3.2232 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx\)
Optimal. Leaf size=147 \[ \frac{4 b (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]
[Out]
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a +
b*x)^(7/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) + (4*b*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^(7/2))/(693*e
*(b*d - a*e)^3*(d + e*x)^(7/2))
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Rubi [A] time = 0.0897558, antiderivative size = 147, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{78, 45, 37} \[ \frac{4 b (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(13/2),x]
[Out]
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a +
b*x)^(7/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) + (4*b*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^(7/2))/(693*e
*(b*d - a*e)^3*(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)^{13/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{(7 b B d+4 A b e-11 a B e) \int \frac{(a+b x)^{5/2}}{(d+e x)^{11/2}} \, dx}{11 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{2 (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac{(2 b (7 b B d+4 A b e-11 a B e)) \int \frac{(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{99 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac{2 (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac{4 b (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.10634, size = 135, normalized size = 0.92 \[ \frac{2 (a+b x)^{7/2} \left (A \left (63 a^2 e^2-14 a b e (11 d+2 e x)+b^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )\right )+B \left (7 a^2 e (2 d+11 e x)-2 a b \left (11 d^2+85 d e x+11 e^2 x^2\right )+7 b^2 d x (11 d+2 e x)\right )\right )}{693 (d+e x)^{11/2} (b d-a e)^3} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(13/2),x]
[Out]
(2*(a + b*x)^(7/2)*(A*(63*a^2*e^2 - 14*a*b*e*(11*d + 2*e*x) + b^2*(99*d^2 + 44*d*e*x + 8*e^2*x^2)) + B*(7*b^2*
d*x*(11*d + 2*e*x) + 7*a^2*e*(2*d + 11*e*x) - 2*a*b*(11*d^2 + 85*d*e*x + 11*e^2*x^2))))/(693*(b*d - a*e)^3*(d
+ e*x)^(11/2))
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Maple [A] time = 0.007, size = 177, normalized size = 1.2 \begin{align*} -{\frac{16\,A{b}^{2}{e}^{2}{x}^{2}-44\,Bab{e}^{2}{x}^{2}+28\,B{b}^{2}de{x}^{2}-56\,Aab{e}^{2}x+88\,A{b}^{2}dex+154\,B{a}^{2}{e}^{2}x-340\,Babdex+154\,B{b}^{2}{d}^{2}x+126\,A{a}^{2}{e}^{2}-308\,Aabde+198\,A{b}^{2}{d}^{2}+28\,B{a}^{2}de-44\,Bab{d}^{2}}{693\,{a}^{3}{e}^{3}-2079\,{a}^{2}bd{e}^{2}+2079\,a{b}^{2}{d}^{2}e-693\,{b}^{3}{d}^{3}} \left ( bx+a \right ) ^{{\frac{7}{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)^(5/2)*(B*x+A)/(e*x+d)^(13/2),x)
[Out]
-2/693*(b*x+a)^(7/2)*(8*A*b^2*e^2*x^2-22*B*a*b*e^2*x^2+14*B*b^2*d*e*x^2-28*A*a*b*e^2*x+44*A*b^2*d*e*x+77*B*a^2
*e^2*x-170*B*a*b*d*e*x+77*B*b^2*d^2*x+63*A*a^2*e^2-154*A*a*b*d*e+99*A*b^2*d^2+14*B*a^2*d*e-22*B*a*b*d^2)/(e*x+
d)^(11/2)/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)
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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)^(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)^(5/2)*(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)**(5/2)*(B*x+A)/(e*x+d)**(13/2),x)
[Out]
Timed out
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Giac [B] time = 2.39362, size = 887, normalized size = 6.03 \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)^(13/2),x, algorithm="giac")
[Out]
-1/2838528*((b*x + a)*(2*(7*B*b^14*d^3*abs(b)*e^6 - 25*B*a*b^13*d^2*abs(b)*e^7 + 4*A*b^14*d^2*abs(b)*e^7 + 29*
B*a^2*b^12*d*abs(b)*e^8 - 8*A*a*b^13*d*abs(b)*e^8 - 11*B*a^3*b^11*abs(b)*e^9 + 4*A*a^2*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*(7*B*b^15*d^4*abs(b)*e^5 - 32*B*a*b^14*d^3*abs(b)*e^6 + 4*A*b^15*d^3*a
bs(b)*e^6 + 54*B*a^2*b^13*d^2*abs(b)*e^7 - 12*A*a*b^14*d^2*abs(b)*e^7 - 40*B*a^3*b^12*d*abs(b)*e^8 + 12*A*a^2*
b^13*d*abs(b)*e^8 + 11*B*a^4*b^11*abs(b)*e^9 - 4*A*a^3*b^12*abs(b)*e^9)/(b^24*d^6*e^12 - 6*a*b^23*d^5*e^13 + 1
5*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*(
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)^(7/2)/(b^2*d + (b
*x + a)*b*e - a*b*e)^(11/2)