3.2235 \(\int \frac{A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=237 \[ \frac{32 b^2 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 \sqrt{d+e x} (b d-a e)^5}+\frac{16 b \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{3/2} (b d-a e)^4}+\frac{12 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{5/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{7 b (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{7/2} (b d-a e)} \]

[Out]

(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(7/2)) + (2*(b*B*d - 8*A
*b*e + 7*a*B*e)*Sqrt[a + b*x])/(7*b*(b*d - a*e)^2*(d + e*x)^(7/2)) + (12*(b*B*d
- 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^3*(d + e*x)^(5/2)) + (16*b*(
b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^4*(d + e*x)^(3/2)) + (
32*b^2*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^5*Sqrt[d + e*x
])

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Rubi [A]  time = 0.448499, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{32 b^2 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 \sqrt{d+e x} (b d-a e)^5}+\frac{16 b \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{3/2} (b d-a e)^4}+\frac{12 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{5/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{7 b (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{7/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(9/2)),x]

[Out]

(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(7/2)) + (2*(b*B*d - 8*A
*b*e + 7*a*B*e)*Sqrt[a + b*x])/(7*b*(b*d - a*e)^2*(d + e*x)^(7/2)) + (12*(b*B*d
- 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^3*(d + e*x)^(5/2)) + (16*b*(
b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^4*(d + e*x)^(3/2)) + (
32*b^2*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^5*Sqrt[d + e*x
])

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Rubi in Sympy [A]  time = 49.8565, size = 231, normalized size = 0.97 \[ \frac{32 b^{2} \sqrt{a + b x} \left (8 A b e - 7 B a e - B b d\right )}{35 \sqrt{d + e x} \left (a e - b d\right )^{5}} - \frac{16 b \sqrt{a + b x} \left (8 A b e - 7 B a e - B b d\right )}{35 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4}} + \frac{12 \sqrt{a + b x} \left (8 A b e - 7 B a e - B b d\right )}{35 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3}} - \frac{2 \sqrt{a + b x} \left (8 A b e - 7 B a e - B b d\right )}{7 b \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2}} + \frac{2 \left (A b - B a\right )}{b \sqrt{a + b x} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**(3/2)/(e*x+d)**(9/2),x)

[Out]

32*b**2*sqrt(a + b*x)*(8*A*b*e - 7*B*a*e - B*b*d)/(35*sqrt(d + e*x)*(a*e - b*d)*
*5) - 16*b*sqrt(a + b*x)*(8*A*b*e - 7*B*a*e - B*b*d)/(35*(d + e*x)**(3/2)*(a*e -
b*d)**4) + 12*sqrt(a + b*x)*(8*A*b*e - 7*B*a*e - B*b*d)/(35*(d + e*x)**(5/2)*(a
*e - b*d)**3) - 2*sqrt(a + b*x)*(8*A*b*e - 7*B*a*e - B*b*d)/(7*b*(d + e*x)**(7/2
)*(a*e - b*d)**2) + 2*(A*b - B*a)/(b*sqrt(a + b*x)*(d + e*x)**(7/2)*(a*e - b*d))

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Mathematica [A]  time = 0.454577, size = 174, normalized size = 0.73 \[ \frac{2 \sqrt{a+b x} \sqrt{d+e x} \left (-\frac{35 b^3 (A b-a B)}{a+b x}+\frac{b^2 (77 a B e-93 A b e+16 b B d)}{d+e x}+\frac{b (b d-a e) (21 a B e-29 A b e+8 b B d)}{(d+e x)^2}+\frac{(b d-a e)^2 (7 a B e-13 A b e+6 b B d)}{(d+e x)^3}+\frac{5 (b d-a e)^3 (B d-A e)}{(d+e x)^4}\right )}{35 (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(9/2)),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[d + e*x]*((-35*b^3*(A*b - a*B))/(a + b*x) + (5*(b*d - a*e)
^3*(B*d - A*e))/(d + e*x)^4 + ((b*d - a*e)^2*(6*b*B*d - 13*A*b*e + 7*a*B*e))/(d
+ e*x)^3 + (b*(b*d - a*e)*(8*b*B*d - 29*A*b*e + 21*a*B*e))/(d + e*x)^2 + (b^2*(1
6*b*B*d - 93*A*b*e + 77*a*B*e))/(d + e*x)))/(35*(b*d - a*e)^5)

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Maple [B]  time = 0.018, size = 505, normalized size = 2.1 \[ -{\frac{-256\,A{b}^{4}{e}^{4}{x}^{4}+224\,Ba{b}^{3}{e}^{4}{x}^{4}+32\,B{b}^{4}d{e}^{3}{x}^{4}-128\,Aa{b}^{3}{e}^{4}{x}^{3}-896\,A{b}^{4}d{e}^{3}{x}^{3}+112\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}+800\,Ba{b}^{3}d{e}^{3}{x}^{3}+112\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+32\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}-448\,Aa{b}^{3}d{e}^{3}{x}^{2}-1120\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-28\,B{a}^{3}b{e}^{4}{x}^{2}+388\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}+1036\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}+140\,B{b}^{4}{d}^{3}e{x}^{2}-16\,A{a}^{3}b{e}^{4}x+112\,A{a}^{2}{b}^{2}d{e}^{3}x-560\,Aa{b}^{3}{d}^{2}{e}^{2}x-560\,A{b}^{4}{d}^{3}ex+14\,B{a}^{4}{e}^{4}x-96\,B{a}^{3}bd{e}^{3}x+476\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x+560\,Ba{b}^{3}{d}^{3}ex+70\,B{b}^{4}{d}^{4}x+10\,A{a}^{4}{e}^{4}-56\,A{a}^{3}bd{e}^{3}+140\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}-280\,Aa{b}^{3}{d}^{3}e-70\,A{b}^{4}{d}^{4}+4\,B{a}^{4}d{e}^{3}-28\,B{a}^{3}b{d}^{2}{e}^{2}+140\,B{a}^{2}{b}^{2}{d}^{3}e+140\,Ba{b}^{3}{d}^{4}}{35\,{a}^{5}{e}^{5}-175\,{a}^{4}bd{e}^{4}+350\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-350\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+175\,a{b}^{4}{d}^{4}e-35\,{b}^{5}{d}^{5}}{\frac{1}{\sqrt{bx+a}}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(9/2),x)

[Out]

-2/35*(-128*A*b^4*e^4*x^4+112*B*a*b^3*e^4*x^4+16*B*b^4*d*e^3*x^4-64*A*a*b^3*e^4*
x^3-448*A*b^4*d*e^3*x^3+56*B*a^2*b^2*e^4*x^3+400*B*a*b^3*d*e^3*x^3+56*B*b^4*d^2*
e^2*x^3+16*A*a^2*b^2*e^4*x^2-224*A*a*b^3*d*e^3*x^2-560*A*b^4*d^2*e^2*x^2-14*B*a^
3*b*e^4*x^2+194*B*a^2*b^2*d*e^3*x^2+518*B*a*b^3*d^2*e^2*x^2+70*B*b^4*d^3*e*x^2-8
*A*a^3*b*e^4*x+56*A*a^2*b^2*d*e^3*x-280*A*a*b^3*d^2*e^2*x-280*A*b^4*d^3*e*x+7*B*
a^4*e^4*x-48*B*a^3*b*d*e^3*x+238*B*a^2*b^2*d^2*e^2*x+280*B*a*b^3*d^3*e*x+35*B*b^
4*d^4*x+5*A*a^4*e^4-28*A*a^3*b*d*e^3+70*A*a^2*b^2*d^2*e^2-140*A*a*b^3*d^3*e-35*A
*b^4*d^4+2*B*a^4*d*e^3-14*B*a^3*b*d^2*e^2+70*B*a^2*b^2*d^3*e+70*B*a*b^3*d^4)/(b*
x+a)^(1/2)/(e*x+d)^(7/2)/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^
3*e^2+5*a*b^4*d^4*e-b^5*d^5)

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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)^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.36355, size = 1197, normalized size = 5.05 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*(e*x + d)^(9/2)),x, algorithm="fricas")

[Out]

2/35*(5*A*a^4*e^4 + 35*(2*B*a*b^3 - A*b^4)*d^4 + 70*(B*a^2*b^2 - 2*A*a*b^3)*d^3*
e - 14*(B*a^3*b - 5*A*a^2*b^2)*d^2*e^2 + 2*(B*a^4 - 14*A*a^3*b)*d*e^3 + 16*(B*b^
4*d*e^3 + (7*B*a*b^3 - 8*A*b^4)*e^4)*x^4 + 8*(7*B*b^4*d^2*e^2 + 2*(25*B*a*b^3 -
28*A*b^4)*d*e^3 + (7*B*a^2*b^2 - 8*A*a*b^3)*e^4)*x^3 + 2*(35*B*b^4*d^3*e + 7*(37
*B*a*b^3 - 40*A*b^4)*d^2*e^2 + (97*B*a^2*b^2 - 112*A*a*b^3)*d*e^3 - (7*B*a^3*b -
8*A*a^2*b^2)*e^4)*x^2 + (35*B*b^4*d^4 + 280*(B*a*b^3 - A*b^4)*d^3*e + 14*(17*B*
a^2*b^2 - 20*A*a*b^3)*d^2*e^2 - 8*(6*B*a^3*b - 7*A*a^2*b^2)*d*e^3 + (7*B*a^4 - 8
*A*a^3*b)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(a*b^5*d^9 - 5*a^2*b^4*d^8*e + 10*
a^3*b^3*d^7*e^2 - 10*a^4*b^2*d^6*e^3 + 5*a^5*b*d^5*e^4 - a^6*d^4*e^5 + (b^6*d^5*
e^4 - 5*a*b^5*d^4*e^5 + 10*a^2*b^4*d^3*e^6 - 10*a^3*b^3*d^2*e^7 + 5*a^4*b^2*d*e^
8 - a^5*b*e^9)*x^5 + (4*b^6*d^6*e^3 - 19*a*b^5*d^5*e^4 + 35*a^2*b^4*d^4*e^5 - 30
*a^3*b^3*d^3*e^6 + 10*a^4*b^2*d^2*e^7 + a^5*b*d*e^8 - a^6*e^9)*x^4 + 2*(3*b^6*d^
7*e^2 - 13*a*b^5*d^6*e^3 + 20*a^2*b^4*d^5*e^4 - 10*a^3*b^3*d^4*e^5 - 5*a^4*b^2*d
^3*e^6 + 7*a^5*b*d^2*e^7 - 2*a^6*d*e^8)*x^3 + 2*(2*b^6*d^8*e - 7*a*b^5*d^7*e^2 +
5*a^2*b^4*d^6*e^3 + 10*a^3*b^3*d^5*e^4 - 20*a^4*b^2*d^4*e^5 + 13*a^5*b*d^3*e^6
- 3*a^6*d^2*e^7)*x^2 + (b^6*d^9 - a*b^5*d^8*e - 10*a^2*b^4*d^7*e^2 + 30*a^3*b^3*
d^6*e^3 - 35*a^4*b^2*d^5*e^4 + 19*a^5*b*d^4*e^5 - 4*a^6*d^3*e^6)*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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.900563, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*(e*x + d)^(9/2)),x, algorithm="giac")

[Out]

Done