 6*a^
5*b*d^7*e^6 + a^6*d^6*e^7)*x)

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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)*(b*x+a)**(1/2)/(e*x+d)**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.523678, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(15/2),x, algorithm="giac")

[Out]

Done