3.2234 \(\int \frac{A+B x}{(a+b x)^{3/2} (d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{5/2} (b d-a e)}+\frac{16 b \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{15 \sqrt{d+e x} (b d-a e)^4}+\frac{8 \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{15 (d+e x)^{3/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{5 b (d+e x)^{5/2} (b d-a e)^2} \]

[Out]

(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(5/2)) + (2*(b*B*d - 6*A
*b*e + 5*a*B*e)*Sqrt[a + b*x])/(5*b*(b*d - a*e)^2*(d + e*x)^(5/2)) + (8*(b*B*d -
6*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^3*(d + e*x)^(3/2)) + (16*b*(b
*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^4*Sqrt[d + e*x])

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Rubi [A]  time = 0.338952, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{5/2} (b d-a e)}+\frac{16 b \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{15 \sqrt{d+e x} (b d-a e)^4}+\frac{8 \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{15 (d+e x)^{3/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{5 b (d+e x)^{5/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(7/2)),x]

[Out]

(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(5/2)) + (2*(b*B*d - 6*A
*b*e + 5*a*B*e)*Sqrt[a + b*x])/(5*b*(b*d - a*e)^2*(d + e*x)^(5/2)) + (8*(b*B*d -
6*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^3*(d + e*x)^(3/2)) + (16*b*(b
*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^4*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 37.0173, size = 180, normalized size = 0.96 \[ - \frac{16 b \sqrt{a + b x} \left (6 A b e - 5 B a e - B b d\right )}{15 \sqrt{d + e x} \left (a e - b d\right )^{4}} + \frac{8 \sqrt{a + b x} \left (6 A b e - 5 B a e - B b d\right )}{15 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}} - \frac{2 \sqrt{a + b x} \left (6 A b e - 5 B a e - B b d\right )}{5 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}} + \frac{2 \left (A b - B a\right )}{b \sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**(3/2)/(e*x+d)**(7/2),x)

[Out]

-16*b*sqrt(a + b*x)*(6*A*b*e - 5*B*a*e - B*b*d)/(15*sqrt(d + e*x)*(a*e - b*d)**4
) + 8*sqrt(a + b*x)*(6*A*b*e - 5*B*a*e - B*b*d)/(15*(d + e*x)**(3/2)*(a*e - b*d)
**3) - 2*sqrt(a + b*x)*(6*A*b*e - 5*B*a*e - B*b*d)/(5*b*(d + e*x)**(5/2)*(a*e -
b*d)**2) + 2*(A*b - B*a)/(b*sqrt(a + b*x)*(d + e*x)**(5/2)*(a*e - b*d))

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Mathematica [A]  time = 0.474398, size = 137, normalized size = 0.73 \[ \frac{2 \sqrt{a+b x} \sqrt{d+e x} \left (-\frac{15 b^2 (A b-a B)}{a+b x}+\frac{b (25 a B e-33 A b e+8 b B d)}{d+e x}+\frac{(b d-a e) (5 a B e-9 A b e+4 b B d)}{(d+e x)^2}+\frac{3 (b d-a e)^2 (B d-A e)}{(d+e x)^3}\right )}{15 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(7/2)),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[d + e*x]*((-15*b^2*(A*b - a*B))/(a + b*x) + (3*(b*d - a*e)
^2*(B*d - A*e))/(d + e*x)^3 + ((b*d - a*e)*(4*b*B*d - 9*A*b*e + 5*a*B*e))/(d + e
*x)^2 + (b*(8*b*B*d - 33*A*b*e + 25*a*B*e))/(d + e*x)))/(15*(b*d - a*e)^4)

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Maple [A]  time = 0.014, size = 322, normalized size = 1.7 \[ -{\frac{96\,A{b}^{3}{e}^{3}{x}^{3}-80\,Ba{b}^{2}{e}^{3}{x}^{3}-16\,B{b}^{3}d{e}^{2}{x}^{3}+48\,Aa{b}^{2}{e}^{3}{x}^{2}+240\,A{b}^{3}d{e}^{2}{x}^{2}-40\,B{a}^{2}b{e}^{3}{x}^{2}-208\,Ba{b}^{2}d{e}^{2}{x}^{2}-40\,B{b}^{3}{d}^{2}e{x}^{2}-12\,A{a}^{2}b{e}^{3}x+120\,Aa{b}^{2}d{e}^{2}x+180\,A{b}^{3}{d}^{2}ex+10\,B{a}^{3}{e}^{3}x-98\,B{a}^{2}bd{e}^{2}x-170\,Ba{b}^{2}{d}^{2}ex-30\,B{b}^{3}{d}^{3}x+6\,A{a}^{3}{e}^{3}-30\,A{a}^{2}bd{e}^{2}+90\,Aa{b}^{2}{d}^{2}e+30\,A{b}^{3}{d}^{3}+4\,B{a}^{3}d{e}^{2}-40\,B{a}^{2}b{d}^{2}e-60\,Ba{b}^{2}{d}^{3}}{15\,{e}^{4}{a}^{4}-60\,b{e}^{3}d{a}^{3}+90\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-60\,a{b}^{3}{d}^{3}e+15\,{b}^{4}{d}^{4}}{\frac{1}{\sqrt{bx+a}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(7/2),x)

[Out]

-2/15*(48*A*b^3*e^3*x^3-40*B*a*b^2*e^3*x^3-8*B*b^3*d*e^2*x^3+24*A*a*b^2*e^3*x^2+
120*A*b^3*d*e^2*x^2-20*B*a^2*b*e^3*x^2-104*B*a*b^2*d*e^2*x^2-20*B*b^3*d^2*e*x^2-
6*A*a^2*b*e^3*x+60*A*a*b^2*d*e^2*x+90*A*b^3*d^2*e*x+5*B*a^3*e^3*x-49*B*a^2*b*d*e
^2*x-85*B*a*b^2*d^2*e*x-15*B*b^3*d^3*x+3*A*a^3*e^3-15*A*a^2*b*d*e^2+45*A*a*b^2*d
^2*e+15*A*b^3*d^3+2*B*a^3*d*e^2-20*B*a^2*b*d^2*e-30*B*a*b^2*d^3)/(b*x+a)^(1/2)/(
e*x+d)^(5/2)/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)

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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)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.19508, size = 787, normalized size = 4.21 \[ -\frac{2 \,{\left (3 \, A a^{3} e^{3} - 15 \,{\left (2 \, B a b^{2} - A b^{3}\right )} d^{3} - 5 \,{\left (4 \, B a^{2} b - 9 \, A a b^{2}\right )} d^{2} e +{\left (2 \, B a^{3} - 15 \, A a^{2} b\right )} d e^{2} - 8 \,{\left (B b^{3} d e^{2} +{\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} e^{3}\right )} x^{3} - 4 \,{\left (5 \, B b^{3} d^{2} e + 2 \,{\left (13 \, B a b^{2} - 15 \, A b^{3}\right )} d e^{2} +{\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} e^{3}\right )} x^{2} -{\left (15 \, B b^{3} d^{3} + 5 \,{\left (17 \, B a b^{2} - 18 \, A b^{3}\right )} d^{2} e +{\left (49 \, B a^{2} b - 60 \, A a b^{2}\right )} d e^{2} -{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{15 \,{\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} +{\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} +{\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \,{\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} +{\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

-2/15*(3*A*a^3*e^3 - 15*(2*B*a*b^2 - A*b^3)*d^3 - 5*(4*B*a^2*b - 9*A*a*b^2)*d^2*
e + (2*B*a^3 - 15*A*a^2*b)*d*e^2 - 8*(B*b^3*d*e^2 + (5*B*a*b^2 - 6*A*b^3)*e^3)*x
^3 - 4*(5*B*b^3*d^2*e + 2*(13*B*a*b^2 - 15*A*b^3)*d*e^2 + (5*B*a^2*b - 6*A*a*b^2
)*e^3)*x^2 - (15*B*b^3*d^3 + 5*(17*B*a*b^2 - 18*A*b^3)*d^2*e + (49*B*a^2*b - 60*
A*a*b^2)*d*e^2 - (5*B*a^3 - 6*A*a^2*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(a*b^
4*d^7 - 4*a^2*b^3*d^6*e + 6*a^3*b^2*d^5*e^2 - 4*a^4*b*d^4*e^3 + a^5*d^3*e^4 + (b
^5*d^4*e^3 - 4*a*b^4*d^3*e^4 + 6*a^2*b^3*d^2*e^5 - 4*a^3*b^2*d*e^6 + a^4*b*e^7)*
x^4 + (3*b^5*d^5*e^2 - 11*a*b^4*d^4*e^3 + 14*a^2*b^3*d^3*e^4 - 6*a^3*b^2*d^2*e^5
- a^4*b*d*e^6 + a^5*e^7)*x^3 + 3*(b^5*d^6*e - 3*a*b^4*d^5*e^2 + 2*a^2*b^3*d^4*e
^3 + 2*a^3*b^2*d^3*e^4 - 3*a^4*b*d^2*e^5 + a^5*d*e^6)*x^2 + (b^5*d^7 - a*b^4*d^6
*e - 6*a^2*b^3*d^5*e^2 + 14*a^3*b^2*d^4*e^3 - 11*a^4*b*d^3*e^4 + 3*a^5*d^2*e^5)*
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)**(3/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.455715, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

Done