3.2244 \(\int \frac{A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=298 \[ -\frac{256 b^2 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{105 \sqrt{d+e x} (b d-a e)^6}-\frac{128 b e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{105 (d+e x)^{3/2} (b d-a e)^5}-\frac{32 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{35 (d+e x)^{5/2} (b d-a e)^4}-\frac{16 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{21 b (d+e x)^{7/2} (b d-a e)^3}-\frac{2 (7 a B e-10 A b e+3 b B d)}{3 b \sqrt{a+b x} (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)} \]

[Out]

(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(7/2)) - (2*(3*b*B*d
- 10*A*b*e + 7*a*B*e))/(3*b*(b*d - a*e)^2*Sqrt[a + b*x]*(d + e*x)^(7/2)) - (16*
e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(21*b*(b*d - a*e)^3*(d + e*x)^(7
/2)) - (32*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^4*(d
+ e*x)^(5/2)) - (128*b*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(105*(b*d
- a*e)^5*(d + e*x)^(3/2)) - (256*b^2*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a +
b*x])/(105*(b*d - a*e)^6*Sqrt[d + e*x])

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Rubi [A]  time = 0.574721, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{256 b^2 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{105 \sqrt{d+e x} (b d-a e)^6}-\frac{128 b e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{105 (d+e x)^{3/2} (b d-a e)^5}-\frac{32 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{35 (d+e x)^{5/2} (b d-a e)^4}-\frac{16 e \sqrt{a+b x} (7 a B e-10 A b e+3 b B d)}{21 b (d+e x)^{7/2} (b d-a e)^3}-\frac{2 (7 a B e-10 A b e+3 b B d)}{3 b \sqrt{a+b x} (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(9/2)),x]

[Out]

(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(7/2)) - (2*(3*b*B*d
- 10*A*b*e + 7*a*B*e))/(3*b*(b*d - a*e)^2*Sqrt[a + b*x]*(d + e*x)^(7/2)) - (16*
e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(21*b*(b*d - a*e)^3*(d + e*x)^(7
/2)) - (32*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^4*(d
+ e*x)^(5/2)) - (128*b*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(105*(b*d
- a*e)^5*(d + e*x)^(3/2)) - (256*b^2*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a +
b*x])/(105*(b*d - a*e)^6*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 64.2006, size = 296, normalized size = 0.99 \[ \frac{256 b^{2} e \sqrt{a + b x} \left (10 A b e - 7 B a e - 3 B b d\right )}{105 \sqrt{d + e x} \left (a e - b d\right )^{6}} - \frac{128 b e \sqrt{a + b x} \left (10 A b e - 7 B a e - 3 B b d\right )}{105 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{5}} + \frac{32 e \sqrt{a + b x} \left (10 A b e - 7 B a e - 3 B b d\right )}{35 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{4}} - \frac{16 e \sqrt{a + b x} \left (10 A b e - 7 B a e - 3 B b d\right )}{21 b \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (10 A b e - 7 B a e - 3 B b d\right )}{3 b \sqrt{a + b x} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2}} + \frac{2 \left (A b - B a\right )}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**(5/2)/(e*x+d)**(9/2),x)

[Out]

256*b**2*e*sqrt(a + b*x)*(10*A*b*e - 7*B*a*e - 3*B*b*d)/(105*sqrt(d + e*x)*(a*e
- b*d)**6) - 128*b*e*sqrt(a + b*x)*(10*A*b*e - 7*B*a*e - 3*B*b*d)/(105*(d + e*x)
**(3/2)*(a*e - b*d)**5) + 32*e*sqrt(a + b*x)*(10*A*b*e - 7*B*a*e - 3*B*b*d)/(35*
(d + e*x)**(5/2)*(a*e - b*d)**4) - 16*e*sqrt(a + b*x)*(10*A*b*e - 7*B*a*e - 3*B*
b*d)/(21*b*(d + e*x)**(7/2)*(a*e - b*d)**3) + 2*(10*A*b*e - 7*B*a*e - 3*B*b*d)/(
3*b*sqrt(a + b*x)*(d + e*x)**(7/2)*(a*e - b*d)**2) + 2*(A*b - B*a)/(3*b*(a + b*x
)**(3/2)*(d + e*x)**(7/2)*(a*e - b*d))

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Mathematica [A]  time = 0.735889, size = 215, normalized size = 0.72 \[ \frac{2 \sqrt{a+b x} \sqrt{d+e x} \left (-\frac{35 b^3 (11 a B e-14 A b e+3 b B d)}{a+b x}-\frac{35 b^3 (A b-a B) (b d-a e)}{(a+b x)^2}+\frac{b^2 e (-511 a B e+790 A b e-279 b B d)}{d+e x}+\frac{b e (b d-a e) (-98 a B e+185 A b e-87 b B d)}{(d+e x)^2}+\frac{3 e (b d-a e)^2 (-7 a B e+20 A b e-13 b B d)}{(d+e x)^3}+\frac{15 e (b d-a e)^3 (A e-B d)}{(d+e x)^4}\right )}{105 (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(9/2)),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[d + e*x]*((-35*b^3*(A*b - a*B)*(b*d - a*e))/(a + b*x)^2 -
(35*b^3*(3*b*B*d - 14*A*b*e + 11*a*B*e))/(a + b*x) + (15*e*(b*d - a*e)^3*(-(B*d)
+ A*e))/(d + e*x)^4 + (3*e*(b*d - a*e)^2*(-13*b*B*d + 20*A*b*e - 7*a*B*e))/(d +
e*x)^3 + (b*e*(b*d - a*e)*(-87*b*B*d + 185*A*b*e - 98*a*B*e))/(d + e*x)^2 + (b^
2*e*(-279*b*B*d + 790*A*b*e - 511*a*B*e))/(d + e*x)))/(105*(b*d - a*e)^6)

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Maple [B]  time = 0.019, size = 722, normalized size = 2.4 \[ -{\frac{-2560\,A{b}^{5}{e}^{5}{x}^{5}+1792\,Ba{b}^{4}{e}^{5}{x}^{5}+768\,B{b}^{5}d{e}^{4}{x}^{5}-3840\,Aa{b}^{4}{e}^{5}{x}^{4}-8960\,A{b}^{5}d{e}^{4}{x}^{4}+2688\,B{a}^{2}{b}^{3}{e}^{5}{x}^{4}+7424\,Ba{b}^{4}d{e}^{4}{x}^{4}+2688\,B{b}^{5}{d}^{2}{e}^{3}{x}^{4}-960\,A{a}^{2}{b}^{3}{e}^{5}{x}^{3}-13440\,Aa{b}^{4}d{e}^{4}{x}^{3}-11200\,A{b}^{5}{d}^{2}{e}^{3}{x}^{3}+672\,B{a}^{3}{b}^{2}{e}^{5}{x}^{3}+9696\,B{a}^{2}{b}^{3}d{e}^{4}{x}^{3}+11872\,Ba{b}^{4}{d}^{2}{e}^{3}{x}^{3}+3360\,B{b}^{5}{d}^{3}{e}^{2}{x}^{3}+160\,A{a}^{3}{b}^{2}{e}^{5}{x}^{2}-3360\,A{a}^{2}{b}^{3}d{e}^{4}{x}^{2}-16800\,Aa{b}^{4}{d}^{2}{e}^{3}{x}^{2}-5600\,A{b}^{5}{d}^{3}{e}^{2}{x}^{2}-112\,B{a}^{4}b{e}^{5}{x}^{2}+2304\,B{a}^{3}{b}^{2}d{e}^{4}{x}^{2}+12768\,B{a}^{2}{b}^{3}{d}^{2}{e}^{3}{x}^{2}+8960\,Ba{b}^{4}{d}^{3}{e}^{2}{x}^{2}+1680\,B{b}^{5}{d}^{4}e{x}^{2}-60\,A{a}^{4}b{e}^{5}x+560\,A{a}^{3}{b}^{2}d{e}^{4}x-4200\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}x-8400\,Aa{b}^{4}{d}^{3}{e}^{2}x-700\,A{b}^{5}{d}^{4}ex+42\,B{a}^{5}{e}^{5}x-374\,B{a}^{4}bd{e}^{4}x+2772\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}x+7140\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}x+3010\,Ba{b}^{4}{d}^{4}ex+210\,B{b}^{5}{d}^{5}x+30\,A{a}^{5}{e}^{5}-210\,A{a}^{4}bd{e}^{4}+700\,A{a}^{3}{b}^{2}{d}^{2}{e}^{3}-2100\,A{a}^{2}{b}^{3}{d}^{3}{e}^{2}-1050\,Aa{b}^{4}{d}^{4}e+70\,A{b}^{5}{d}^{5}+12\,B{a}^{5}d{e}^{4}-112\,B{a}^{4}b{d}^{2}{e}^{3}+840\,B{a}^{3}{b}^{2}{d}^{3}{e}^{2}+1680\,B{a}^{2}{b}^{3}{d}^{4}e+140\,Ba{b}^{4}{d}^{5}}{105\,{a}^{6}{e}^{6}-630\,{a}^{5}bd{e}^{5}+1575\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-2100\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+1575\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-630\,a{b}^{5}{d}^{5}e+105\,{b}^{6}{d}^{6}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(9/2),x)

[Out]

-2/105*(-1280*A*b^5*e^5*x^5+896*B*a*b^4*e^5*x^5+384*B*b^5*d*e^4*x^5-1920*A*a*b^4
*e^5*x^4-4480*A*b^5*d*e^4*x^4+1344*B*a^2*b^3*e^5*x^4+3712*B*a*b^4*d*e^4*x^4+1344
*B*b^5*d^2*e^3*x^4-480*A*a^2*b^3*e^5*x^3-6720*A*a*b^4*d*e^4*x^3-5600*A*b^5*d^2*e
^3*x^3+336*B*a^3*b^2*e^5*x^3+4848*B*a^2*b^3*d*e^4*x^3+5936*B*a*b^4*d^2*e^3*x^3+1
680*B*b^5*d^3*e^2*x^3+80*A*a^3*b^2*e^5*x^2-1680*A*a^2*b^3*d*e^4*x^2-8400*A*a*b^4
*d^2*e^3*x^2-2800*A*b^5*d^3*e^2*x^2-56*B*a^4*b*e^5*x^2+1152*B*a^3*b^2*d*e^4*x^2+
6384*B*a^2*b^3*d^2*e^3*x^2+4480*B*a*b^4*d^3*e^2*x^2+840*B*b^5*d^4*e*x^2-30*A*a^4
*b*e^5*x+280*A*a^3*b^2*d*e^4*x-2100*A*a^2*b^3*d^2*e^3*x-4200*A*a*b^4*d^3*e^2*x-3
50*A*b^5*d^4*e*x+21*B*a^5*e^5*x-187*B*a^4*b*d*e^4*x+1386*B*a^3*b^2*d^2*e^3*x+357
0*B*a^2*b^3*d^3*e^2*x+1505*B*a*b^4*d^4*e*x+105*B*b^5*d^5*x+15*A*a^5*e^5-105*A*a^
4*b*d*e^4+350*A*a^3*b^2*d^2*e^3-1050*A*a^2*b^3*d^3*e^2-525*A*a*b^4*d^4*e+35*A*b^
5*d^5+6*B*a^5*d*e^4-56*B*a^4*b*d^2*e^3+420*B*a^3*b^2*d^3*e^2+840*B*a^2*b^3*d^4*e
+70*B*a*b^4*d^5)/(b*x+a)^(3/2)/(e*x+d)^(7/2)/(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d
^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^6)

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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)^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.61097, size = 1742, normalized size = 5.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(5/2)*(e*x + d)^(9/2)),x, algorithm="fricas")

[Out]

-2/105*(15*A*a^5*e^5 + 35*(2*B*a*b^4 + A*b^5)*d^5 + 105*(8*B*a^2*b^3 - 5*A*a*b^4
)*d^4*e + 210*(2*B*a^3*b^2 - 5*A*a^2*b^3)*d^3*e^2 - 14*(4*B*a^4*b - 25*A*a^3*b^2
)*d^2*e^3 + 3*(2*B*a^5 - 35*A*a^4*b)*d*e^4 + 128*(3*B*b^5*d*e^4 + (7*B*a*b^4 - 1
0*A*b^5)*e^5)*x^5 + 64*(21*B*b^5*d^2*e^3 + 2*(29*B*a*b^4 - 35*A*b^5)*d*e^4 + 3*(
7*B*a^2*b^3 - 10*A*a*b^4)*e^5)*x^4 + 16*(105*B*b^5*d^3*e^2 + 7*(53*B*a*b^4 - 50*
A*b^5)*d^2*e^3 + 3*(101*B*a^2*b^3 - 140*A*a*b^4)*d*e^4 + 3*(7*B*a^3*b^2 - 10*A*a
^2*b^3)*e^5)*x^3 + 8*(105*B*b^5*d^4*e + 70*(8*B*a*b^4 - 5*A*b^5)*d^3*e^2 + 42*(1
9*B*a^2*b^3 - 25*A*a*b^4)*d^2*e^3 + 6*(24*B*a^3*b^2 - 35*A*a^2*b^3)*d*e^4 - (7*B
*a^4*b - 10*A*a^3*b^2)*e^5)*x^2 + (105*B*b^5*d^5 + 35*(43*B*a*b^4 - 10*A*b^5)*d^
4*e + 210*(17*B*a^2*b^3 - 20*A*a*b^4)*d^3*e^2 + 42*(33*B*a^3*b^2 - 50*A*a^2*b^3)
*d^2*e^3 - (187*B*a^4*b - 280*A*a^3*b^2)*d*e^4 + 3*(7*B*a^5 - 10*A*a^4*b)*e^5)*x
)*sqrt(b*x + a)*sqrt(e*x + d)/(a^2*b^6*d^10 - 6*a^3*b^5*d^9*e + 15*a^4*b^4*d^8*e
^2 - 20*a^5*b^3*d^7*e^3 + 15*a^6*b^2*d^6*e^4 - 6*a^7*b*d^5*e^5 + a^8*d^4*e^6 + (
b^8*d^6*e^4 - 6*a*b^7*d^5*e^5 + 15*a^2*b^6*d^4*e^6 - 20*a^3*b^5*d^3*e^7 + 15*a^4
*b^4*d^2*e^8 - 6*a^5*b^3*d*e^9 + a^6*b^2*e^10)*x^6 + 2*(2*b^8*d^7*e^3 - 11*a*b^7
*d^6*e^4 + 24*a^2*b^6*d^5*e^5 - 25*a^3*b^5*d^4*e^6 + 10*a^4*b^4*d^3*e^7 + 3*a^5*
b^3*d^2*e^8 - 4*a^6*b^2*d*e^9 + a^7*b*e^10)*x^5 + (6*b^8*d^8*e^2 - 28*a*b^7*d^7*
e^3 + 43*a^2*b^6*d^6*e^4 - 6*a^3*b^5*d^5*e^5 - 55*a^4*b^4*d^4*e^6 + 64*a^5*b^3*d
^3*e^7 - 27*a^6*b^2*d^2*e^8 + 2*a^7*b*d*e^9 + a^8*e^10)*x^4 + 4*(b^8*d^9*e - 3*a
*b^7*d^8*e^2 - 2*a^2*b^6*d^7*e^3 + 19*a^3*b^5*d^6*e^4 - 30*a^4*b^4*d^5*e^5 + 19*
a^5*b^3*d^4*e^6 - 2*a^6*b^2*d^3*e^7 - 3*a^7*b*d^2*e^8 + a^8*d*e^9)*x^3 + (b^8*d^
10 + 2*a*b^7*d^9*e - 27*a^2*b^6*d^8*e^2 + 64*a^3*b^5*d^7*e^3 - 55*a^4*b^4*d^6*e^
4 - 6*a^5*b^3*d^5*e^5 + 43*a^6*b^2*d^4*e^6 - 28*a^7*b*d^3*e^7 + 6*a^8*d^2*e^8)*x
^2 + 2*(a*b^7*d^10 - 4*a^2*b^6*d^9*e + 3*a^3*b^5*d^8*e^2 + 10*a^4*b^4*d^7*e^3 -
25*a^5*b^3*d^6*e^4 + 24*a^6*b^2*d^5*e^5 - 11*a^7*b*d^4*e^6 + 2*a^8*d^3*e^7)*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)**(5/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 2.65207, 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)^(5/2)*(e*x + d)^(9/2)),x, algorithm="giac")

[Out]

Done