3.2241 \(\int \frac{A+B x}{(a+b x)^{5/2} (d+e x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.270384, 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 \[ -\frac{2 (B d-A e)}{e (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)}-\frac{4 \sqrt{d+e x} (a B e-4 A b e+3 b B d)}{3 \sqrt{a+b x} (b d-a e)^3}+\frac{2 \sqrt{d+e x} (a B e-4 A b e+3 b B d)}{3 e (a+b x)^{3/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.5774, size = 133, normalized size = 0.96 \[ - \frac{4 \sqrt{d + e x} \left (4 A b e - B a e - 3 B b d\right )}{3 \sqrt{a + b x} \left (a e - b d\right )^{3}} - \frac{2 \sqrt{d + e x} \left (4 A b e - B a e - 3 B b d\right )}{3 e \left (a + b x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (A e - B d\right )}{e \left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.474624, size = 100, normalized size = 0.72 \[ \frac{2 \sqrt{a+b x} \sqrt{d+e x} \left (-\frac{(A b-a B) (b d-a e)}{(a+b x)^2}+\frac{-2 a B e+5 A b e-3 b B d}{a+b x}+\frac{3 e (A e-B d)}{d+e x}\right )}{3 (b d-a e)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 177, normalized size = 1.3 \[ -{\frac{16\,A{b}^{2}{e}^{2}{x}^{2}-4\,Bab{e}^{2}{x}^{2}-12\,B{b}^{2}de{x}^{2}+24\,Aab{e}^{2}x+8\,A{b}^{2}dex-6\,B{a}^{2}{e}^{2}x-20\,Babdex-6\,B{b}^{2}{d}^{2}x+6\,A{a}^{2}{e}^{2}+12\,Aabde-2\,A{b}^{2}{d}^{2}-12\,B{a}^{2}de-4\,Bab{d}^{2}}{3\,{a}^{3}{e}^{3}-9\,{a}^{2}bd{e}^{2}+9\,a{b}^{2}{d}^{2}e-3\,{b}^{3}{d}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.673711, size = 455, normalized size = 3.27 \[ \frac{2 \,{\left (3 \, A a^{2} e^{2} -{\left (2 \, B a b + A b^{2}\right )} d^{2} - 6 \,{\left (B a^{2} - A a b\right )} d e - 2 \,{\left (3 \, B b^{2} d e +{\left (B a b - 4 \, A b^{2}\right )} e^{2}\right )} x^{2} -{\left (3 \, B b^{2} d^{2} + 2 \,{\left (5 \, B a b - 2 \, A b^{2}\right )} d e + 3 \,{\left (B a^{2} - 4 \, A a b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} +{\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} +{\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} +{\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.413303, size = 763, normalized size = 5.49 \[ -\frac{2 \,{\left (B b^{2} d e - A b^{2} e^{2}\right )} \sqrt{b x + a}}{{\left (b^{3} d^{3}{\left | b \right |} - 3 \, a b^{2} d^{2}{\left | b \right |} e + 3 \, a^{2} b d{\left | b \right |} e^{2} - a^{3}{\left | b \right |} e^{3}\right )} \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} - \frac{4 \,{\left (3 \, B b^{\frac{13}{2}} d^{3} e^{\frac{1}{2}} - 4 \, B a b^{\frac{11}{2}} d^{2} e^{\frac{3}{2}} - 5 \, A b^{\frac{13}{2}} d^{2} e^{\frac{3}{2}} - 6 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2} B b^{\frac{9}{2}} d^{2} e^{\frac{1}{2}} - B a^{2} b^{\frac{9}{2}} d e^{\frac{5}{2}} + 10 \, A a b^{\frac{11}{2}} d e^{\frac{5}{2}} + 12 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2} A b^{\frac{9}{2}} d e^{\frac{3}{2}} + 3 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{4} B b^{\frac{5}{2}} d e^{\frac{1}{2}} + 2 \, B a^{3} b^{\frac{7}{2}} e^{\frac{7}{2}} - 5 \, A a^{2} b^{\frac{9}{2}} e^{\frac{7}{2}} + 6 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2} B a^{2} b^{\frac{5}{2}} e^{\frac{5}{2}} - 12 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2} A a b^{\frac{7}{2}} e^{\frac{5}{2}} - 3 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{4} A b^{\frac{5}{2}} e^{\frac{3}{2}}\right )}}{3 \,{\left (b^{2} d^{2}{\left | b \right |} - 2 \, a b d{\left | b \right |} e + a^{2}{\left | b \right |} e^{2}\right )}{\left (b^{2} d - a b e -{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(B*b^2*d*e - A*b^2*e^2)*sqrt(b*x + a)/((b^3*d^3*abs(b) - 3*a*b^2*d^2*abs(b)*e
+ 3*a^2*b*d*abs(b)*e^2 - a^3*abs(b)*e^3)*sqrt(b^2*d + (b*x + a)*b*e - a*b*e)) -
4/3*(3*B*b^(13/2)*d^3*e^(1/2) - 4*B*a*b^(11/2)*d^2*e^(3/2) - 5*A*b^(13/2)*d^2*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^(9/2)*d^2*e^(1/2) - B*a^2*b^(9/2)*d*e^(5/2) + 10*A*a*b^(11/2)*d*e^(5/2) +
12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b^
(9/2)*d*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*b^(5/2)*d*e^(1/2) + 2*B*a^3*b^(7/2)*e^(7/2) - 5*A*a^2*b^(9/2)*e^(7
/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*a^2*b^(5/2)*e^(5/2) - 12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x +
a)*b*e - a*b*e))^2*A*a*b^(7/2)*e^(5/2) - 3*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt
(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*b^(5/2)*e^(3/2))/((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) - sq
rt(b^2*d + (b*x + a)*b*e - a*b*e))^2)^3)