3.2243 \(\int \frac{A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=246 \[ -\frac{2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}-\frac{32 b e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{15 \sqrt{d+e x} (b d-a e)^5}-\frac{16 e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{15 (d+e x)^{3/2} (b d-a e)^4}-\frac{4 e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{5 b (d+e x)^{5/2} (b d-a e)^3}-\frac{2 (5 a B e-8 A b e+3 b B d)}{3 b \sqrt{a+b x} (d+e x)^{5/2} (b d-a e)^2} \]

[Out]

(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(5/2)) - (2*(3*b*B*d
- 8*A*b*e + 5*a*B*e))/(3*b*(b*d - a*e)^2*Sqrt[a + b*x]*(d + e*x)^(5/2)) - (4*e*
(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(5*b*(b*d - a*e)^3*(d + e*x)^(5/2))
- (16*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^4*(d + e*x
)^(3/2)) - (32*b*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^
5*Sqrt[d + e*x])

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Rubi [A]  time = 0.478994, antiderivative size = 246, 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 \[ -\frac{2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}-\frac{32 b e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{15 \sqrt{d+e x} (b d-a e)^5}-\frac{16 e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{15 (d+e x)^{3/2} (b d-a e)^4}-\frac{4 e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{5 b (d+e x)^{5/2} (b d-a e)^3}-\frac{2 (5 a B e-8 A b e+3 b B d)}{3 b \sqrt{a+b x} (d+e x)^{5/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(7/2)),x]

[Out]

(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(5/2)) - (2*(3*b*B*d
- 8*A*b*e + 5*a*B*e))/(3*b*(b*d - a*e)^2*Sqrt[a + b*x]*(d + e*x)^(5/2)) - (4*e*
(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(5*b*(b*d - a*e)^3*(d + e*x)^(5/2))
- (16*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^4*(d + e*x
)^(3/2)) - (32*b*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^
5*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 48.6957, size = 241, normalized size = 0.98 \[ - \frac{32 b e \sqrt{a + b x} \left (8 A b e - 5 B a e - 3 B b d\right )}{15 \sqrt{d + e x} \left (a e - b d\right )^{5}} + \frac{16 e \sqrt{a + b x} \left (8 A b e - 5 B a e - 3 B b d\right )}{15 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4}} - \frac{4 e \sqrt{a + b x} \left (8 A b e - 5 B a e - 3 B b d\right )}{5 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (8 A b e - 5 B a e - 3 B b d\right )}{3 b \sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}} + \frac{2 \left (A b - B a\right )}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**(5/2)/(e*x+d)**(7/2),x)

[Out]

-32*b*e*sqrt(a + b*x)*(8*A*b*e - 5*B*a*e - 3*B*b*d)/(15*sqrt(d + e*x)*(a*e - b*d
)**5) + 16*e*sqrt(a + b*x)*(8*A*b*e - 5*B*a*e - 3*B*b*d)/(15*(d + e*x)**(3/2)*(a
*e - b*d)**4) - 4*e*sqrt(a + b*x)*(8*A*b*e - 5*B*a*e - 3*B*b*d)/(5*b*(d + e*x)**
(5/2)*(a*e - b*d)**3) + 2*(8*A*b*e - 5*B*a*e - 3*B*b*d)/(3*b*sqrt(a + b*x)*(d +
e*x)**(5/2)*(a*e - b*d)**2) + 2*(A*b - B*a)/(3*b*(a + b*x)**(3/2)*(d + e*x)**(5/
2)*(a*e - b*d))

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Mathematica [A]  time = 0.511272, size = 176, normalized size = 0.72 \[ \frac{2 \sqrt{a+b x} \sqrt{d+e x} \left (-\frac{5 b^2 (8 a B e-11 A b e+3 b B d)}{a+b x}-\frac{5 b^2 (A b-a B) (b d-a e)}{(a+b x)^2}+\frac{b e (-40 a B e+73 A b e-33 b B d)}{d+e x}+\frac{e (b d-a e) (-5 a B e+14 A b e-9 b B d)}{(d+e x)^2}+\frac{3 e (b d-a e)^2 (A e-B d)}{(d+e x)^3}\right )}{15 (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(7/2)),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[d + e*x]*((-5*b^2*(A*b - a*B)*(b*d - a*e))/(a + b*x)^2 - (
5*b^2*(3*b*B*d - 11*A*b*e + 8*a*B*e))/(a + b*x) + (3*e*(b*d - a*e)^2*(-(B*d) + A
*e))/(d + e*x)^3 + (e*(b*d - a*e)*(-9*b*B*d + 14*A*b*e - 5*a*B*e))/(d + e*x)^2 +
(b*e*(-33*b*B*d + 73*A*b*e - 40*a*B*e))/(d + e*x)))/(15*(b*d - a*e)^5)

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Maple [B]  time = 0.017, size = 505, normalized size = 2.1 \[ -{\frac{256\,A{b}^{4}{e}^{4}{x}^{4}-160\,Ba{b}^{3}{e}^{4}{x}^{4}-96\,B{b}^{4}d{e}^{3}{x}^{4}+384\,Aa{b}^{3}{e}^{4}{x}^{3}+640\,A{b}^{4}d{e}^{3}{x}^{3}-240\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}-544\,Ba{b}^{3}d{e}^{3}{x}^{3}-240\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+96\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}+960\,Aa{b}^{3}d{e}^{3}{x}^{2}+480\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-60\,B{a}^{3}b{e}^{4}{x}^{2}-636\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}-660\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}-180\,B{b}^{4}{d}^{3}e{x}^{2}-16\,A{a}^{3}b{e}^{4}x+240\,A{a}^{2}{b}^{2}d{e}^{3}x+720\,Aa{b}^{3}{d}^{2}{e}^{2}x+80\,A{b}^{4}{d}^{3}ex+10\,B{a}^{4}{e}^{4}x-144\,B{a}^{3}bd{e}^{3}x-540\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x-320\,Ba{b}^{3}{d}^{3}ex-30\,B{b}^{4}{d}^{4}x+6\,A{a}^{4}{e}^{4}-40\,A{a}^{3}bd{e}^{3}+180\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}+120\,Aa{b}^{3}{d}^{3}e-10\,A{b}^{4}{d}^{4}+4\,B{a}^{4}d{e}^{3}-60\,B{a}^{3}b{d}^{2}{e}^{2}-180\,B{a}^{2}{b}^{2}{d}^{3}e-20\,Ba{b}^{3}{d}^{4}}{15\,{a}^{5}{e}^{5}-75\,{a}^{4}bd{e}^{4}+150\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-150\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+75\,a{b}^{4}{d}^{4}e-15\,{b}^{5}{d}^{5}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(7/2),x)

[Out]

-2/15*(128*A*b^4*e^4*x^4-80*B*a*b^3*e^4*x^4-48*B*b^4*d*e^3*x^4+192*A*a*b^3*e^4*x
^3+320*A*b^4*d*e^3*x^3-120*B*a^2*b^2*e^4*x^3-272*B*a*b^3*d*e^3*x^3-120*B*b^4*d^2
*e^2*x^3+48*A*a^2*b^2*e^4*x^2+480*A*a*b^3*d*e^3*x^2+240*A*b^4*d^2*e^2*x^2-30*B*a
^3*b*e^4*x^2-318*B*a^2*b^2*d*e^3*x^2-330*B*a*b^3*d^2*e^2*x^2-90*B*b^4*d^3*e*x^2-
8*A*a^3*b*e^4*x+120*A*a^2*b^2*d*e^3*x+360*A*a*b^3*d^2*e^2*x+40*A*b^4*d^3*e*x+5*B
*a^4*e^4*x-72*B*a^3*b*d*e^3*x-270*B*a^2*b^2*d^2*e^2*x-160*B*a*b^3*d^3*e*x-15*B*b
^4*d^4*x+3*A*a^4*e^4-20*A*a^3*b*d*e^3+90*A*a^2*b^2*d^2*e^2+60*A*a*b^3*d^3*e-5*A*
b^4*d^4+2*B*a^4*d*e^3-30*B*a^3*b*d^2*e^2-90*B*a^2*b^2*d^3*e-10*B*a*b^3*d^4)/(b*x
+a)^(3/2)/(e*x+d)^(5/2)/(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)^(5/2)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.59262, size = 1237, normalized size = 5.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(5/2)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

2/15*(3*A*a^4*e^4 - 5*(2*B*a*b^3 + A*b^4)*d^4 - 30*(3*B*a^2*b^2 - 2*A*a*b^3)*d^3
*e - 30*(B*a^3*b - 3*A*a^2*b^2)*d^2*e^2 + 2*(B*a^4 - 10*A*a^3*b)*d*e^3 - 16*(3*B
*b^4*d*e^3 + (5*B*a*b^3 - 8*A*b^4)*e^4)*x^4 - 8*(15*B*b^4*d^2*e^2 + 2*(17*B*a*b^
3 - 20*A*b^4)*d*e^3 + 3*(5*B*a^2*b^2 - 8*A*a*b^3)*e^4)*x^3 - 6*(15*B*b^4*d^3*e +
5*(11*B*a*b^3 - 8*A*b^4)*d^2*e^2 + (53*B*a^2*b^2 - 80*A*a*b^3)*d*e^3 + (5*B*a^3
*b - 8*A*a^2*b^2)*e^4)*x^2 - (15*B*b^4*d^4 + 40*(4*B*a*b^3 - A*b^4)*d^3*e + 90*(
3*B*a^2*b^2 - 4*A*a*b^3)*d^2*e^2 + 24*(3*B*a^3*b - 5*A*a^2*b^2)*d*e^3 - (5*B*a^4
- 8*A*a^3*b)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(a^2*b^5*d^8 - 5*a^3*b^4*d^7*e
+ 10*a^4*b^3*d^6*e^2 - 10*a^5*b^2*d^5*e^3 + 5*a^6*b*d^4*e^4 - a^7*d^3*e^5 + (b^
7*d^5*e^3 - 5*a*b^6*d^4*e^4 + 10*a^2*b^5*d^3*e^5 - 10*a^3*b^4*d^2*e^6 + 5*a^4*b^
3*d*e^7 - a^5*b^2*e^8)*x^5 + (3*b^7*d^6*e^2 - 13*a*b^6*d^5*e^3 + 20*a^2*b^5*d^4*
e^4 - 10*a^3*b^4*d^3*e^5 - 5*a^4*b^3*d^2*e^6 + 7*a^5*b^2*d*e^7 - 2*a^6*b*e^8)*x^
4 + (3*b^7*d^7*e - 9*a*b^6*d^6*e^2 + a^2*b^5*d^5*e^3 + 25*a^3*b^4*d^4*e^4 - 35*a
^4*b^3*d^3*e^5 + 17*a^5*b^2*d^2*e^6 - a^6*b*d*e^7 - a^7*e^8)*x^3 + (b^7*d^8 + a*
b^6*d^7*e - 17*a^2*b^5*d^6*e^2 + 35*a^3*b^4*d^5*e^3 - 25*a^4*b^3*d^4*e^4 - a^5*b
^2*d^3*e^5 + 9*a^6*b*d^2*e^6 - 3*a^7*d*e^7)*x^2 + (2*a*b^6*d^8 - 7*a^2*b^5*d^7*e
+ 5*a^3*b^4*d^6*e^2 + 10*a^4*b^3*d^5*e^3 - 20*a^5*b^2*d^4*e^4 + 13*a^6*b*d^3*e^
5 - 3*a^7*d^2*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)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.31823, 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)^(7/2)),x, algorithm="giac")

[Out]

Done