3.2227 \(\int \frac{A+B x}{\sqrt{a+b x} (d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=251 \[ \frac{32 b^3 \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{315 e \sqrt{d+e x} (b d-a e)^5}+\frac{16 b^2 \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac{4 b \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 \sqrt{a+b x} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*Sqrt[a + b*x])/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(b*B*d + 8
*A*b*e - 9*a*B*e)*Sqrt[a + b*x])/(63*e*(b*d - a*e)^2*(d + e*x)^(7/2)) + (4*b*(b*
B*d + 8*A*b*e - 9*a*B*e)*Sqrt[a + b*x])/(105*e*(b*d - a*e)^3*(d + e*x)^(5/2)) +
(16*b^2*(b*B*d + 8*A*b*e - 9*a*B*e)*Sqrt[a + b*x])/(315*e*(b*d - a*e)^4*(d + e*x
)^(3/2)) + (32*b^3*(b*B*d + 8*A*b*e - 9*a*B*e)*Sqrt[a + b*x])/(315*e*(b*d - a*e)
^5*Sqrt[d + e*x])

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Rubi [A]  time = 0.441817, antiderivative size = 251, 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 \[ \frac{32 b^3 \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{315 e \sqrt{d+e x} (b d-a e)^5}+\frac{16 b^2 \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac{4 b \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 \sqrt{a+b x} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(11/2)),x]

[Out]

(-2*(B*d - A*e)*Sqrt[a + b*x])/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(b*B*d + 8
*A*b*e - 9*a*B*e)*Sqrt[a + b*x])/(63*e*(b*d - a*e)^2*(d + e*x)^(7/2)) + (4*b*(b*
B*d + 8*A*b*e - 9*a*B*e)*Sqrt[a + b*x])/(105*e*(b*d - a*e)^3*(d + e*x)^(5/2)) +
(16*b^2*(b*B*d + 8*A*b*e - 9*a*B*e)*Sqrt[a + b*x])/(315*e*(b*d - a*e)^4*(d + e*x
)^(3/2)) + (32*b^3*(b*B*d + 8*A*b*e - 9*a*B*e)*Sqrt[a + b*x])/(315*e*(b*d - a*e)
^5*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 50.4658, size = 240, normalized size = 0.96 \[ - \frac{32 b^{3} \sqrt{a + b x} \left (8 A b e - 9 B a e + B b d\right )}{315 e \sqrt{d + e x} \left (a e - b d\right )^{5}} + \frac{16 b^{2} \sqrt{a + b x} \left (8 A b e - 9 B a e + B b d\right )}{315 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4}} - \frac{4 b \sqrt{a + b x} \left (8 A b e - 9 B a e + B b d\right )}{105 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3}} + \frac{2 \sqrt{a + b x} \left (8 A b e - 9 B a e + B b d\right )}{63 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2}} - \frac{2 \sqrt{a + b x} \left (A e - B d\right )}{9 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x+d)**(11/2)/(b*x+a)**(1/2),x)

[Out]

-32*b**3*sqrt(a + b*x)*(8*A*b*e - 9*B*a*e + B*b*d)/(315*e*sqrt(d + e*x)*(a*e - b
*d)**5) + 16*b**2*sqrt(a + b*x)*(8*A*b*e - 9*B*a*e + B*b*d)/(315*e*(d + e*x)**(3
/2)*(a*e - b*d)**4) - 4*b*sqrt(a + b*x)*(8*A*b*e - 9*B*a*e + B*b*d)/(105*e*(d +
e*x)**(5/2)*(a*e - b*d)**3) + 2*sqrt(a + b*x)*(8*A*b*e - 9*B*a*e + B*b*d)/(63*e*
(d + e*x)**(7/2)*(a*e - b*d)**2) - 2*sqrt(a + b*x)*(A*e - B*d)/(9*e*(d + e*x)**(
9/2)*(a*e - b*d))

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Mathematica [A]  time = 0.32935, size = 185, normalized size = 0.74 \[ \frac{2 \sqrt{a+b x} \left (-16 b^3 (d+e x)^4 (-9 a B e+8 A b e+b B d)+8 b^2 (d+e x)^3 (a e-b d) (-9 a B e+8 A b e+b B d)-6 b (d+e x)^2 (b d-a e)^2 (-9 a B e+8 A b e+b B d)+5 (d+e x) (a e-b d)^3 (-9 a B e+8 A b e+b B d)+35 (b d-a e)^4 (B d-A e)\right )}{315 e (d+e x)^{9/2} (a e-b d)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(11/2)),x]

[Out]

(2*Sqrt[a + b*x]*(35*(b*d - a*e)^4*(B*d - A*e) + 5*(-(b*d) + a*e)^3*(b*B*d + 8*A
*b*e - 9*a*B*e)*(d + e*x) - 6*b*(b*d - a*e)^2*(b*B*d + 8*A*b*e - 9*a*B*e)*(d + e
*x)^2 + 8*b^2*(-(b*d) + a*e)*(b*B*d + 8*A*b*e - 9*a*B*e)*(d + e*x)^3 - 16*b^3*(b
*B*d + 8*A*b*e - 9*a*B*e)*(d + e*x)^4))/(315*e*(-(b*d) + a*e)^5*(d + e*x)^(9/2))

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Maple [B]  time = 0.017, size = 505, normalized size = 2. \[ -{\frac{256\,A{b}^{4}{e}^{4}{x}^{4}-288\,Ba{b}^{3}{e}^{4}{x}^{4}+32\,B{b}^{4}d{e}^{3}{x}^{4}-128\,Aa{b}^{3}{e}^{4}{x}^{3}+1152\,A{b}^{4}d{e}^{3}{x}^{3}+144\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}-1312\,Ba{b}^{3}d{e}^{3}{x}^{3}+144\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+96\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}-576\,Aa{b}^{3}d{e}^{3}{x}^{2}+2016\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-108\,B{a}^{3}b{e}^{4}{x}^{2}+660\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}-2340\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}+252\,B{b}^{4}{d}^{3}e{x}^{2}-80\,A{a}^{3}b{e}^{4}x+432\,A{a}^{2}{b}^{2}d{e}^{3}x-1008\,Aa{b}^{3}{d}^{2}{e}^{2}x+1680\,A{b}^{4}{d}^{3}ex+90\,B{a}^{4}{e}^{4}x-496\,B{a}^{3}bd{e}^{3}x+1188\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x-2016\,Ba{b}^{3}{d}^{3}ex+210\,B{b}^{4}{d}^{4}x+70\,A{a}^{4}{e}^{4}-360\,A{a}^{3}bd{e}^{3}+756\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}-840\,Aa{b}^{3}{d}^{3}e+630\,A{b}^{4}{d}^{4}+20\,B{a}^{4}d{e}^{3}-108\,B{a}^{3}b{d}^{2}{e}^{2}+252\,B{a}^{2}{b}^{2}{d}^{3}e-420\,Ba{b}^{3}{d}^{4}}{315\,{a}^{5}{e}^{5}-1575\,{a}^{4}bd{e}^{4}+3150\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-3150\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+1575\,a{b}^{4}{d}^{4}e-315\,{b}^{5}{d}^{5}}\sqrt{bx+a} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^(11/2)/(b*x+a)^(1/2),x)

[Out]

-2/315*(b*x+a)^(1/2)*(128*A*b^4*e^4*x^4-144*B*a*b^3*e^4*x^4+16*B*b^4*d*e^3*x^4-6
4*A*a*b^3*e^4*x^3+576*A*b^4*d*e^3*x^3+72*B*a^2*b^2*e^4*x^3-656*B*a*b^3*d*e^3*x^3
+72*B*b^4*d^2*e^2*x^3+48*A*a^2*b^2*e^4*x^2-288*A*a*b^3*d*e^3*x^2+1008*A*b^4*d^2*
e^2*x^2-54*B*a^3*b*e^4*x^2+330*B*a^2*b^2*d*e^3*x^2-1170*B*a*b^3*d^2*e^2*x^2+126*
B*b^4*d^3*e*x^2-40*A*a^3*b*e^4*x+216*A*a^2*b^2*d*e^3*x-504*A*a*b^3*d^2*e^2*x+840
*A*b^4*d^3*e*x+45*B*a^4*e^4*x-248*B*a^3*b*d*e^3*x+594*B*a^2*b^2*d^2*e^2*x-1008*B
*a*b^3*d^3*e*x+105*B*b^4*d^4*x+35*A*a^4*e^4-180*A*a^3*b*d*e^3+378*A*a^2*b^2*d^2*
e^2-420*A*a*b^3*d^3*e+315*A*b^4*d^4+10*B*a^4*d*e^3-54*B*a^3*b*d^2*e^2+126*B*a^2*
b^2*d^3*e-210*B*a*b^3*d^4)/(e*x+d)^(9/2)/(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)/(sqrt(b*x + a)*(e*x + d)^(11/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.61028, size = 1133, normalized size = 4.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*(e*x + d)^(11/2)),x, algorithm="fricas")

[Out]

2/315*(35*A*a^4*e^4 - 105*(2*B*a*b^3 - 3*A*b^4)*d^4 + 42*(3*B*a^2*b^2 - 10*A*a*b
^3)*d^3*e - 54*(B*a^3*b - 7*A*a^2*b^2)*d^2*e^2 + 10*(B*a^4 - 18*A*a^3*b)*d*e^3 +
16*(B*b^4*d*e^3 - (9*B*a*b^3 - 8*A*b^4)*e^4)*x^4 + 8*(9*B*b^4*d^2*e^2 - 2*(41*B
*a*b^3 - 36*A*b^4)*d*e^3 + (9*B*a^2*b^2 - 8*A*a*b^3)*e^4)*x^3 + 6*(21*B*b^4*d^3*
e - 3*(65*B*a*b^3 - 56*A*b^4)*d^2*e^2 + (55*B*a^2*b^2 - 48*A*a*b^3)*d*e^3 - (9*B
*a^3*b - 8*A*a^2*b^2)*e^4)*x^2 + (105*B*b^4*d^4 - 168*(6*B*a*b^3 - 5*A*b^4)*d^3*
e + 18*(33*B*a^2*b^2 - 28*A*a*b^3)*d^2*e^2 - 8*(31*B*a^3*b - 27*A*a^2*b^2)*d*e^3
+ 5*(9*B*a^4 - 8*A*a^3*b)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^5*d^10 - 5*a*b
^4*d^9*e + 10*a^2*b^3*d^8*e^2 - 10*a^3*b^2*d^7*e^3 + 5*a^4*b*d^6*e^4 - a^5*d^5*e
^5 + (b^5*d^5*e^5 - 5*a*b^4*d^4*e^6 + 10*a^2*b^3*d^3*e^7 - 10*a^3*b^2*d^2*e^8 +
5*a^4*b*d*e^9 - a^5*e^10)*x^5 + 5*(b^5*d^6*e^4 - 5*a*b^4*d^5*e^5 + 10*a^2*b^3*d^
4*e^6 - 10*a^3*b^2*d^3*e^7 + 5*a^4*b*d^2*e^8 - a^5*d*e^9)*x^4 + 10*(b^5*d^7*e^3
- 5*a*b^4*d^6*e^4 + 10*a^2*b^3*d^5*e^5 - 10*a^3*b^2*d^4*e^6 + 5*a^4*b*d^3*e^7 -
a^5*d^2*e^8)*x^3 + 10*(b^5*d^8*e^2 - 5*a*b^4*d^7*e^3 + 10*a^2*b^3*d^6*e^4 - 10*a
^3*b^2*d^5*e^5 + 5*a^4*b*d^4*e^6 - a^5*d^3*e^7)*x^2 + 5*(b^5*d^9*e - 5*a*b^4*d^8
*e^2 + 10*a^2*b^3*d^7*e^3 - 10*a^3*b^2*d^6*e^4 + 5*a^4*b*d^5*e^5 - a^5*d^4*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)/(e*x+d)**(11/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.336046, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*(e*x + d)^(11/2)),x, algorithm="giac")

[Out]

Done