3.2209 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=255 \[ \frac{32 b^3 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{3465 e (d+e x)^{3/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{1155 e (d+e x)^{5/2} (b d-a e)^4}+\frac{4 b (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{231 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a +
b*x)^(3/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) + (4*b*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(231*e
*(b*d - a*e)^3*(d + e*x)^(7/2)) + (16*b^2*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(1155*e*(b*d - a*e)^
4*(d + e*x)^(5/2)) + (32*b^3*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(3465*e*(b*d - a*e)^5*(d + e*x)^(
3/2))

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Rubi [A]  time = 0.168754, antiderivative size = 255, 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} \[ \frac{32 b^3 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{3465 e (d+e x)^{3/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{1155 e (d+e x)^{5/2} (b d-a e)^4}+\frac{4 b (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{231 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a +
b*x)^(3/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) + (4*b*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(231*e
*(b*d - a*e)^3*(d + e*x)^(7/2)) + (16*b^2*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(1155*e*(b*d - a*e)^
4*(d + e*x)^(5/2)) + (32*b^3*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(3465*e*(b*d - a*e)^5*(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)^{13/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{(3 b B d+8 A b e-11 a B e) \int \frac{\sqrt{a+b x}}{(d+e x)^{11/2}} \, dx}{11 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac{(2 b (3 b B d+8 A b e-11 a B e)) \int \frac{\sqrt{a+b x}}{(d+e x)^{9/2}} \, dx}{33 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac{4 b (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{231 e (b d-a e)^3 (d+e x)^{7/2}}+\frac{\left (8 b^2 (3 b B d+8 A b e-11 a B e)\right ) \int \frac{\sqrt{a+b x}}{(d+e x)^{7/2}} \, dx}{231 e (b d-a e)^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac{4 b (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{231 e (b d-a e)^3 (d+e x)^{7/2}}+\frac{16 b^2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{1155 e (b d-a e)^4 (d+e x)^{5/2}}+\frac{\left (16 b^3 (3 b B d+8 A b e-11 a B e)\right ) \int \frac{\sqrt{a+b x}}{(d+e x)^{5/2}} \, dx}{1155 e (b d-a e)^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac{4 b (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{231 e (b d-a e)^3 (d+e x)^{7/2}}+\frac{16 b^2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{1155 e (b d-a e)^4 (d+e x)^{5/2}}+\frac{32 b^3 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{3465 e (b d-a e)^5 (d+e x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.35348, size = 135, normalized size = 0.53 \[ \frac{2 (a+b x)^{3/2} \left (315 (B d-A e)-\frac{(d+e x) \left (2 b (d+e x) \left (4 b (d+e x) (-3 a e+5 b d+2 b e x)+15 (b d-a e)^2\right )+35 (b d-a e)^3\right ) (-11 a B e+8 A b e+3 b B d)}{(b d-a e)^4}\right )}{3465 e (d+e x)^{11/2} (a e-b d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(2*(a + b*x)^(3/2)*(315*(B*d - A*e) - ((3*b*B*d + 8*A*b*e - 11*a*B*e)*(d + e*x)*(35*(b*d - a*e)^3 + 2*b*(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)^4))/(3465*e*(-(b*d) + a*e)*(d +
e*x)^(11/2))

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Maple [B]  time = 0.012, size = 505, normalized size = 2. \begin{align*} -{\frac{256\,A{b}^{4}{e}^{4}{x}^{4}-352\,Ba{b}^{3}{e}^{4}{x}^{4}+96\,B{b}^{4}d{e}^{3}{x}^{4}-384\,Aa{b}^{3}{e}^{4}{x}^{3}+1408\,A{b}^{4}d{e}^{3}{x}^{3}+528\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}-2080\,Ba{b}^{3}d{e}^{3}{x}^{3}+528\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+480\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}-2112\,Aa{b}^{3}d{e}^{3}{x}^{2}+3168\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-660\,B{a}^{3}b{e}^{4}{x}^{2}+3084\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}-5148\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}+1188\,B{b}^{4}{d}^{3}e{x}^{2}-560\,A{a}^{3}b{e}^{4}x+2640\,A{a}^{2}{b}^{2}d{e}^{3}x-4752\,Aa{b}^{3}{d}^{2}{e}^{2}x+3696\,A{b}^{4}{d}^{3}ex+770\,B{a}^{4}{e}^{4}x-3840\,B{a}^{3}bd{e}^{3}x+7524\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x-6864\,Ba{b}^{3}{d}^{3}ex+1386\,B{b}^{4}{d}^{4}x+630\,A{a}^{4}{e}^{4}-3080\,A{a}^{3}bd{e}^{3}+5940\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}-5544\,Aa{b}^{3}{d}^{3}e+2310\,A{b}^{4}{d}^{4}+140\,B{a}^{4}d{e}^{3}-660\,B{a}^{3}b{d}^{2}{e}^{2}+1188\,B{a}^{2}{b}^{2}{d}^{3}e-924\,Ba{b}^{3}{d}^{4}}{3465\,{a}^{5}{e}^{5}-17325\,{a}^{4}bd{e}^{4}+34650\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-34650\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+17325\,a{b}^{4}{d}^{4}e-3465\,{b}^{5}{d}^{5}} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{11}{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)^(13/2),x)

[Out]

-2/3465*(b*x+a)^(3/2)*(128*A*b^4*e^4*x^4-176*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+704*A*b^4*
d*e^3*x^3+264*B*a^2*b^2*e^4*x^3-1040*B*a*b^3*d*e^3*x^3+264*B*b^4*d^2*e^2*x^3+240*A*a^2*b^2*e^4*x^2-1056*A*a*b^
3*d*e^3*x^2+1584*A*b^4*d^2*e^2*x^2-330*B*a^3*b*e^4*x^2+1542*B*a^2*b^2*d*e^3*x^2-2574*B*a*b^3*d^2*e^2*x^2+594*B
*b^4*d^3*e*x^2-280*A*a^3*b*e^4*x+1320*A*a^2*b^2*d*e^3*x-2376*A*a*b^3*d^2*e^2*x+1848*A*b^4*d^3*e*x+385*B*a^4*e^
4*x-1920*B*a^3*b*d*e^3*x+3762*B*a^2*b^2*d^2*e^2*x-3432*B*a*b^3*d^3*e*x+693*B*b^4*d^4*x+315*A*a^4*e^4-1540*A*a^
3*b*d*e^3+2970*A*a^2*b^2*d^2*e^2-2772*A*a*b^3*d^3*e+1155*A*b^4*d^4+70*B*a^4*d*e^3-330*B*a^3*b*d^2*e^2+594*B*a^
2*b^2*d^3*e-462*B*a*b^3*d^4)/(e*x+d)^(11/2)/(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)*(b*x+a)^(1/2)/(e*x+d)^(13/2),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)^(13/2),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)**(13/2),x)

[Out]

Timed out

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Giac [B]  time = 3.54078, size = 1262, normalized size = 4.95 \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)^(13/2),x, algorithm="giac")

[Out]

-1/14192640*((2*(4*(b*x + a)*(2*(3*B*b^12*d*abs(b)*e^8 - 11*B*a*b^11*abs(b)*e^9 + 8*A*b^12*abs(b)*e^9)*(b*x +
a)/(b^24*d^6*e^12 - 6*a*b^23*d^5*e^13 + 15*a^2*b^22*d^4*e^14 - 20*a^3*b^21*d^3*e^15 + 15*a^4*b^20*d^2*e^16 - 6
*a^5*b^19*d*e^17 + a^6*b^18*e^18) + 11*(3*B*b^13*d^2*abs(b)*e^7 - 14*B*a*b^12*d*abs(b)*e^8 + 8*A*b^13*d*abs(b)
*e^8 + 11*B*a^2*b^11*abs(b)*e^9 - 8*A*a*b^12*abs(b)*e^9)/(b^24*d^6*e^12 - 6*a*b^23*d^5*e^13 + 15*a^2*b^22*d^4*
e^14 - 20*a^3*b^21*d^3*e^15 + 15*a^4*b^20*d^2*e^16 - 6*a^5*b^19*d*e^17 + a^6*b^18*e^18)) + 99*(3*B*b^14*d^3*ab
s(b)*e^6 - 17*B*a*b^13*d^2*abs(b)*e^7 + 8*A*b^14*d^2*abs(b)*e^7 + 25*B*a^2*b^12*d*abs(b)*e^8 - 16*A*a*b^13*d*a
bs(b)*e^8 - 11*B*a^3*b^11*abs(b)*e^9 + 8*A*a^2*b^12*abs(b)*e^9)/(b^24*d^6*e^12 - 6*a*b^23*d^5*e^13 + 15*a^2*b^
22*d^4*e^14 - 20*a^3*b^21*d^3*e^15 + 15*a^4*b^20*d^2*e^16 - 6*a^5*b^19*d*e^17 + a^6*b^18*e^18))*(b*x + a) + 23
1*(3*B*b^15*d^4*abs(b)*e^5 - 20*B*a*b^14*d^3*abs(b)*e^6 + 8*A*b^15*d^3*abs(b)*e^6 + 42*B*a^2*b^13*d^2*abs(b)*e
^7 - 24*A*a*b^14*d^2*abs(b)*e^7 - 36*B*a^3*b^12*d*abs(b)*e^8 + 24*A*a^2*b^13*d*abs(b)*e^8 + 11*B*a^4*b^11*abs(
b)*e^9 - 8*A*a^3*b^12*abs(b)*e^9)/(b^24*d^6*e^12 - 6*a*b^23*d^5*e^13 + 15*a^2*b^22*d^4*e^14 - 20*a^3*b^21*d^3*
e^15 + 15*a^4*b^20*d^2*e^16 - 6*a^5*b^19*d*e^17 + a^6*b^18*e^18))*(b*x + a) - 1155*(B*a*b^15*d^4*abs(b)*e^5 -
A*b^16*d^4*abs(b)*e^5 - 4*B*a^2*b^14*d^3*abs(b)*e^6 + 4*A*a*b^15*d^3*abs(b)*e^6 + 6*B*a^3*b^13*d^2*abs(b)*e^7
- 6*A*a^2*b^14*d^2*abs(b)*e^7 - 4*B*a^4*b^12*d*abs(b)*e^8 + 4*A*a^3*b^13*d*abs(b)*e^8 + B*a^5*b^11*abs(b)*e^9
- A*a^4*b^12*abs(b)*e^9)/(b^24*d^6*e^12 - 6*a*b^23*d^5*e^13 + 15*a^2*b^22*d^4*e^14 - 20*a^3*b^21*d^3*e^15 + 15
*a^4*b^20*d^2*e^16 - 6*a^5*b^19*d*e^17 + a^6*b^18*e^18))*(b*x + a)^(3/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(11/2
)