3.2221 \(\int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=255 \[ \frac{32 b^3 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac{4 b (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{429 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a +
b*x)^(5/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2)) + (4*b*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(429
*e*(b*d - a*e)^3*(d + e*x)^(9/2)) + (16*b^2*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(3003*e*(b*d - a*e
)^4*(d + e*x)^(7/2)) + (32*b^3*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(15015*e*(b*d - a*e)^5*(d + e*x
)^(5/2))

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Rubi [A]  time = 0.163884, 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} \[ \frac{32 b^3 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac{4 b (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{429 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a +
b*x)^(5/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2)) + (4*b*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(429
*e*(b*d - a*e)^3*(d + e*x)^(9/2)) + (16*b^2*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(3003*e*(b*d - a*e
)^4*(d + e*x)^(7/2)) + (32*b^3*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(15015*e*(b*d - a*e)^5*(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)^{15/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{(5 b B d+8 A b e-13 a B e) \int \frac{(a+b x)^{3/2}}{(d+e x)^{13/2}} \, dx}{13 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{(6 b (5 b B d+8 A b e-13 a B e)) \int \frac{(a+b x)^{3/2}}{(d+e x)^{11/2}} \, dx}{143 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{4 b (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{9/2}}+\frac{\left (8 b^2 (5 b B d+8 A b e-13 a B e)\right ) \int \frac{(a+b x)^{3/2}}{(d+e x)^{9/2}} \, dx}{429 e (b d-a e)^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{4 b (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{9/2}}+\frac{16 b^2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac{\left (16 b^3 (5 b B d+8 A b e-13 a B e)\right ) \int \frac{(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{3003 e (b d-a e)^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{4 b (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{9/2}}+\frac{16 b^2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac{32 b^3 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.380652, size = 135, normalized size = 0.53 \[ \frac{2 (a+b x)^{5/2} \left (1155 (B d-A e)-\frac{(d+e x) \left (2 b (d+e x) \left (4 b (d+e x) (-5 a e+7 b d+2 b e x)+35 (b d-a e)^2\right )+105 (b d-a e)^3\right ) (-13 a B e+8 A b e+5 b B d)}{(b d-a e)^4}\right )}{15015 e (d+e x)^{13/2} (a e-b d)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(2*(a + b*x)^(5/2)*(1155*(B*d - A*e) - ((5*b*B*d + 8*A*b*e - 13*a*B*e)*(d + e*x)*(105*(b*d - a*e)^3 + 2*b*(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)^4))/(15015*e*(-(b*d) + a*e)*(
d + e*x)^(13/2))

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Maple [B]  time = 0.01, size = 505, normalized size = 2. \begin{align*} -{\frac{256\,A{b}^{4}{e}^{4}{x}^{4}-416\,Ba{b}^{3}{e}^{4}{x}^{4}+160\,B{b}^{4}d{e}^{3}{x}^{4}-640\,Aa{b}^{3}{e}^{4}{x}^{3}+1664\,A{b}^{4}d{e}^{3}{x}^{3}+1040\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}-3104\,Ba{b}^{3}d{e}^{3}{x}^{3}+1040\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+1120\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}-4160\,Aa{b}^{3}d{e}^{3}{x}^{2}+4576\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-1820\,B{a}^{3}b{e}^{4}{x}^{2}+7460\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}-10036\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}+2860\,B{b}^{4}{d}^{3}e{x}^{2}-1680\,A{a}^{3}b{e}^{4}x+7280\,A{a}^{2}{b}^{2}d{e}^{3}x-11440\,Aa{b}^{3}{d}^{2}{e}^{2}x+6864\,A{b}^{4}{d}^{3}ex+2730\,B{a}^{4}{e}^{4}x-12880\,B{a}^{3}bd{e}^{3}x+23140\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x-18304\,Ba{b}^{3}{d}^{3}ex+4290\,B{b}^{4}{d}^{4}x+2310\,A{a}^{4}{e}^{4}-10920\,A{a}^{3}bd{e}^{3}+20020\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}-17160\,Aa{b}^{3}{d}^{3}e+6006\,A{b}^{4}{d}^{4}+420\,B{a}^{4}d{e}^{3}-1820\,B{a}^{3}b{d}^{2}{e}^{2}+2860\,B{a}^{2}{b}^{2}{d}^{3}e-1716\,Ba{b}^{3}{d}^{4}}{15015\,{a}^{5}{e}^{5}-75075\,{a}^{4}bd{e}^{4}+150150\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-150150\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+75075\,a{b}^{4}{d}^{4}e-15015\,{b}^{5}{d}^{5}} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{13}{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)^(15/2),x)

[Out]

-2/15015*(b*x+a)^(5/2)*(128*A*b^4*e^4*x^4-208*B*a*b^3*e^4*x^4+80*B*b^4*d*e^3*x^4-320*A*a*b^3*e^4*x^3+832*A*b^4
*d*e^3*x^3+520*B*a^2*b^2*e^4*x^3-1552*B*a*b^3*d*e^3*x^3+520*B*b^4*d^2*e^2*x^3+560*A*a^2*b^2*e^4*x^2-2080*A*a*b
^3*d*e^3*x^2+2288*A*b^4*d^2*e^2*x^2-910*B*a^3*b*e^4*x^2+3730*B*a^2*b^2*d*e^3*x^2-5018*B*a*b^3*d^2*e^2*x^2+1430
*B*b^4*d^3*e*x^2-840*A*a^3*b*e^4*x+3640*A*a^2*b^2*d*e^3*x-5720*A*a*b^3*d^2*e^2*x+3432*A*b^4*d^3*e*x+1365*B*a^4
*e^4*x-6440*B*a^3*b*d*e^3*x+11570*B*a^2*b^2*d^2*e^2*x-9152*B*a*b^3*d^3*e*x+2145*B*b^4*d^4*x+1155*A*a^4*e^4-546
0*A*a^3*b*d*e^3+10010*A*a^2*b^2*d^2*e^2-8580*A*a*b^3*d^3*e+3003*A*b^4*d^4+210*B*a^4*d*e^3-910*B*a^3*b*d^2*e^2+
1430*B*a^2*b^2*d^3*e-858*B*a*b^3*d^4)/(e*x+d)^(13/2)/(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)^(3/2)*(B*x+A)/(e*x+d)^(15/2),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)^(15/2),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)**(15/2),x)

[Out]

Timed out

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Giac [B]  time = 4.01254, size = 1567, normalized size = 6.15 \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)^(15/2),x, algorithm="giac")

[Out]

-1/11070259200*((2*(4*(b*x + a)*(2*(5*B*b^15*d^2*abs(b)*e^9 - 18*B*a*b^14*d*abs(b)*e^10 + 8*A*b^15*d*abs(b)*e^
10 + 13*B*a^2*b^13*abs(b)*e^11 - 8*A*a*b^14*abs(b)*e^11)*(b*x + a)/(b^28*d^7*e^14 - 7*a*b^27*d^6*e^15 + 21*a^2
*b^26*d^5*e^16 - 35*a^3*b^25*d^4*e^17 + 35*a^4*b^24*d^3*e^18 - 21*a^5*b^23*d^2*e^19 + 7*a^6*b^22*d*e^20 - a^7*
b^21*e^21) + 13*(5*B*b^16*d^3*abs(b)*e^8 - 23*B*a*b^15*d^2*abs(b)*e^9 + 8*A*b^16*d^2*abs(b)*e^9 + 31*B*a^2*b^1
4*d*abs(b)*e^10 - 16*A*a*b^15*d*abs(b)*e^10 - 13*B*a^3*b^13*abs(b)*e^11 + 8*A*a^2*b^14*abs(b)*e^11)/(b^28*d^7*
e^14 - 7*a*b^27*d^6*e^15 + 21*a^2*b^26*d^5*e^16 - 35*a^3*b^25*d^4*e^17 + 35*a^4*b^24*d^3*e^18 - 21*a^5*b^23*d^
2*e^19 + 7*a^6*b^22*d*e^20 - a^7*b^21*e^21)) + 143*(5*B*b^17*d^4*abs(b)*e^7 - 28*B*a*b^16*d^3*abs(b)*e^8 + 8*A
*b^17*d^3*abs(b)*e^8 + 54*B*a^2*b^15*d^2*abs(b)*e^9 - 24*A*a*b^16*d^2*abs(b)*e^9 - 44*B*a^3*b^14*d*abs(b)*e^10
+ 24*A*a^2*b^15*d*abs(b)*e^10 + 13*B*a^4*b^13*abs(b)*e^11 - 8*A*a^3*b^14*abs(b)*e^11)/(b^28*d^7*e^14 - 7*a*b^
27*d^6*e^15 + 21*a^2*b^26*d^5*e^16 - 35*a^3*b^25*d^4*e^17 + 35*a^4*b^24*d^3*e^18 - 21*a^5*b^23*d^2*e^19 + 7*a^
6*b^22*d*e^20 - a^7*b^21*e^21))*(b*x + a) + 429*(5*B*b^18*d^5*abs(b)*e^6 - 33*B*a*b^17*d^4*abs(b)*e^7 + 8*A*b^
18*d^4*abs(b)*e^7 + 82*B*a^2*b^16*d^3*abs(b)*e^8 - 32*A*a*b^17*d^3*abs(b)*e^8 - 98*B*a^3*b^15*d^2*abs(b)*e^9 +
48*A*a^2*b^16*d^2*abs(b)*e^9 + 57*B*a^4*b^14*d*abs(b)*e^10 - 32*A*a^3*b^15*d*abs(b)*e^10 - 13*B*a^5*b^13*abs(
b)*e^11 + 8*A*a^4*b^14*abs(b)*e^11)/(b^28*d^7*e^14 - 7*a*b^27*d^6*e^15 + 21*a^2*b^26*d^5*e^16 - 35*a^3*b^25*d^
4*e^17 + 35*a^4*b^24*d^3*e^18 - 21*a^5*b^23*d^2*e^19 + 7*a^6*b^22*d*e^20 - a^7*b^21*e^21))*(b*x + a) - 3003*(B
*a*b^18*d^5*abs(b)*e^6 - A*b^19*d^5*abs(b)*e^6 - 5*B*a^2*b^17*d^4*abs(b)*e^7 + 5*A*a*b^18*d^4*abs(b)*e^7 + 10*
B*a^3*b^16*d^3*abs(b)*e^8 - 10*A*a^2*b^17*d^3*abs(b)*e^8 - 10*B*a^4*b^15*d^2*abs(b)*e^9 + 10*A*a^3*b^16*d^2*ab
s(b)*e^9 + 5*B*a^5*b^14*d*abs(b)*e^10 - 5*A*a^4*b^15*d*abs(b)*e^10 - B*a^6*b^13*abs(b)*e^11 + A*a^5*b^14*abs(b
)*e^11)/(b^28*d^7*e^14 - 7*a*b^27*d^6*e^15 + 21*a^2*b^26*d^5*e^16 - 35*a^3*b^25*d^4*e^17 + 35*a^4*b^24*d^3*e^1
8 - 21*a^5*b^23*d^2*e^19 + 7*a^6*b^22*d*e^20 - a^7*b^21*e^21))*(b*x + a)^(5/2)/(b^2*d + (b*x + a)*b*e - a*b*e)
^(13/2)