3.2233 \(\int \frac{A+B x}{(a+b x)^{3/2} (d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{3/2} (b d-a e)}+\frac{4 \sqrt{a+b x} (3 a B e-4 A b e+b B d)}{3 \sqrt{d+e x} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (3 a B e-4 A b e+b B d)}{3 b (d+e x)^{3/2} (b d-a e)^2} \]

[Out]

(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(3/2)) + (2*(b*B*d - 4*A
*b*e + 3*a*B*e)*Sqrt[a + b*x])/(3*b*(b*d - a*e)^2*(d + e*x)^(3/2)) + (4*(b*B*d -
4*A*b*e + 3*a*B*e)*Sqrt[a + b*x])/(3*(b*d - a*e)^3*Sqrt[d + e*x])

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Rubi [A]  time = 0.257098, antiderivative size = 139, 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 \[ -\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{3/2} (b d-a e)}+\frac{4 \sqrt{a+b x} (3 a B e-4 A b e+b B d)}{3 \sqrt{d+e x} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (3 a B e-4 A b e+b B d)}{3 b (d+e x)^{3/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(5/2)),x]

[Out]

(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(3/2)) + (2*(b*B*d - 4*A
*b*e + 3*a*B*e)*Sqrt[a + b*x])/(3*b*(b*d - a*e)^2*(d + e*x)^(3/2)) + (4*(b*B*d -
4*A*b*e + 3*a*B*e)*Sqrt[a + b*x])/(3*(b*d - a*e)^3*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 25.7412, size = 131, normalized size = 0.94 \[ \frac{4 \sqrt{a + b x} \left (4 A b e - 3 B a e - B b d\right )}{3 \sqrt{d + e x} \left (a e - b d\right )^{3}} - \frac{2 \sqrt{a + b x} \left (4 A b e - 3 B a e - B b d\right )}{3 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} + \frac{2 \left (A b - B a\right )}{b \sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**(3/2)/(e*x+d)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.438901, size = 99, normalized size = 0.71 \[ \frac{2 \sqrt{a+b x} \sqrt{d+e x} \left (\frac{3 a B e-5 A b e+2 b B d}{d+e x}+\frac{(b d-a e) (B d-A e)}{(d+e x)^2}-\frac{3 b (A b-a B)}{a+b x}\right )}{3 (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(5/2)),x]

[Out]

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

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Maple [A]  time = 0.012, size = 176, normalized size = 1.3 \[ -{\frac{-16\,A{b}^{2}{e}^{2}{x}^{2}+12\,Bab{e}^{2}{x}^{2}+4\,B{b}^{2}de{x}^{2}-8\,Aab{e}^{2}x-24\,A{b}^{2}dex+6\,B{a}^{2}{e}^{2}x+20\,Babdex+6\,B{b}^{2}{d}^{2}x+2\,A{a}^{2}{e}^{2}-12\,Aabde-6\,A{b}^{2}{d}^{2}+4\,B{a}^{2}de+12\,Bab{d}^{2}}{3\,{a}^{3}{e}^{3}-9\,{a}^{2}bd{e}^{2}+9\,a{b}^{2}{d}^{2}e-3\,{b}^{3}{d}^{3}}{\frac{1}{\sqrt{bx+a}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(5/2),x)

[Out]

-2/3*(-8*A*b^2*e^2*x^2+6*B*a*b*e^2*x^2+2*B*b^2*d*e*x^2-4*A*a*b*e^2*x-12*A*b^2*d*
e*x+3*B*a^2*e^2*x+10*B*a*b*d*e*x+3*B*b^2*d^2*x+A*a^2*e^2-6*A*a*b*d*e-3*A*b^2*d^2
+2*B*a^2*d*e+6*B*a*b*d^2)/(b*x+a)^(1/2)/(e*x+d)^(3/2)/(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)/((b*x + a)^(3/2)*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.678851, size = 454, normalized size = 3.27 \[ \frac{2 \,{\left (A a^{2} e^{2} + 3 \,{\left (2 \, B a b - A b^{2}\right )} d^{2} + 2 \,{\left (B a^{2} - 3 \, A a b\right )} d e + 2 \,{\left (B b^{2} d e +{\left (3 \, B a b - 4 \, A b^{2}\right )} e^{2}\right )} x^{2} +{\left (3 \, B b^{2} d^{2} + 2 \,{\left (5 \, B a b - 6 \, A b^{2}\right )} d e +{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} +{\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} +{\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} +{\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b*x+a)**(3/2)/(e*x+d)**(5/2),x)

[Out]

Integral((A + B*x)/((a + b*x)**(3/2)*(d + e*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.324066, size = 551, normalized size = 3.96 \[ -\frac{\sqrt{b x + a}{\left (\frac{{\left (2 \, B b^{6} d^{3}{\left | b \right |} e^{2} - B a b^{5} d^{2}{\left | b \right |} e^{3} - 5 \, A b^{6} d^{2}{\left | b \right |} e^{3} - 4 \, B a^{2} b^{4} d{\left | b \right |} e^{4} + 10 \, A a b^{5} d{\left | b \right |} e^{4} + 3 \, B a^{3} b^{3}{\left | b \right |} e^{5} - 5 \, A a^{2} b^{4}{\left | b \right |} e^{5}\right )}{\left (b x + a\right )}}{b^{8} d^{2} e^{4} - 2 \, a b^{7} d e^{5} + a^{2} b^{6} e^{6}} + \frac{3 \,{\left (B b^{7} d^{4}{\left | b \right |} e - 2 \, B a b^{6} d^{3}{\left | b \right |} e^{2} - 2 \, A b^{7} d^{3}{\left | b \right |} e^{2} + 6 \, A a b^{6} d^{2}{\left | b \right |} e^{3} + 2 \, B a^{3} b^{4} d{\left | b \right |} e^{4} - 6 \, A a^{2} b^{5} d{\left | b \right |} e^{4} - B a^{4} b^{3}{\left | b \right |} e^{5} + 2 \, A a^{3} b^{4}{\left | b \right |} e^{5}\right )}}{b^{8} d^{2} e^{4} - 2 \, a b^{7} d e^{5} + a^{2} b^{6} e^{6}}\right )}}{48 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{3}{2}}} + \frac{4 \,{\left (B a b^{\frac{5}{2}} e^{\frac{1}{2}} - A b^{\frac{7}{2}} e^{\frac{1}{2}}\right )}}{{\left (b^{2} d^{2}{\left | b \right |} - 2 \, a b d{\left | b \right |} e + a^{2}{\left | b \right |} e^{2}\right )}{\left (b^{2} d - a b e -{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

-1/48*sqrt(b*x + a)*((2*B*b^6*d^3*abs(b)*e^2 - B*a*b^5*d^2*abs(b)*e^3 - 5*A*b^6*
d^2*abs(b)*e^3 - 4*B*a^2*b^4*d*abs(b)*e^4 + 10*A*a*b^5*d*abs(b)*e^4 + 3*B*a^3*b^
3*abs(b)*e^5 - 5*A*a^2*b^4*abs(b)*e^5)*(b*x + a)/(b^8*d^2*e^4 - 2*a*b^7*d*e^5 +
a^2*b^6*e^6) + 3*(B*b^7*d^4*abs(b)*e - 2*B*a*b^6*d^3*abs(b)*e^2 - 2*A*b^7*d^3*ab
s(b)*e^2 + 6*A*a*b^6*d^2*abs(b)*e^3 + 2*B*a^3*b^4*d*abs(b)*e^4 - 6*A*a^2*b^5*d*a
bs(b)*e^4 - B*a^4*b^3*abs(b)*e^5 + 2*A*a^3*b^4*abs(b)*e^5)/(b^8*d^2*e^4 - 2*a*b^
7*d*e^5 + a^2*b^6*e^6))/(b^2*d + (b*x + a)*b*e - a*b*e)^(3/2) + 4*(B*a*b^(5/2)*e
^(1/2) - A*b^(7/2)*e^(1/2))/((b^2*d^2*abs(b) - 2*a*b*d*abs(b)*e + a^2*abs(b)*e^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))