3.2232 \(\int \frac{A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=89 \[ \frac{2 \sqrt{d+e x} (a B e-2 A b e+b B d)}{e \sqrt{a+b x} (b d-a e)^2}-\frac{2 (B d-A e)}{e \sqrt{a+b x} \sqrt{d+e x} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.178216, antiderivative size = 89, 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 \[ \frac{2 \sqrt{d+e x} (a B e-2 A b e+b B d)}{e \sqrt{a+b x} (b d-a e)^2}-\frac{2 (B d-A e)}{e \sqrt{a+b x} \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.0069, size = 78, normalized size = 0.88 \[ \frac{4 \sqrt{a + b x} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{b \sqrt{d + e x} \left (a e - b d\right )^{2}} + \frac{2 \left (A b - B a\right )}{b \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**(3/2)/(e*x+d)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.130139, size = 61, normalized size = 0.69 \[ \frac{2 B (2 a d+a e x+b d x)-2 A (a e+b (d+2 e x))}{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 72, normalized size = 0.8 \[ -2\,{\frac{2\,Abex-Baex-Bbdx+Aae+Abd-2\,Bad}{\sqrt{bx+a}\sqrt{ex+d} \left ({a}^{2}{e}^{2}-2\,bead+{b}^{2}{d}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(3/2),x)

[Out]

-2*(2*A*b*e*x-B*a*e*x-B*b*d*x+A*a*e+A*b*d-2*B*a*d)/(b*x+a)^(1/2)/(e*x+d)^(1/2)/(
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)/((b*x + a)^(3/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.481613, size = 200, normalized size = 2.25 \[ -\frac{2 \,{\left (A a e -{\left (2 \, B a - A b\right )} d -{\left (B b d +{\left (B a - 2 \, A b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} +{\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b*x+a)**(3/2)/(e*x+d)**(3/2),x)

[Out]

Integral((A + B*x)/((a + b*x)**(3/2)*(d + e*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.258091, size = 227, normalized size = 2.55 \[ \frac{2 \,{\left (B b^{2} d - A b^{2} e\right )} \sqrt{b x + a}}{{\left (b^{2} d^{2}{\left | b \right |} - 2 \, a b d{\left | b \right |} e + a^{2}{\left | b \right |} e^{2}\right )} \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} + \frac{4 \,{\left (B a b^{\frac{3}{2}} e^{\frac{1}{2}} - A b^{\frac{5}{2}} e^{\frac{1}{2}}\right )}}{{\left (b^{2} d - a b e -{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{\left (b d{\left | b \right |} - a{\left | b \right |} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

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