3.2208 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=198 \[ \frac{16 b^2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac{8 b (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{21 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^
(3/2))/(21*e*(b*d - a*e)^2*(d + e*x)^(7/2)) + (8*b*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(3/2))/(105*e*(b*d -
a*e)^3*(d + e*x)^(5/2)) + (16*b^2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(3/2))/(315*e*(b*d - a*e)^4*(d + e*x)^
(3/2))

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Rubi [A]  time = 0.116684, antiderivative size = 198, 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, Rules used = {78, 45, 37} \[ \frac{16 b^2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac{8 b (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{21 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^
(3/2))/(21*e*(b*d - a*e)^2*(d + e*x)^(7/2)) + (8*b*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(3/2))/(105*e*(b*d -
a*e)^3*(d + e*x)^(5/2)) + (16*b^2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(3/2))/(315*e*(b*d - a*e)^4*(d + e*x)^
(3/2))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{(b B d+2 A b e-3 a B e) \int \frac{\sqrt{a+b x}}{(d+e x)^{9/2}} \, dx}{3 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{21 e (b d-a e)^2 (d+e x)^{7/2}}+\frac{(4 b (b B d+2 A b e-3 a B e)) \int \frac{\sqrt{a+b x}}{(d+e x)^{7/2}} \, dx}{21 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{21 e (b d-a e)^2 (d+e x)^{7/2}}+\frac{8 b (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac{\left (8 b^2 (b B d+2 A b e-3 a B e)\right ) \int \frac{\sqrt{a+b x}}{(d+e x)^{5/2}} \, dx}{105 e (b d-a e)^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{21 e (b d-a e)^2 (d+e x)^{7/2}}+\frac{8 b (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac{16 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{315 e (b d-a e)^4 (d+e x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.283789, size = 113, normalized size = 0.57 \[ \frac{2 (a+b x)^{3/2} \left (105 (B d-A e)-\frac{3 (d+e x) \left (4 b (d+e x) (-3 a e+5 b d+2 b e x)+15 (b d-a e)^2\right ) (-3 a B e+2 A b e+b B d)}{(b d-a e)^3}\right )}{945 e (d+e x)^{9/2} (a e-b d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^(3/2)*(105*(B*d - A*e) - (3*(b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)*(15*(b*d - a*e)^2 + 4*b*(d + e*
x)*(5*b*d - 3*a*e + 2*b*e*x)))/(b*d - a*e)^3))/(945*e*(-(b*d) + a*e)*(d + e*x)^(9/2))

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Maple [A]  time = 0.008, size = 322, normalized size = 1.6 \begin{align*} -{\frac{-32\,A{b}^{3}{e}^{3}{x}^{3}+48\,Ba{b}^{2}{e}^{3}{x}^{3}-16\,B{b}^{3}d{e}^{2}{x}^{3}+48\,Aa{b}^{2}{e}^{3}{x}^{2}-144\,A{b}^{3}d{e}^{2}{x}^{2}-72\,B{a}^{2}b{e}^{3}{x}^{2}+240\,Ba{b}^{2}d{e}^{2}{x}^{2}-72\,B{b}^{3}{d}^{2}e{x}^{2}-60\,A{a}^{2}b{e}^{3}x+216\,Aa{b}^{2}d{e}^{2}x-252\,A{b}^{3}{d}^{2}ex+90\,B{a}^{3}{e}^{3}x-354\,B{a}^{2}bd{e}^{2}x+486\,Ba{b}^{2}{d}^{2}ex-126\,B{b}^{3}{d}^{3}x+70\,A{a}^{3}{e}^{3}-270\,A{a}^{2}bd{e}^{2}+378\,Aa{b}^{2}{d}^{2}e-210\,A{b}^{3}{d}^{3}+20\,B{a}^{3}d{e}^{2}-72\,B{a}^{2}b{d}^{2}e+84\,Ba{b}^{2}{d}^{3}}{315\,{e}^{4}{a}^{4}-1260\,b{e}^{3}d{a}^{3}+1890\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-1260\,a{b}^{3}{d}^{3}e+315\,{b}^{4}{d}^{4}} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(b*x+a)^(3/2)*(-16*A*b^3*e^3*x^3+24*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^2-72*A*b^3*d*e^2
*x^2-36*B*a^2*b*e^3*x^2+120*B*a*b^2*d*e^2*x^2-36*B*b^3*d^2*e*x^2-30*A*a^2*b*e^3*x+108*A*a*b^2*d*e^2*x-126*A*b^
3*d^2*e*x+45*B*a^3*e^3*x-177*B*a^2*b*d*e^2*x+243*B*a*b^2*d^2*e*x-63*B*b^3*d^3*x+35*A*a^3*e^3-135*A*a^2*b*d*e^2
+189*A*a*b^2*d^2*e-105*A*b^3*d^3+10*B*a^3*d*e^2-36*B*a^2*b*d^2*e+42*B*a*b^2*d^3)/(e*x+d)^(9/2)/(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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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Giac [B]  time = 2.64485, size = 857, normalized size = 4.33 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

-1/322560*((4*(b*x + a)*(2*(B*b^10*d*abs(b)*e^6 - 3*B*a*b^9*abs(b)*e^7 + 2*A*b^10*abs(b)*e^7)*(b*x + a)/(b^20*
d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^12 - 10*a^3*b^17*d^2*e^13 + 5*a^4*b^16*d*e^14 - a^5*b^15*e^15
) + 9*(B*b^11*d^2*abs(b)*e^5 - 4*B*a*b^10*d*abs(b)*e^6 + 2*A*b^11*d*abs(b)*e^6 + 3*B*a^2*b^9*abs(b)*e^7 - 2*A*
a*b^10*abs(b)*e^7)/(b^20*d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^12 - 10*a^3*b^17*d^2*e^13 + 5*a^4*b^
16*d*e^14 - a^5*b^15*e^15)) + 63*(B*b^12*d^3*abs(b)*e^4 - 5*B*a*b^11*d^2*abs(b)*e^5 + 2*A*b^12*d^2*abs(b)*e^5
+ 7*B*a^2*b^10*d*abs(b)*e^6 - 4*A*a*b^11*d*abs(b)*e^6 - 3*B*a^3*b^9*abs(b)*e^7 + 2*A*a^2*b^10*abs(b)*e^7)/(b^2
0*d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^12 - 10*a^3*b^17*d^2*e^13 + 5*a^4*b^16*d*e^14 - a^5*b^15*e^
15))*(b*x + a) - 105*(B*a*b^12*d^3*abs(b)*e^4 - A*b^13*d^3*abs(b)*e^4 - 3*B*a^2*b^11*d^2*abs(b)*e^5 + 3*A*a*b^
12*d^2*abs(b)*e^5 + 3*B*a^3*b^10*d*abs(b)*e^6 - 3*A*a^2*b^11*d*abs(b)*e^6 - B*a^4*b^9*abs(b)*e^7 + A*a^3*b^10*
abs(b)*e^7)/(b^20*d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^12 - 10*a^3*b^17*d^2*e^13 + 5*a^4*b^16*d*e^
14 - a^5*b^15*e^15))*(b*x + a)^(3/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(9/2)