3.2191 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=198 \[ \frac{16 b^2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac{8 b (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{21 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (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)) + (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)) + (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))

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Rubi [A]  time = 0.351453, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{16 b^2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac{8 b (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{21 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (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)) + (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)) + (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))

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Rubi in Sympy [A]  time = 34.6386, size = 185, normalized size = 0.93 \[ - \frac{32 b^{2} \left (a + b x\right )^{\frac{3}{2}} \left (- A b e + \frac{B \left (3 a e - b d\right )}{2}\right )}{315 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4}} - \frac{8 b \left (a + b x\right )^{\frac{3}{2}} \left (2 A b e - 3 B a e + B b d\right )}{105 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3}} - \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (- A b e + \frac{B \left (3 a e - b d\right )}{2}\right )}{21 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (A e - B d\right )}{9 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(11/2),x)

[Out]

-32*b**2*(a + b*x)**(3/2)*(-A*b*e + B*(3*a*e - b*d)/2)/(315*e*(d + e*x)**(3/2)*(
a*e - b*d)**4) - 8*b*(a + b*x)**(3/2)*(2*A*b*e - 3*B*a*e + B*b*d)/(105*e*(d + e*
x)**(5/2)*(a*e - b*d)**3) - 4*(a + b*x)**(3/2)*(-A*b*e + B*(3*a*e - b*d)/2)/(21*
e*(d + e*x)**(7/2)*(a*e - b*d)**2) - 2*(a + b*x)**(3/2)*(A*e - B*d)/(9*e*(d + e*
x)**(9/2)*(a*e - b*d))

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Mathematica [A]  time = 0.449448, size = 177, normalized size = 0.89 \[ \frac{2 \sqrt{a+b x} \left (\frac{8 b^3 (d+e x)^4 (-3 a B e+2 A b e+b B d)}{(b d-a e)^4}+\frac{4 b^2 (d+e x)^3 (-3 a B e+2 A b e+b B d)}{(b d-a e)^3}+\frac{3 b (d+e x)^2 (-3 a B e+2 A b e+b B d)}{(b d-a e)^2}-\frac{5 (d+e x) (9 a B e+A b e-10 b B d)}{a e-b d}+35 (B d-A e)\right )}{315 e^2 (d+e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

(2*Sqrt[a + b*x]*(35*(B*d - A*e) - (5*(-10*b*B*d + A*b*e + 9*a*B*e)*(d + e*x))/(
-(b*d) + a*e) + (3*b*(b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)^2)/(b*d - a*e)^2 + (4
*b^2*(b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)^3)/(b*d - a*e)^3 + (8*b^3*(b*B*d + 2*
A*b*e - 3*a*B*e)*(d + e*x)^4)/(b*d - a*e)^4))/(315*e^2*(d + e*x)^(9/2))

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Maple [A]  time = 0.014, size = 322, normalized size = 1.6 \[ -{\frac{-32\,A{b}^{3}{e}^{3}{x}^{3}+48\,Ba{b}^{2}{e}^{3}{x}^{3}-16\,B{b}^{3}d{e}^{2}{x}^{3}+48\,Aa{b}^{2}{e}^{3}{x}^{2}-144\,A{b}^{3}d{e}^{2}{x}^{2}-72\,B{a}^{2}b{e}^{3}{x}^{2}+240\,Ba{b}^{2}d{e}^{2}{x}^{2}-72\,B{b}^{3}{d}^{2}e{x}^{2}-60\,A{a}^{2}b{e}^{3}x+216\,Aa{b}^{2}d{e}^{2}x-252\,A{b}^{3}{d}^{2}ex+90\,B{a}^{3}{e}^{3}x-354\,B{a}^{2}bd{e}^{2}x+486\,Ba{b}^{2}{d}^{2}ex-126\,B{b}^{3}{d}^{3}x+70\,A{a}^{3}{e}^{3}-270\,A{a}^{2}bd{e}^{2}+378\,Aa{b}^{2}{d}^{2}e-210\,A{b}^{3}{d}^{3}+20\,B{a}^{3}d{e}^{2}-72\,B{a}^{2}b{d}^{2}e+84\,Ba{b}^{2}{d}^{3}}{315\,{e}^{4}{a}^{4}-1260\,b{e}^{3}d{a}^{3}+1890\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-1260\,a{b}^{3}{d}^{3}e+315\,{b}^{4}{d}^{4}} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(11/2),x)

[Out]

-2/315*(b*x+a)^(3/2)*(-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)/(e*x+d)^(9/2)/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+
b^4*d^4)

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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)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.50234, size = 959, normalized size = 4.84 \[ -\frac{2 \,{\left (35 \, A a^{4} e^{3} - 8 \,{\left (B b^{4} d e^{2} -{\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} e^{3}\right )} x^{4} + 21 \,{\left (2 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} d^{3} - 9 \,{\left (4 \, B a^{3} b - 21 \, A a^{2} b^{2}\right )} d^{2} e + 5 \,{\left (2 \, B a^{4} - 27 \, A a^{3} b\right )} d e^{2} - 4 \,{\left (9 \, B b^{4} d^{2} e - 2 \,{\left (14 \, B a b^{3} - 9 \, A b^{4}\right )} d e^{2} +{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} e^{3}\right )} x^{3} - 3 \,{\left (21 \, B b^{4} d^{3} - 3 \,{\left (23 \, B a b^{3} - 14 \, A b^{4}\right )} d^{2} e +{\left (19 \, B a^{2} b^{2} - 12 \, A a b^{3}\right )} d e^{2} -{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} e^{3}\right )} x^{2} -{\left (21 \,{\left (B a b^{3} + 5 \, A b^{4}\right )} d^{3} - 9 \,{\left (23 \, B a^{2} b^{2} + 7 \, A a b^{3}\right )} d^{2} e +{\left (167 \, B a^{3} b + 27 \, A a^{2} b^{2}\right )} d e^{2} - 5 \,{\left (9 \, B a^{4} + A a^{3} b\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{315 \,{\left (b^{4} d^{9} - 4 \, a b^{3} d^{8} e + 6 \, a^{2} b^{2} d^{7} e^{2} - 4 \, a^{3} b d^{6} e^{3} + a^{4} d^{5} e^{4} +{\left (b^{4} d^{4} e^{5} - 4 \, a b^{3} d^{3} e^{6} + 6 \, a^{2} b^{2} d^{2} e^{7} - 4 \, a^{3} b d e^{8} + a^{4} e^{9}\right )} x^{5} + 5 \,{\left (b^{4} d^{5} e^{4} - 4 \, a b^{3} d^{4} e^{5} + 6 \, a^{2} b^{2} d^{3} e^{6} - 4 \, a^{3} b d^{2} e^{7} + a^{4} d e^{8}\right )} x^{4} + 10 \,{\left (b^{4} d^{6} e^{3} - 4 \, a b^{3} d^{5} e^{4} + 6 \, a^{2} b^{2} d^{4} e^{5} - 4 \, a^{3} b d^{3} e^{6} + a^{4} d^{2} e^{7}\right )} x^{3} + 10 \,{\left (b^{4} d^{7} e^{2} - 4 \, a b^{3} d^{6} e^{3} + 6 \, a^{2} b^{2} d^{5} e^{4} - 4 \, a^{3} b d^{4} e^{5} + a^{4} d^{3} e^{6}\right )} x^{2} + 5 \,{\left (b^{4} d^{8} e - 4 \, a b^{3} d^{7} e^{2} + 6 \, a^{2} b^{2} d^{6} e^{3} - 4 \, a^{3} b d^{5} e^{4} + a^{4} d^{4} e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

-2/315*(35*A*a^4*e^3 - 8*(B*b^4*d*e^2 - (3*B*a*b^3 - 2*A*b^4)*e^3)*x^4 + 21*(2*B
*a^2*b^2 - 5*A*a*b^3)*d^3 - 9*(4*B*a^3*b - 21*A*a^2*b^2)*d^2*e + 5*(2*B*a^4 - 27
*A*a^3*b)*d*e^2 - 4*(9*B*b^4*d^2*e - 2*(14*B*a*b^3 - 9*A*b^4)*d*e^2 + (3*B*a^2*b
^2 - 2*A*a*b^3)*e^3)*x^3 - 3*(21*B*b^4*d^3 - 3*(23*B*a*b^3 - 14*A*b^4)*d^2*e + (
19*B*a^2*b^2 - 12*A*a*b^3)*d*e^2 - (3*B*a^3*b - 2*A*a^2*b^2)*e^3)*x^2 - (21*(B*a
*b^3 + 5*A*b^4)*d^3 - 9*(23*B*a^2*b^2 + 7*A*a*b^3)*d^2*e + (167*B*a^3*b + 27*A*a
^2*b^2)*d*e^2 - 5*(9*B*a^4 + A*a^3*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(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 + (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)*x^5 + 5*(
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)
*x^4 + 10*(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)*x^3 + 10*(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)*x^2 + 5*(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)*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)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.331064, size = 857, normalized size = 4.33 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

-1/322560*((4*(b*x + a)*(2*(B*b^10*d*abs(b)*e^6 - 3*B*a*b^9*abs(b)*e^7 + 2*A*b^1
0*abs(b)*e^7)*(b*x + a)/(b^20*d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^1
2 - 10*a^3*b^17*d^2*e^13 + 5*a^4*b^16*d*e^14 - a^5*b^15*e^15) + 9*(B*b^11*d^2*ab
s(b)*e^5 - 4*B*a*b^10*d*abs(b)*e^6 + 2*A*b^11*d*abs(b)*e^6 + 3*B*a^2*b^9*abs(b)*
e^7 - 2*A*a*b^10*abs(b)*e^7)/(b^20*d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^
3*e^12 - 10*a^3*b^17*d^2*e^13 + 5*a^4*b^16*d*e^14 - a^5*b^15*e^15)) + 63*(B*b^12
*d^3*abs(b)*e^4 - 5*B*a*b^11*d^2*abs(b)*e^5 + 2*A*b^12*d^2*abs(b)*e^5 + 7*B*a^2*
b^10*d*abs(b)*e^6 - 4*A*a*b^11*d*abs(b)*e^6 - 3*B*a^3*b^9*abs(b)*e^7 + 2*A*a^2*b
^10*abs(b)*e^7)/(b^20*d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^12 - 10*a
^3*b^17*d^2*e^13 + 5*a^4*b^16*d*e^14 - a^5*b^15*e^15))*(b*x + a) - 105*(B*a*b^12
*d^3*abs(b)*e^4 - A*b^13*d^3*abs(b)*e^4 - 3*B*a^2*b^11*d^2*abs(b)*e^5 + 3*A*a*b^
12*d^2*abs(b)*e^5 + 3*B*a^3*b^10*d*abs(b)*e^6 - 3*A*a^2*b^11*d*abs(b)*e^6 - B*a^
4*b^9*abs(b)*e^7 + A*a^3*b^10*abs(b)*e^7)/(b^20*d^5*e^10 - 5*a*b^19*d^4*e^11 + 1
0*a^2*b^18*d^3*e^12 - 10*a^3*b^17*d^2*e^13 + 5*a^4*b^16*d*e^14 - a^5*b^15*e^15))
*(b*x + a)^(3/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(9/2)