3.2234 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx\)

Optimal. Leaf size=255 \[ \frac{32 b^3 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{45045 e (d+e x)^{7/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{6435 e (d+e x)^{9/2} (b d-a e)^4}+\frac{4 b (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{715 e (d+e x)^{11/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{195 e (d+e x)^{13/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + (2*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a +
b*x)^(7/2))/(195*e*(b*d - a*e)^2*(d + e*x)^(13/2)) + (4*b*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(715
*e*(b*d - a*e)^3*(d + e*x)^(11/2)) + (16*b^2*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(6435*e*(b*d - a*
e)^4*(d + e*x)^(9/2)) + (32*b^3*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(45045*e*(b*d - a*e)^5*(d + e*
x)^(7/2))

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Rubi [A]  time = 0.163508, 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} \[ \frac{32 b^3 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{45045 e (d+e x)^{7/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{6435 e (d+e x)^{9/2} (b d-a e)^4}+\frac{4 b (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{715 e (d+e x)^{11/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{195 e (d+e x)^{13/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + (2*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a +
b*x)^(7/2))/(195*e*(b*d - a*e)^2*(d + e*x)^(13/2)) + (4*b*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(715
*e*(b*d - a*e)^3*(d + e*x)^(11/2)) + (16*b^2*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(6435*e*(b*d - a*
e)^4*(d + e*x)^(9/2)) + (32*b^3*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(45045*e*(b*d - a*e)^5*(d + e*
x)^(7/2))

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)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac{(7 b B d+8 A b e-15 a B e) \int \frac{(a+b x)^{5/2}}{(d+e x)^{15/2}} \, dx}{15 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac{2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{195 e (b d-a e)^2 (d+e x)^{13/2}}+\frac{(2 b (7 b B d+8 A b e-15 a B e)) \int \frac{(a+b x)^{5/2}}{(d+e x)^{13/2}} \, dx}{65 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac{2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{195 e (b d-a e)^2 (d+e x)^{13/2}}+\frac{4 b (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{715 e (b d-a e)^3 (d+e x)^{11/2}}+\frac{\left (8 b^2 (7 b B d+8 A b e-15 a B e)\right ) \int \frac{(a+b x)^{5/2}}{(d+e x)^{11/2}} \, dx}{715 e (b d-a e)^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac{2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{195 e (b d-a e)^2 (d+e x)^{13/2}}+\frac{4 b (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{715 e (b d-a e)^3 (d+e x)^{11/2}}+\frac{16 b^2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{6435 e (b d-a e)^4 (d+e x)^{9/2}}+\frac{\left (16 b^3 (7 b B d+8 A b e-15 a B e)\right ) \int \frac{(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{6435 e (b d-a e)^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac{2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{195 e (b d-a e)^2 (d+e x)^{13/2}}+\frac{4 b (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{715 e (b d-a e)^3 (d+e x)^{11/2}}+\frac{16 b^2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{6435 e (b d-a e)^4 (d+e x)^{9/2}}+\frac{32 b^3 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{45045 e (b d-a e)^5 (d+e x)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.421758, size = 135, normalized size = 0.53 \[ \frac{2 (a+b x)^{7/2} \left (3003 (B d-A e)-\frac{(d+e x) \left (2 b (d+e x) \left (4 b (d+e x) (-7 a e+9 b d+2 b e x)+63 (b d-a e)^2\right )+231 (b d-a e)^3\right ) (-15 a B e+8 A b e+7 b B d)}{(b d-a e)^4}\right )}{45045 e (d+e x)^{15/2} (a e-b d)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(2*(a + b*x)^(7/2)*(3003*(B*d - A*e) - ((7*b*B*d + 8*A*b*e - 15*a*B*e)*(d + e*x)*(231*(b*d - a*e)^3 + 2*b*(d +
e*x)*(63*(b*d - a*e)^2 + 4*b*(d + e*x)*(9*b*d - 7*a*e + 2*b*e*x))))/(b*d - a*e)^4))/(45045*e*(-(b*d) + a*e)*(
d + e*x)^(15/2))

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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}-480\,Ba{b}^{3}{e}^{4}{x}^{4}+224\,B{b}^{4}d{e}^{3}{x}^{4}-896\,Aa{b}^{3}{e}^{4}{x}^{3}+1920\,A{b}^{4}d{e}^{3}{x}^{3}+1680\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}-4384\,Ba{b}^{3}d{e}^{3}{x}^{3}+1680\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+2016\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}-6720\,Aa{b}^{3}d{e}^{3}{x}^{2}+6240\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-3780\,B{a}^{3}b{e}^{4}{x}^{2}+14364\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}-17580\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}+5460\,B{b}^{4}{d}^{3}e{x}^{2}-3696\,A{a}^{3}b{e}^{4}x+15120\,A{a}^{2}{b}^{2}d{e}^{3}x-21840\,Aa{b}^{3}{d}^{2}{e}^{2}x+11440\,A{b}^{4}{d}^{3}ex+6930\,B{a}^{4}{e}^{4}x-31584\,B{a}^{3}bd{e}^{3}x+54180\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x-40560\,Ba{b}^{3}{d}^{3}ex+10010\,B{b}^{4}{d}^{4}x+6006\,A{a}^{4}{e}^{4}-27720\,A{a}^{3}bd{e}^{3}+49140\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}-40040\,Aa{b}^{3}{d}^{3}e+12870\,A{b}^{4}{d}^{4}+924\,B{a}^{4}d{e}^{3}-3780\,B{a}^{3}b{d}^{2}{e}^{2}+5460\,B{a}^{2}{b}^{2}{d}^{3}e-2860\,Ba{b}^{3}{d}^{4}}{45045\,{a}^{5}{e}^{5}-225225\,{a}^{4}bd{e}^{4}+450450\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-450450\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+225225\,a{b}^{4}{d}^{4}e-45045\,{b}^{5}{d}^{5}} \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( ex+d \right ) ^{-{\frac{15}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(17/2),x)

[Out]

-2/45045*(b*x+a)^(7/2)*(128*A*b^4*e^4*x^4-240*B*a*b^3*e^4*x^4+112*B*b^4*d*e^3*x^4-448*A*a*b^3*e^4*x^3+960*A*b^
4*d*e^3*x^3+840*B*a^2*b^2*e^4*x^3-2192*B*a*b^3*d*e^3*x^3+840*B*b^4*d^2*e^2*x^3+1008*A*a^2*b^2*e^4*x^2-3360*A*a
*b^3*d*e^3*x^2+3120*A*b^4*d^2*e^2*x^2-1890*B*a^3*b*e^4*x^2+7182*B*a^2*b^2*d*e^3*x^2-8790*B*a*b^3*d^2*e^2*x^2+2
730*B*b^4*d^3*e*x^2-1848*A*a^3*b*e^4*x+7560*A*a^2*b^2*d*e^3*x-10920*A*a*b^3*d^2*e^2*x+5720*A*b^4*d^3*e*x+3465*
B*a^4*e^4*x-15792*B*a^3*b*d*e^3*x+27090*B*a^2*b^2*d^2*e^2*x-20280*B*a*b^3*d^3*e*x+5005*B*b^4*d^4*x+3003*A*a^4*
e^4-13860*A*a^3*b*d*e^3+24570*A*a^2*b^2*d^2*e^2-20020*A*a*b^3*d^3*e+6435*A*b^4*d^4+462*B*a^4*d*e^3-1890*B*a^3*
b*d^2*e^2+2730*B*a^2*b^2*d^3*e-1430*B*a*b^3*d^4)/(e*x+d)^(15/2)/(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)^(5/2)*(B*x+A)/(e*x+d)^(17/2),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)^(5/2)*(B*x+A)/(e*x+d)^(17/2),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)**(5/2)*(B*x+A)/(e*x+d)**(17/2),x)

[Out]

Timed out

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Giac [B]  time = 6.24365, size = 1864, normalized size = 7.31 \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)^(5/2)*(B*x+A)/(e*x+d)^(17/2),x, algorithm="giac")

[Out]

-1/2952069120*((2*(4*(b*x + a)*(2*(7*B*b^18*d^3*abs(b)*e^10 - 29*B*a*b^17*d^2*abs(b)*e^11 + 8*A*b^18*d^2*abs(b
)*e^11 + 37*B*a^2*b^16*d*abs(b)*e^12 - 16*A*a*b^17*d*abs(b)*e^12 - 15*B*a^3*b^15*abs(b)*e^13 + 8*A*a^2*b^16*ab
s(b)*e^13)*(b*x + a)/(b^32*d^8*e^16 - 8*a*b^31*d^7*e^17 + 28*a^2*b^30*d^6*e^18 - 56*a^3*b^29*d^5*e^19 + 70*a^4
*b^28*d^4*e^20 - 56*a^5*b^27*d^3*e^21 + 28*a^6*b^26*d^2*e^22 - 8*a^7*b^25*d*e^23 + a^8*b^24*e^24) + 15*(7*B*b^
19*d^4*abs(b)*e^9 - 36*B*a*b^18*d^3*abs(b)*e^10 + 8*A*b^19*d^3*abs(b)*e^10 + 66*B*a^2*b^17*d^2*abs(b)*e^11 - 2
4*A*a*b^18*d^2*abs(b)*e^11 - 52*B*a^3*b^16*d*abs(b)*e^12 + 24*A*a^2*b^17*d*abs(b)*e^12 + 15*B*a^4*b^15*abs(b)*
e^13 - 8*A*a^3*b^16*abs(b)*e^13)/(b^32*d^8*e^16 - 8*a*b^31*d^7*e^17 + 28*a^2*b^30*d^6*e^18 - 56*a^3*b^29*d^5*e
^19 + 70*a^4*b^28*d^4*e^20 - 56*a^5*b^27*d^3*e^21 + 28*a^6*b^26*d^2*e^22 - 8*a^7*b^25*d*e^23 + a^8*b^24*e^24))
+ 195*(7*B*b^20*d^5*abs(b)*e^8 - 43*B*a*b^19*d^4*abs(b)*e^9 + 8*A*b^20*d^4*abs(b)*e^9 + 102*B*a^2*b^18*d^3*ab
s(b)*e^10 - 32*A*a*b^19*d^3*abs(b)*e^10 - 118*B*a^3*b^17*d^2*abs(b)*e^11 + 48*A*a^2*b^18*d^2*abs(b)*e^11 + 67*
B*a^4*b^16*d*abs(b)*e^12 - 32*A*a^3*b^17*d*abs(b)*e^12 - 15*B*a^5*b^15*abs(b)*e^13 + 8*A*a^4*b^16*abs(b)*e^13)
/(b^32*d^8*e^16 - 8*a*b^31*d^7*e^17 + 28*a^2*b^30*d^6*e^18 - 56*a^3*b^29*d^5*e^19 + 70*a^4*b^28*d^4*e^20 - 56*
a^5*b^27*d^3*e^21 + 28*a^6*b^26*d^2*e^22 - 8*a^7*b^25*d*e^23 + a^8*b^24*e^24))*(b*x + a) + 715*(7*B*b^21*d^6*a
bs(b)*e^7 - 50*B*a*b^20*d^5*abs(b)*e^8 + 8*A*b^21*d^5*abs(b)*e^8 + 145*B*a^2*b^19*d^4*abs(b)*e^9 - 40*A*a*b^20
*d^4*abs(b)*e^9 - 220*B*a^3*b^18*d^3*abs(b)*e^10 + 80*A*a^2*b^19*d^3*abs(b)*e^10 + 185*B*a^4*b^17*d^2*abs(b)*e
^11 - 80*A*a^3*b^18*d^2*abs(b)*e^11 - 82*B*a^5*b^16*d*abs(b)*e^12 + 40*A*a^4*b^17*d*abs(b)*e^12 + 15*B*a^6*b^1
5*abs(b)*e^13 - 8*A*a^5*b^16*abs(b)*e^13)/(b^32*d^8*e^16 - 8*a*b^31*d^7*e^17 + 28*a^2*b^30*d^6*e^18 - 56*a^3*b
^29*d^5*e^19 + 70*a^4*b^28*d^4*e^20 - 56*a^5*b^27*d^3*e^21 + 28*a^6*b^26*d^2*e^22 - 8*a^7*b^25*d*e^23 + a^8*b^
24*e^24))*(b*x + a) - 6435*(B*a*b^21*d^6*abs(b)*e^7 - A*b^22*d^6*abs(b)*e^7 - 6*B*a^2*b^20*d^5*abs(b)*e^8 + 6*
A*a*b^21*d^5*abs(b)*e^8 + 15*B*a^3*b^19*d^4*abs(b)*e^9 - 15*A*a^2*b^20*d^4*abs(b)*e^9 - 20*B*a^4*b^18*d^3*abs(
b)*e^10 + 20*A*a^3*b^19*d^3*abs(b)*e^10 + 15*B*a^5*b^17*d^2*abs(b)*e^11 - 15*A*a^4*b^18*d^2*abs(b)*e^11 - 6*B*
a^6*b^16*d*abs(b)*e^12 + 6*A*a^5*b^17*d*abs(b)*e^12 + B*a^7*b^15*abs(b)*e^13 - A*a^6*b^16*abs(b)*e^13)/(b^32*d
^8*e^16 - 8*a*b^31*d^7*e^17 + 28*a^2*b^30*d^6*e^18 - 56*a^3*b^29*d^5*e^19 + 70*a^4*b^28*d^4*e^20 - 56*a^5*b^27
*d^3*e^21 + 28*a^6*b^26*d^2*e^22 - 8*a^7*b^25*d*e^23 + a^8*b^24*e^24))*(b*x + a)^(7/2)/(b^2*d + (b*x + a)*b*e
- a*b*e)^(15/2)