3.2190 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=147 \[ \frac{4 b (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) + (2*(3*b*B*d
+ 4*A*b*e - 7*a*B*e)*(a + b*x)^(3/2))/(35*e*(b*d - a*e)^2*(d + e*x)^(5/2)) + (4
*b*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(a + b*x)^(3/2))/(105*e*(b*d - a*e)^3*(d + e*x)
^(3/2))

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Rubi [A]  time = 0.267357, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{4 b (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) + (2*(3*b*B*d
+ 4*A*b*e - 7*a*B*e)*(a + b*x)^(3/2))/(35*e*(b*d - a*e)^2*(d + e*x)^(5/2)) + (4
*b*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(a + b*x)^(3/2))/(105*e*(b*d - a*e)^3*(d + e*x)
^(3/2))

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Rubi in Sympy [A]  time = 24.397, size = 138, normalized size = 0.94 \[ - \frac{4 b \left (a + b x\right )^{\frac{3}{2}} \left (4 A b e - 7 B a e + 3 B b d\right )}{105 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (4 A b e - 7 B a e + 3 B b d\right )}{35 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (A e - B d\right )}{7 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(9/2),x)

[Out]

-4*b*(a + b*x)**(3/2)*(4*A*b*e - 7*B*a*e + 3*B*b*d)/(105*e*(d + e*x)**(3/2)*(a*e
- b*d)**3) + 2*(a + b*x)**(3/2)*(4*A*b*e - 7*B*a*e + 3*B*b*d)/(35*e*(d + e*x)**
(5/2)*(a*e - b*d)**2) - 2*(a + b*x)**(3/2)*(A*e - B*d)/(7*e*(d + e*x)**(7/2)*(a*
e - b*d))

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Mathematica [A]  time = 0.241897, size = 135, normalized size = 0.92 \[ \frac{2 (a+b x)^{3/2} \left (A \left (15 a^2 e^2-6 a b e (7 d+2 e x)+b^2 \left (35 d^2+28 d e x+8 e^2 x^2\right )\right )+B \left (3 a^2 e (2 d+7 e x)-2 a b \left (7 d^2+29 d e x+7 e^2 x^2\right )+3 b^2 d x (7 d+2 e x)\right )\right )}{105 (d+e x)^{7/2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

(2*(a + b*x)^(3/2)*(B*(3*b^2*d*x*(7*d + 2*e*x) + 3*a^2*e*(2*d + 7*e*x) - 2*a*b*(
7*d^2 + 29*d*e*x + 7*e^2*x^2)) + A*(15*a^2*e^2 - 6*a*b*e*(7*d + 2*e*x) + b^2*(35
*d^2 + 28*d*e*x + 8*e^2*x^2))))/(105*(b*d - a*e)^3*(d + e*x)^(7/2))

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Maple [A]  time = 0.011, size = 177, normalized size = 1.2 \[ -{\frac{16\,A{b}^{2}{e}^{2}{x}^{2}-28\,Bab{e}^{2}{x}^{2}+12\,B{b}^{2}de{x}^{2}-24\,Aab{e}^{2}x+56\,A{b}^{2}dex+42\,B{a}^{2}{e}^{2}x-116\,Babdex+42\,B{b}^{2}{d}^{2}x+30\,A{a}^{2}{e}^{2}-84\,Aabde+70\,A{b}^{2}{d}^{2}+12\,B{a}^{2}de-28\,Bab{d}^{2}}{105\,{a}^{3}{e}^{3}-315\,{a}^{2}bd{e}^{2}+315\,a{b}^{2}{d}^{2}e-105\,{b}^{3}{d}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(9/2),x)

[Out]

-2/105*(b*x+a)^(3/2)*(8*A*b^2*e^2*x^2-14*B*a*b*e^2*x^2+6*B*b^2*d*e*x^2-12*A*a*b*
e^2*x+28*A*b^2*d*e*x+21*B*a^2*e^2*x-58*B*a*b*d*e*x+21*B*b^2*d^2*x+15*A*a^2*e^2-4
2*A*a*b*d*e+35*A*b^2*d^2+6*B*a^2*d*e-14*B*a*b*d^2)/(e*x+d)^(7/2)/(a^3*e^3-3*a^2*
b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)

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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)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.963663, size = 594, normalized size = 4.04 \[ \frac{2 \,{\left (15 \, A a^{3} e^{2} + 2 \,{\left (3 \, B b^{3} d e -{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x^{3} - 7 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} d^{2} + 6 \,{\left (B a^{3} - 7 \, A a^{2} b\right )} d e +{\left (21 \, B b^{3} d^{2} - 4 \,{\left (13 \, B a b^{2} - 7 \, A b^{3}\right )} d e +{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x^{2} +{\left (7 \,{\left (B a b^{2} + 5 \, A b^{3}\right )} d^{2} - 2 \,{\left (26 \, B a^{2} b + 7 \, A a b^{2}\right )} d e + 3 \,{\left (7 \, B a^{3} + A a^{2} b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{105 \,{\left (b^{3} d^{7} - 3 \, a b^{2} d^{6} e + 3 \, a^{2} b d^{5} e^{2} - a^{3} d^{4} e^{3} +{\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{4} + 4 \,{\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{3} + 6 \,{\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x^{2} + 4 \,{\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} e^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(9/2),x, algorithm="fricas")

[Out]

2/105*(15*A*a^3*e^2 + 2*(3*B*b^3*d*e - (7*B*a*b^2 - 4*A*b^3)*e^2)*x^3 - 7*(2*B*a
^2*b - 5*A*a*b^2)*d^2 + 6*(B*a^3 - 7*A*a^2*b)*d*e + (21*B*b^3*d^2 - 4*(13*B*a*b^
2 - 7*A*b^3)*d*e + (7*B*a^2*b - 4*A*a*b^2)*e^2)*x^2 + (7*(B*a*b^2 + 5*A*b^3)*d^2
- 2*(26*B*a^2*b + 7*A*a*b^2)*d*e + 3*(7*B*a^3 + A*a^2*b)*e^2)*x)*sqrt(b*x + a)*
sqrt(e*x + d)/(b^3*d^7 - 3*a*b^2*d^6*e + 3*a^2*b*d^5*e^2 - a^3*d^4*e^3 + (b^3*d^
3*e^4 - 3*a*b^2*d^2*e^5 + 3*a^2*b*d*e^6 - a^3*e^7)*x^4 + 4*(b^3*d^4*e^3 - 3*a*b^
2*d^3*e^4 + 3*a^2*b*d^2*e^5 - a^3*d*e^6)*x^3 + 6*(b^3*d^5*e^2 - 3*a*b^2*d^4*e^3
+ 3*a^2*b*d^3*e^4 - a^3*d^2*e^5)*x^2 + 4*(b^3*d^6*e - 3*a*b^2*d^5*e^2 + 3*a^2*b*
d^4*e^3 - a^3*d^3*e^4)*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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283433, size = 528, normalized size = 3.59 \[ -\frac{{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (3 \, B b^{8} d{\left | b \right |} e^{4} - 7 \, B a b^{7}{\left | b \right |} e^{5} + 4 \, A b^{8}{\left | b \right |} e^{5}\right )}{\left (b x + a\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}} + \frac{7 \,{\left (3 \, B b^{9} d^{2}{\left | b \right |} e^{3} - 10 \, B a b^{8} d{\left | b \right |} e^{4} + 4 \, A b^{9} d{\left | b \right |} e^{4} + 7 \, B a^{2} b^{7}{\left | b \right |} e^{5} - 4 \, A a b^{8}{\left | b \right |} e^{5}\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}}\right )} - \frac{35 \,{\left (B a b^{9} d^{2}{\left | b \right |} e^{3} - A b^{10} d^{2}{\left | b \right |} e^{3} - 2 \, B a^{2} b^{8} d{\left | b \right |} e^{4} + 2 \, A a b^{9} d{\left | b \right |} e^{4} + B a^{3} b^{7}{\left | b \right |} e^{5} - A a^{2} b^{8}{\left | b \right |} e^{5}\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{80640 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(9/2),x, algorithm="giac")

[Out]

-1/80640*((b*x + a)*(2*(3*B*b^8*d*abs(b)*e^4 - 7*B*a*b^7*abs(b)*e^5 + 4*A*b^8*ab
s(b)*e^5)*(b*x + a)/(b^16*d^4*e^8 - 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a
^3*b^13*d*e^11 + a^4*b^12*e^12) + 7*(3*B*b^9*d^2*abs(b)*e^3 - 10*B*a*b^8*d*abs(b
)*e^4 + 4*A*b^9*d*abs(b)*e^4 + 7*B*a^2*b^7*abs(b)*e^5 - 4*A*a*b^8*abs(b)*e^5)/(b
^16*d^4*e^8 - 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b
^12*e^12)) - 35*(B*a*b^9*d^2*abs(b)*e^3 - A*b^10*d^2*abs(b)*e^3 - 2*B*a^2*b^8*d*
abs(b)*e^4 + 2*A*a*b^9*d*abs(b)*e^4 + B*a^3*b^7*abs(b)*e^5 - A*a^2*b^8*abs(b)*e^
5)/(b^16*d^4*e^8 - 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 +
a^4*b^12*e^12))*(b*x + a)^(3/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7/2)