3.2233 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx\)
Optimal. Leaf size=201 \[ \frac{16 b^2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^4}+\frac{8 b (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]
[Out]
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a +
b*x)^(7/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2)) + (8*b*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(128
7*e*(b*d - a*e)^3*(d + e*x)^(9/2)) + (16*b^2*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(9009*e*(b*d - a*
e)^4*(d + e*x)^(7/2))
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Rubi [A] time = 0.12509, 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} \[ \frac{16 b^2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^4}+\frac{8 b (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(15/2),x]
[Out]
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a +
b*x)^(7/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2)) + (8*b*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(128
7*e*(b*d - a*e)^3*(d + e*x)^(9/2)) + (16*b^2*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(9009*e*(b*d - a*
e)^4*(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)^{15/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{(7 b B d+6 A b e-13 a B e) \int \frac{(a+b x)^{5/2}}{(d+e x)^{13/2}} \, dx}{13 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{(4 b (7 b B d+6 A b e-13 a B e)) \int \frac{(a+b x)^{5/2}}{(d+e x)^{11/2}} \, dx}{143 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{8 b (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac{\left (8 b^2 (7 b B d+6 A b e-13 a B e)\right ) \int \frac{(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{1287 e (b d-a e)^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac{2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac{8 b (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac{16 b^2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{9009 e (b d-a e)^4 (d+e x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.341575, size = 114, normalized size = 0.57 \[ \frac{2 (a+b x)^{7/2} \left (693 (B d-A e)-\frac{(d+e x) \left (4 b (d+e x) (-7 a e+9 b d+2 b e x)+63 (b d-a e)^2\right ) (-13 a B e+6 A b e+7 b B d)}{(b d-a e)^3}\right )}{9009 e (d+e x)^{13/2} (a e-b d)} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(15/2),x]
[Out]
(2*(a + b*x)^(7/2)*(693*(B*d - A*e) - ((7*b*B*d + 6*A*b*e - 13*a*B*e)*(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)^3))/(9009*e*(-(b*d) + a*e)*(d + e*x)^(13/2))
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Maple [A] time = 0.009, size = 322, normalized size = 1.6 \begin{align*} -{\frac{-96\,A{b}^{3}{e}^{3}{x}^{3}+208\,Ba{b}^{2}{e}^{3}{x}^{3}-112\,B{b}^{3}d{e}^{2}{x}^{3}+336\,Aa{b}^{2}{e}^{3}{x}^{2}-624\,A{b}^{3}d{e}^{2}{x}^{2}-728\,B{a}^{2}b{e}^{3}{x}^{2}+1744\,Ba{b}^{2}d{e}^{2}{x}^{2}-728\,B{b}^{3}{d}^{2}e{x}^{2}-756\,A{a}^{2}b{e}^{3}x+2184\,Aa{b}^{2}d{e}^{2}x-1716\,A{b}^{3}{d}^{2}ex+1638\,B{a}^{3}{e}^{3}x-5614\,B{a}^{2}bd{e}^{2}x+6266\,Ba{b}^{2}{d}^{2}ex-2002\,B{b}^{3}{d}^{3}x+1386\,A{a}^{3}{e}^{3}-4914\,A{a}^{2}bd{e}^{2}+6006\,Aa{b}^{2}{d}^{2}e-2574\,A{b}^{3}{d}^{3}+252\,B{a}^{3}d{e}^{2}-728\,B{a}^{2}b{d}^{2}e+572\,Ba{b}^{2}{d}^{3}}{9009\,{e}^{4}{a}^{4}-36036\,b{e}^{3}d{a}^{3}+54054\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-36036\,a{b}^{3}{d}^{3}e+9009\,{b}^{4}{d}^{4}} \left ( bx+a \right ) ^{{\frac{7}{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)^(5/2)*(B*x+A)/(e*x+d)^(15/2),x)
[Out]
-2/9009*(b*x+a)^(7/2)*(-48*A*b^3*e^3*x^3+104*B*a*b^2*e^3*x^3-56*B*b^3*d*e^2*x^3+168*A*a*b^2*e^3*x^2-312*A*b^3*
d*e^2*x^2-364*B*a^2*b*e^3*x^2+872*B*a*b^2*d*e^2*x^2-364*B*b^3*d^2*e*x^2-378*A*a^2*b*e^3*x+1092*A*a*b^2*d*e^2*x
-858*A*b^3*d^2*e*x+819*B*a^3*e^3*x-2807*B*a^2*b*d*e^2*x+3133*B*a*b^2*d^2*e*x-1001*B*b^3*d^3*x+693*A*a^3*e^3-24
57*A*a^2*b*d*e^2+3003*A*a*b^2*d^2*e-1287*A*b^3*d^3+126*B*a^3*d*e^2-364*B*a^2*b*d^2*e+286*B*a*b^2*d^3)/(e*x+d)^
(13/2)/(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)^(5/2)*(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)^(5/2)*(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)**(5/2)*(B*x+A)/(e*x+d)**(15/2),x)
[Out]
Timed out
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Giac [B] time = 4.82224, size = 1342, normalized size = 6.68 \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)^(15/2),x, algorithm="giac")
[Out]
-1/6642155520*((4*(b*x + a)*(2*(7*B*b^16*d^3*abs(b)*e^8 - 27*B*a*b^15*d^2*abs(b)*e^9 + 6*A*b^16*d^2*abs(b)*e^9
+ 33*B*a^2*b^14*d*abs(b)*e^10 - 12*A*a*b^15*d*abs(b)*e^10 - 13*B*a^3*b^13*abs(b)*e^11 + 6*A*a^2*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*(7*B*b^17*d^4*abs(b)*e^7 - 34*B*a*b^
16*d^3*abs(b)*e^8 + 6*A*b^17*d^3*abs(b)*e^8 + 60*B*a^2*b^15*d^2*abs(b)*e^9 - 18*A*a*b^16*d^2*abs(b)*e^9 - 46*B
*a^3*b^14*d*abs(b)*e^10 + 18*A*a^2*b^15*d*abs(b)*e^10 + 13*B*a^4*b^13*abs(b)*e^11 - 6*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)) + 143*(7*B*b^18*d^5*abs(b)*e^6 - 41*B*a*b^17*d^4*abs(b)
*e^7 + 6*A*b^18*d^4*abs(b)*e^7 + 94*B*a^2*b^16*d^3*abs(b)*e^8 - 24*A*a*b^17*d^3*abs(b)*e^8 - 106*B*a^3*b^15*d^
2*abs(b)*e^9 + 36*A*a^2*b^16*d^2*abs(b)*e^9 + 59*B*a^4*b^14*d*abs(b)*e^10 - 24*A*a^3*b^15*d*abs(b)*e^10 - 13*B
*a^5*b^13*abs(b)*e^11 + 6*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) - 1287*(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*ab
s(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*abs(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^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)^(7/2)/(b^2*d + (b*x + a
)*b*e - a*b*e)^(13/2)