3.2189 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=95 \[ \frac{2 (a+b x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) + (2*(3*b*B*d
+ 2*A*b*e - 5*a*B*e)*(a + b*x)^(3/2))/(15*e*(b*d - a*e)^2*(d + e*x)^(3/2))

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Rubi [A]  time = 0.166895, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 (a+b x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) + (2*(3*b*B*d
+ 2*A*b*e - 5*a*B*e)*(a + b*x)^(3/2))/(15*e*(b*d - a*e)^2*(d + e*x)^(3/2))

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Rubi in Sympy [A]  time = 12.5278, size = 85, normalized size = 0.89 \[ - \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (- A b e + \frac{B \left (5 a e - 3 b d\right )}{2}\right )}{15 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (A e - B d\right )}{5 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(7/2),x)

[Out]

-4*(a + b*x)**(3/2)*(-A*b*e + B*(5*a*e - 3*b*d)/2)/(15*e*(d + e*x)**(3/2)*(a*e -
b*d)**2) - 2*(a + b*x)**(3/2)*(A*e - B*d)/(5*e*(d + e*x)**(5/2)*(a*e - b*d))

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Mathematica [A]  time = 0.152472, size = 66, normalized size = 0.69 \[ \frac{2 (a+b x)^{3/2} (A (-3 a e+5 b d+2 b e x)+B (-2 a d-5 a e x+3 b d x))}{15 (d+e x)^{5/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(2*(a + b*x)^(3/2)*(B*(-2*a*d + 3*b*d*x - 5*a*e*x) + A*(5*b*d - 3*a*e + 2*b*e*x)
))/(15*(b*d - a*e)^2*(d + e*x)^(5/2))

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Maple [A]  time = 0.011, size = 74, normalized size = 0.8 \[ -{\frac{-4\,Abex+10\,Baex-6\,Bbdx+6\,Aae-10\,Abd+4\,Bad}{15\,{a}^{2}{e}^{2}-30\,bead+15\,{b}^{2}{d}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(7/2),x)

[Out]

-2/15*(b*x+a)^(3/2)*(-2*A*b*e*x+5*B*a*e*x-3*B*b*d*x+3*A*a*e-5*A*b*d+2*B*a*d)/(e*
x+d)^(5/2)/(a^2*e^2-2*a*b*d*e+b^2*d^2)

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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)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.531964, size = 298, normalized size = 3.14 \[ -\frac{2 \,{\left (3 \, A a^{2} e -{\left (3 \, B b^{2} d -{\left (5 \, B a b - 2 \, A b^{2}\right )} e\right )} x^{2} +{\left (2 \, B a^{2} - 5 \, A a b\right )} d -{\left ({\left (B a b + 5 \, A b^{2}\right )} d -{\left (5 \, B a^{2} + A a b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{15 \,{\left (b^{2} d^{5} - 2 \, a b d^{4} e + a^{2} d^{3} e^{2} +{\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} x^{3} + 3 \,{\left (b^{2} d^{3} e^{2} - 2 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x^{2} + 3 \,{\left (b^{2} d^{4} e - 2 \, a b d^{3} e^{2} + a^{2} d^{2} e^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(3*A*a^2*e - (3*B*b^2*d - (5*B*a*b - 2*A*b^2)*e)*x^2 + (2*B*a^2 - 5*A*a*b)
*d - ((B*a*b + 5*A*b^2)*d - (5*B*a^2 + A*a*b)*e)*x)*sqrt(b*x + a)*sqrt(e*x + d)/
(b^2*d^5 - 2*a*b*d^4*e + a^2*d^3*e^2 + (b^2*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*x^3
+ 3*(b^2*d^3*e^2 - 2*a*b*d^2*e^3 + a^2*d*e^4)*x^2 + 3*(b^2*d^4*e - 2*a*b*d^3*e^
2 + a^2*d^2*e^3)*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)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.25549, size = 281, normalized size = 2.96 \[ -\frac{{\left (b x + a\right )}^{\frac{3}{2}}{\left (\frac{{\left (3 \, B b^{6} d{\left | b \right |} e^{2} - 5 \, B a b^{5}{\left | b \right |} e^{3} + 2 \, A b^{6}{\left | b \right |} e^{3}\right )}{\left (b x + a\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}} - \frac{5 \,{\left (B a b^{6} d{\left | b \right |} e^{2} - A b^{7} d{\left | b \right |} e^{2} - B a^{2} b^{5}{\left | b \right |} e^{3} + A a b^{6}{\left | b \right |} e^{3}\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}}\right )}}{960 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

-1/960*(b*x + a)^(3/2)*((3*B*b^6*d*abs(b)*e^2 - 5*B*a*b^5*abs(b)*e^3 + 2*A*b^6*a
bs(b)*e^3)*(b*x + a)/(b^12*d^3*e^6 - 3*a*b^11*d^2*e^7 + 3*a^2*b^10*d*e^8 - a^3*b
^9*e^9) - 5*(B*a*b^6*d*abs(b)*e^2 - A*b^7*d*abs(b)*e^2 - B*a^2*b^5*abs(b)*e^3 +
A*a*b^6*abs(b)*e^3)/(b^12*d^3*e^6 - 3*a*b^11*d^2*e^7 + 3*a^2*b^10*d*e^8 - a^3*b^
9*e^9))/(b^2*d + (b*x + a)*b*e - a*b*e)^(5/2)