3.2240 \(\int \frac{A+B x}{(a+b x)^{5/2} \sqrt{d+e x}} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.166848, 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 \[ -\frac{2 \sqrt{d+e x} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}-\frac{2 \sqrt{d+e x} (-a B e-2 A b e+3 b B d)}{3 b \sqrt{a+b x} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)^(5/2)*Sqrt[d + e*x]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.4704, size = 83, normalized size = 0.87 \[ \frac{2 \sqrt{d + e x} \left (2 A b e + B a e - 3 B b d\right )}{3 b \sqrt{a + b x} \left (a e - b d\right )^{2}} + \frac{2 \sqrt{d + e x} \left (A b - B a\right )}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**(5/2)/(e*x+d)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.130946, size = 65, normalized size = 0.68 \[ \frac{2 \sqrt{d+e x} (A (3 a e-b d+2 b e x)+B (-2 a d+a e x-3 b d x))}{3 (a+b x)^{3/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)^(5/2)*Sqrt[d + e*x]),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 73, normalized size = 0.8 \[{\frac{4\,Abex+2\,Baex-6\,Bbdx+6\,Aae-2\,Abd-4\,Bad}{3\,{a}^{2}{e}^{2}-6\,bead+3\,{b}^{2}{d}^{2}}\sqrt{ex+d} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(1/2),x)

[Out]

2/3*(e*x+d)^(1/2)*(2*A*b*e*x+B*a*e*x-3*B*b*d*x+3*A*a*e-A*b*d-2*B*a*d)/(b*x+a)^(3
/2)/(a^2*e^2-2*a*b*d*e+b^2*d^2)

_______________________________________________________________________________________

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)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.464886, size = 190, normalized size = 2. \[ \frac{2 \,{\left (3 \, A a e -{\left (2 \, B a + A b\right )} d -{\left (3 \, B b d -{\left (B a + 2 \, A b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (a^{2} b^{2} d^{2} - 2 \, a^{3} b d e + a^{4} e^{2} +{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(5/2)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

2/3*(3*A*a*e - (2*B*a + A*b)*d - (3*B*b*d - (B*a + 2*A*b)*e)*x)*sqrt(b*x + a)*sq
rt(e*x + d)/(a^2*b^2*d^2 - 2*a^3*b*d*e + a^4*e^2 + (b^4*d^2 - 2*a*b^3*d*e + a^2*
b^2*e^2)*x^2 + 2*(a*b^3*d^2 - 2*a^2*b^2*d*e + a^3*b*e^2)*x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (a + b x\right )^{\frac{5}{2}} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b*x+a)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Integral((A + B*x)/((a + b*x)**(5/2)*sqrt(d + e*x)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.255452, size = 352, normalized size = 3.71 \[ -\frac{4 \,{\left (3 \, B b^{\frac{9}{2}} d^{2} e^{\frac{1}{2}} - 4 \, B a b^{\frac{7}{2}} d e^{\frac{3}{2}} - 2 \, A b^{\frac{9}{2}} d e^{\frac{3}{2}} - 6 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2} B b^{\frac{5}{2}} d e^{\frac{1}{2}} + B a^{2} b^{\frac{5}{2}} e^{\frac{5}{2}} + 2 \, A a b^{\frac{7}{2}} e^{\frac{5}{2}} + 6 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2} A b^{\frac{5}{2}} e^{\frac{3}{2}} + 3 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{4} B \sqrt{b} e^{\frac{1}{2}}\right )}}{3 \,{\left (b^{2} d - a b e -{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(5/2)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

-4/3*(3*B*b^(9/2)*d^2*e^(1/2) - 4*B*a*b^(7/2)*d*e^(3/2) - 2*A*b^(9/2)*d*e^(3/2)
- 6*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*b^
(5/2)*d*e^(1/2) + B*a^2*b^(5/2)*e^(5/2) + 2*A*a*b^(7/2)*e^(5/2) + 6*(sqrt(b*x +
a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b^(5/2)*e^(3/2) +
3*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*sqrt
(b)*e^(1/2))/((b^2*d - a*b*e - (sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*
x + a)*b*e - a*b*e))^2)^3*abs(b))