3.2220 \(\int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=201 \[ \frac{16 b^2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{3465 e (d+e x)^{5/2} (b d-a e)^4}+\frac{8 b (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a +
b*x)^(5/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) + (8*b*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(693*e
*(b*d - a*e)^3*(d + e*x)^(7/2)) + (16*b^2*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(3465*e*(b*d - a*e)^
4*(d + e*x)^(5/2))

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Rubi [A]  time = 0.122331, antiderivative size = 201, 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} \[ \frac{16 b^2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{3465 e (d+e x)^{5/2} (b d-a e)^4}+\frac{8 b (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a +
b*x)^(5/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) + (8*b*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(693*e
*(b*d - a*e)^3*(d + e*x)^(7/2)) + (16*b^2*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(3465*e*(b*d - a*e)^
4*(d + e*x)^(5/2))

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{(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{(5 b B d+6 A b e-11 a B e) \int \frac{(a+b x)^{3/2}}{(d+e x)^{11/2}} \, dx}{11 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac{(4 b (5 b B d+6 A b e-11 a B e)) \int \frac{(a+b x)^{3/2}}{(d+e x)^{9/2}} \, dx}{99 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac{8 b (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}+\frac{\left (8 b^2 (5 b B d+6 A b e-11 a B e)\right ) \int \frac{(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{693 e (b d-a e)^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac{8 b (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}+\frac{16 b^2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{3465 e (b d-a e)^4 (d+e x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.314565, size = 114, normalized size = 0.57 \[ \frac{2 (a+b x)^{5/2} \left (315 (B d-A e)-\frac{(d+e x) \left (4 b (d+e x) (-5 a e+7 b d+2 b e x)+35 (b d-a e)^2\right ) (-11 a B e+6 A b e+5 b B d)}{(b d-a e)^3}\right )}{3465 e (d+e x)^{11/2} (a e-b d)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(2*(a + b*x)^(5/2)*(315*(B*d - A*e) - ((5*b*B*d + 6*A*b*e - 11*a*B*e)*(d + e*x)*(35*(b*d - a*e)^2 + 4*b*(d + e
*x)*(7*b*d - 5*a*e + 2*b*e*x)))/(b*d - a*e)^3))/(3465*e*(-(b*d) + a*e)*(d + e*x)^(11/2))

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Maple [A]  time = 0.008, size = 322, normalized size = 1.6 \begin{align*} -{\frac{-96\,A{b}^{3}{e}^{3}{x}^{3}+176\,Ba{b}^{2}{e}^{3}{x}^{3}-80\,B{b}^{3}d{e}^{2}{x}^{3}+240\,Aa{b}^{2}{e}^{3}{x}^{2}-528\,A{b}^{3}d{e}^{2}{x}^{2}-440\,B{a}^{2}b{e}^{3}{x}^{2}+1168\,Ba{b}^{2}d{e}^{2}{x}^{2}-440\,B{b}^{3}{d}^{2}e{x}^{2}-420\,A{a}^{2}b{e}^{3}x+1320\,Aa{b}^{2}d{e}^{2}x-1188\,A{b}^{3}{d}^{2}ex+770\,B{a}^{3}{e}^{3}x-2770\,B{a}^{2}bd{e}^{2}x+3278\,Ba{b}^{2}{d}^{2}ex-990\,B{b}^{3}{d}^{3}x+630\,A{a}^{3}{e}^{3}-2310\,A{a}^{2}bd{e}^{2}+2970\,Aa{b}^{2}{d}^{2}e-1386\,A{b}^{3}{d}^{3}+140\,B{a}^{3}d{e}^{2}-440\,B{a}^{2}b{d}^{2}e+396\,Ba{b}^{2}{d}^{3}}{3465\,{e}^{4}{a}^{4}-13860\,b{e}^{3}d{a}^{3}+20790\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-13860\,a{b}^{3}{d}^{3}e+3465\,{b}^{4}{d}^{4}} \left ( bx+a \right ) ^{{\frac{5}{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)^(3/2)*(B*x+A)/(e*x+d)^(13/2),x)

[Out]

-2/3465*(b*x+a)^(5/2)*(-48*A*b^3*e^3*x^3+88*B*a*b^2*e^3*x^3-40*B*b^3*d*e^2*x^3+120*A*a*b^2*e^3*x^2-264*A*b^3*d
*e^2*x^2-220*B*a^2*b*e^3*x^2+584*B*a*b^2*d*e^2*x^2-220*B*b^3*d^2*e*x^2-210*A*a^2*b*e^3*x+660*A*a*b^2*d*e^2*x-5
94*A*b^3*d^2*e*x+385*B*a^3*e^3*x-1385*B*a^2*b*d*e^2*x+1639*B*a*b^2*d^2*e*x-495*B*b^3*d^3*x+315*A*a^3*e^3-1155*
A*a^2*b*d*e^2+1485*A*a*b^2*d^2*e-693*A*b^3*d^3+70*B*a^3*d*e^2-220*B*a^2*b*d^2*e+198*B*a*b^2*d^3)/(e*x+d)^(11/2
)/(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)^(3/2)*(B*x+A)/(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)^(3/2)*(B*x+A)/(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)**(3/2)*(B*x+A)/(e*x+d)**(13/2),x)

[Out]

Timed out

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Giac [B]  time = 3.25536, size = 1094, normalized size = 5.44 \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)^(3/2)*(B*x+A)/(e*x+d)^(13/2),x, algorithm="giac")

[Out]

-1/14192640*((4*(b*x + a)*(2*(5*B*b^13*d^2*abs(b)*e^7 - 16*B*a*b^12*d*abs(b)*e^8 + 6*A*b^13*d*abs(b)*e^8 + 11*
B*a^2*b^11*abs(b)*e^9 - 6*A*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*(5*B*b^14*d^3*abs
(b)*e^6 - 21*B*a*b^13*d^2*abs(b)*e^7 + 6*A*b^14*d^2*abs(b)*e^7 + 27*B*a^2*b^12*d*abs(b)*e^8 - 12*A*a*b^13*d*ab
s(b)*e^8 - 11*B*a^3*b^11*abs(b)*e^9 + 6*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^2
2*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*(5*B*b^15*
d^4*abs(b)*e^5 - 26*B*a*b^14*d^3*abs(b)*e^6 + 6*A*b^15*d^3*abs(b)*e^6 + 48*B*a^2*b^13*d^2*abs(b)*e^7 - 18*A*a*
b^14*d^2*abs(b)*e^7 - 38*B*a^3*b^12*d*abs(b)*e^8 + 18*A*a^2*b^13*d*abs(b)*e^8 + 11*B*a^4*b^11*abs(b)*e^9 - 6*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) - 693*(B*a*b^15*d^4*abs(b)*e^5 - A*b^16*d^4*ab
s(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^1
4*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)^(5/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(11/2)