3.2244 \(\int \frac{A+B x}{\sqrt{a+b x} (d+e x)^{11/2}} \, dx\)

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 = 0.161543, 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, Rules used = {78, 45, 37} \[ \frac{32 b^3 \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{315 e \sqrt{d+e x} (b d-a e)^5}+\frac{16 b^2 \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac{4 b \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-9 a B e+8 A b e+b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 \sqrt{a+b x} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

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])

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{A+B x}{\sqrt{a+b x} (d+e x)^{11/2}} \, dx &=-\frac{2 (B d-A e) \sqrt{a+b x}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{(b B d+8 A b e-9 a B e) \int \frac{1}{\sqrt{a+b x} (d+e x)^{9/2}} \, dx}{9 e (b d-a e)}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{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}}+\frac{(2 b (b B d+8 A b e-9 a B e)) \int \frac{1}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx}{21 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{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}}+\frac{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}}+\frac{\left (8 b^2 (b B d+8 A b e-9 a B e)\right ) \int \frac{1}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx}{105 e (b d-a e)^3}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{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}}+\frac{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}}+\frac{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}}+\frac{\left (16 b^3 (b B d+8 A b e-9 a B e)\right ) \int \frac{1}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx}{315 e (b d-a e)^4}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{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}}+\frac{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}}+\frac{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}}+\frac{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}}\\ \end{align*}

Mathematica [A]  time = 0.292434, size = 134, normalized size = 0.53 \[ \frac{2 \sqrt{a+b x} \left (35 (B d-A e)-\frac{(d+e x) \left (2 b (d+e x) \left (4 b (d+e x) (-a e+3 b d+2 b e x)+3 (b d-a e)^2\right )+5 (b d-a e)^3\right ) (-9 a B e+8 A b e+b B d)}{(b d-a e)^4}\right )}{315 e (d+e x)^{9/2} (a e-b d)} \]

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

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Maple [B]  time = 0.009, size = 505, normalized size = 2. \begin{align*} -{\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}}}} \end{align*}

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^4-64*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(-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)/(e*x+d)^(11/2)/(b*x+a)^(1/2),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)/(e*x+d)^(11/2)/(b*x+a)^(1/2),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)/(e*x+d)**(11/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 3.47743, size = 1176, normalized size = 4.69 \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)/(e*x+d)^(11/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/322560*((2*(4*(b*x + a)*(2*(B*b^10*d*abs(b)*e^7 - 9*B*a*b^9*abs(b)*e^8 + 8*A*b^10*abs(b)*e^8)*(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^6 - 10*B*a*b^10*d*abs(b)*e^7 + 8*A*b^11*d*abs(b)*e^7 + 9*B*a^2*b^9*abs(b)*e^8 -
8*A*a*b^10*abs(b)*e^8)/(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^5 - 11*B*a*b^11*d^2*abs(b)*e^6 + 8*A*b^12*d^2*abs(b)
*e^6 + 19*B*a^2*b^10*d*abs(b)*e^7 - 16*A*a*b^11*d*abs(b)*e^7 - 9*B*a^3*b^9*abs(b)*e^8 + 8*A*a^2*b^10*abs(b)*e^
8)/(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) + 105*(B*b^13*d^4*abs(b)*e^4 - 12*B*a*b^12*d^3*abs(b)*e^5 + 8*A*b^13*d^3*abs(b)*e^5 + 30
*B*a^2*b^11*d^2*abs(b)*e^6 - 24*A*a*b^12*d^2*abs(b)*e^6 - 28*B*a^3*b^10*d*abs(b)*e^7 + 24*A*a^2*b^11*d*abs(b)*
e^7 + 9*B*a^4*b^9*abs(b)*e^8 - 8*A*a^3*b^10*abs(b)*e^8)/(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) - 315*(B*a*b^13*d^4*abs(b)*e^4 - A*
b^14*d^4*abs(b)*e^4 - 4*B*a^2*b^12*d^3*abs(b)*e^5 + 4*A*a*b^13*d^3*abs(b)*e^5 + 6*B*a^3*b^11*d^2*abs(b)*e^6 -
6*A*a^2*b^12*d^2*abs(b)*e^6 - 4*B*a^4*b^10*d*abs(b)*e^7 + 4*A*a^3*b^11*d*abs(b)*e^7 + B*a^5*b^9*abs(b)*e^8 - A
*a^4*b^10*abs(b)*e^8)/(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))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(9/2)