3.2219 \(\int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=147 \[ \frac {4 b (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{315 e (d+e x)^{5/2} (b d-a e)^3}+\frac {2 (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

[Out]

-2/9*(-A*e+B*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)/(e*x+d)^(9/2)+2/63*(4*A*b*e-9*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+
b*d)^2/(e*x+d)^(7/2)+4/315*b*(4*A*b*e-9*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^3/(e*x+d)^(5/2)

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Rubi [A]  time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, 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} \[ \frac {4 b (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{315 e (d+e x)^{5/2} (b d-a e)^3}+\frac {2 (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(5*b*B*d + 4*A*b*e - 9*a*B*e)*(a + b*x
)^(5/2))/(63*e*(b*d - a*e)^2*(d + e*x)^(7/2)) + (4*b*(5*b*B*d + 4*A*b*e - 9*a*B*e)*(a + b*x)^(5/2))/(315*e*(b*
d - a*e)^3*(d + e*x)^(5/2))

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]

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 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]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {(5 b B d+4 A b e-9 a B e) \int \frac {(a+b x)^{3/2}}{(d+e x)^{9/2}} \, dx}{9 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (5 b B d+4 A b e-9 a B e) (a+b x)^{5/2}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {(2 b (5 b B d+4 A b e-9 a B e)) \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{63 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (5 b B d+4 A b e-9 a B e) (a+b x)^{5/2}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {4 b (5 b B d+4 A b e-9 a B e) (a+b x)^{5/2}}{315 e (b d-a e)^3 (d+e x)^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 135, normalized size = 0.92 \[ \frac {2 (a+b x)^{5/2} \left (A \left (35 a^2 e^2-10 a b e (9 d+2 e x)+b^2 \left (63 d^2+36 d e x+8 e^2 x^2\right )\right )+B \left (5 a^2 e (2 d+9 e x)-2 a b \left (9 d^2+53 d e x+9 e^2 x^2\right )+5 b^2 d x (9 d+2 e x)\right )\right )}{315 (d+e x)^{9/2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

(2*(a + b*x)^(5/2)*(A*(35*a^2*e^2 - 10*a*b*e*(9*d + 2*e*x) + b^2*(63*d^2 + 36*d*e*x + 8*e^2*x^2)) + B*(5*b^2*d
*x*(9*d + 2*e*x) + 5*a^2*e*(2*d + 9*e*x) - 2*a*b*(9*d^2 + 53*d*e*x + 9*e^2*x^2))))/(315*(b*d - a*e)^3*(d + e*x
)^(9/2))

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fricas [B]  time = 110.90, size = 567, normalized size = 3.86 \[ \frac {2 \, {\left (35 \, A a^{4} e^{2} + 2 \, {\left (5 \, B b^{4} d e - {\left (9 \, B a b^{3} - 4 \, A b^{4}\right )} e^{2}\right )} x^{4} + {\left (45 \, B b^{4} d^{2} - 2 \, {\left (43 \, B a b^{3} - 18 \, A b^{4}\right )} d e + {\left (9 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{2}\right )} x^{3} - 9 \, {\left (2 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d^{2} + 10 \, {\left (B a^{4} - 9 \, A a^{3} b\right )} d e + 3 \, {\left (3 \, {\left (8 \, B a b^{3} + 7 \, A b^{4}\right )} d^{2} - 2 \, {\left (32 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d e + {\left (24 \, B a^{3} b + A a^{2} b^{2}\right )} e^{2}\right )} x^{2} + {\left (9 \, {\left (B a^{2} b^{2} + 14 \, A a b^{3}\right )} d^{2} - 2 \, {\left (43 \, B a^{3} b + 72 \, A a^{2} b^{2}\right )} d e + 5 \, {\left (9 \, B a^{4} + 10 \, A a^{3} b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{315 \, {\left (b^{3} d^{8} - 3 \, a b^{2} d^{7} e + 3 \, a^{2} b d^{6} e^{2} - a^{3} d^{5} e^{3} + {\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} x^{5} + 5 \, {\left (b^{3} d^{4} e^{4} - 3 \, a b^{2} d^{3} e^{5} + 3 \, a^{2} b d^{2} e^{6} - a^{3} d e^{7}\right )} x^{4} + 10 \, {\left (b^{3} d^{5} e^{3} - 3 \, a b^{2} d^{4} e^{4} + 3 \, a^{2} b d^{3} e^{5} - a^{3} d^{2} e^{6}\right )} x^{3} + 10 \, {\left (b^{3} d^{6} e^{2} - 3 \, a b^{2} d^{5} e^{3} + 3 \, a^{2} b d^{4} e^{4} - a^{3} d^{3} e^{5}\right )} x^{2} + 5 \, {\left (b^{3} d^{7} e - 3 \, a b^{2} d^{6} e^{2} + 3 \, a^{2} b d^{5} e^{3} - a^{3} d^{4} e^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

2/315*(35*A*a^4*e^2 + 2*(5*B*b^4*d*e - (9*B*a*b^3 - 4*A*b^4)*e^2)*x^4 + (45*B*b^4*d^2 - 2*(43*B*a*b^3 - 18*A*b
^4)*d*e + (9*B*a^2*b^2 - 4*A*a*b^3)*e^2)*x^3 - 9*(2*B*a^3*b - 7*A*a^2*b^2)*d^2 + 10*(B*a^4 - 9*A*a^3*b)*d*e +
3*(3*(8*B*a*b^3 + 7*A*b^4)*d^2 - 2*(32*B*a^2*b^2 + 3*A*a*b^3)*d*e + (24*B*a^3*b + A*a^2*b^2)*e^2)*x^2 + (9*(B*
a^2*b^2 + 14*A*a*b^3)*d^2 - 2*(43*B*a^3*b + 72*A*a^2*b^2)*d*e + 5*(9*B*a^4 + 10*A*a^3*b)*e^2)*x)*sqrt(b*x + a)
*sqrt(e*x + d)/(b^3*d^8 - 3*a*b^2*d^7*e + 3*a^2*b*d^6*e^2 - a^3*d^5*e^3 + (b^3*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a
^2*b*d*e^7 - a^3*e^8)*x^5 + 5*(b^3*d^4*e^4 - 3*a*b^2*d^3*e^5 + 3*a^2*b*d^2*e^6 - a^3*d*e^7)*x^4 + 10*(b^3*d^5*
e^3 - 3*a*b^2*d^4*e^4 + 3*a^2*b*d^3*e^5 - a^3*d^2*e^6)*x^3 + 10*(b^3*d^6*e^2 - 3*a*b^2*d^5*e^3 + 3*a^2*b*d^4*e
^4 - a^3*d^3*e^5)*x^2 + 5*(b^3*d^7*e - 3*a*b^2*d^6*e^2 + 3*a^2*b*d^5*e^3 - a^3*d^4*e^4)*x)

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giac [B]  time = 4.55, size = 483, normalized size = 3.29 \[ \frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (5 \, B b^{11} d^{2} {\left | b \right |} e^{5} - 14 \, B a b^{10} d {\left | b \right |} e^{6} + 4 \, A b^{11} d {\left | b \right |} e^{6} + 9 \, B a^{2} b^{9} {\left | b \right |} e^{7} - 4 \, A a b^{10} {\left | b \right |} e^{7}\right )} {\left (b x + a\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}} + \frac {9 \, {\left (5 \, B b^{12} d^{3} {\left | b \right |} e^{4} - 19 \, B a b^{11} d^{2} {\left | b \right |} e^{5} + 4 \, A b^{12} d^{2} {\left | b \right |} e^{5} + 23 \, B a^{2} b^{10} d {\left | b \right |} e^{6} - 8 \, A a b^{11} d {\left | b \right |} e^{6} - 9 \, B a^{3} b^{9} {\left | b \right |} e^{7} + 4 \, A a^{2} b^{10} {\left | b \right |} e^{7}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} - \frac {63 \, {\left (B a b^{12} d^{3} {\left | b \right |} e^{4} - A b^{13} d^{3} {\left | b \right |} e^{4} - 3 \, B a^{2} b^{11} d^{2} {\left | b \right |} e^{5} + 3 \, A a b^{12} d^{2} {\left | b \right |} e^{5} + 3 \, B a^{3} b^{10} d {\left | b \right |} e^{6} - 3 \, A a^{2} b^{11} d {\left | b \right |} e^{6} - B a^{4} b^{9} {\left | b \right |} e^{7} + A a^{3} b^{10} {\left | b \right |} e^{7}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} {\left (b x + a\right )}^{\frac {5}{2}}}{315 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

2/315*((b*x + a)*(2*(5*B*b^11*d^2*abs(b)*e^5 - 14*B*a*b^10*d*abs(b)*e^6 + 4*A*b^11*d*abs(b)*e^6 + 9*B*a^2*b^9*
abs(b)*e^7 - 4*A*a*b^10*abs(b)*e^7)*(b*x + a)/(b^6*d^4*e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3*b^3*d
*e^7 + a^4*b^2*e^8) + 9*(5*B*b^12*d^3*abs(b)*e^4 - 19*B*a*b^11*d^2*abs(b)*e^5 + 4*A*b^12*d^2*abs(b)*e^5 + 23*B
*a^2*b^10*d*abs(b)*e^6 - 8*A*a*b^11*d*abs(b)*e^6 - 9*B*a^3*b^9*abs(b)*e^7 + 4*A*a^2*b^10*abs(b)*e^7)/(b^6*d^4*
e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3*b^3*d*e^7 + a^4*b^2*e^8)) - 63*(B*a*b^12*d^3*abs(b)*e^4 - A*
b^13*d^3*abs(b)*e^4 - 3*B*a^2*b^11*d^2*abs(b)*e^5 + 3*A*a*b^12*d^2*abs(b)*e^5 + 3*B*a^3*b^10*d*abs(b)*e^6 - 3*
A*a^2*b^11*d*abs(b)*e^6 - B*a^4*b^9*abs(b)*e^7 + A*a^3*b^10*abs(b)*e^7)/(b^6*d^4*e^4 - 4*a*b^5*d^3*e^5 + 6*a^2
*b^4*d^2*e^6 - 4*a^3*b^3*d*e^7 + a^4*b^2*e^8))*(b*x + a)^(5/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(9/2)

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maple [A]  time = 0.01, size = 177, normalized size = 1.20 \[ -\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (8 A \,b^{2} e^{2} x^{2}-18 B a b \,e^{2} x^{2}+10 B \,b^{2} d e \,x^{2}-20 A a b \,e^{2} x +36 A \,b^{2} d e x +45 B \,a^{2} e^{2} x -106 B a b d e x +45 B \,b^{2} d^{2} x +35 A \,a^{2} e^{2}-90 A a b d e +63 A \,b^{2} d^{2}+10 B \,a^{2} d e -18 B a b \,d^{2}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x)

[Out]

-2/315*(b*x+a)^(5/2)*(8*A*b^2*e^2*x^2-18*B*a*b*e^2*x^2+10*B*b^2*d*e*x^2-20*A*a*b*e^2*x+36*A*b^2*d*e*x+45*B*a^2
*e^2*x-106*B*a*b*d*e*x+45*B*b^2*d^2*x+35*A*a^2*e^2-90*A*a*b*d*e+63*A*b^2*d^2+10*B*a^2*d*e-18*B*a*b*d^2)/(e*x+d
)^(9/2)/(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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [B]  time = 2.37, size = 402, normalized size = 2.73 \[ -\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (20\,B\,a^4\,d\,e+70\,A\,a^4\,e^2-36\,B\,a^3\,b\,d^2-180\,A\,a^3\,b\,d\,e+126\,A\,a^2\,b^2\,d^2\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}+\frac {x^2\,\sqrt {a+b\,x}\,\left (144\,B\,a^3\,b\,e^2-384\,B\,a^2\,b^2\,d\,e+6\,A\,a^2\,b^2\,e^2+144\,B\,a\,b^3\,d^2-36\,A\,a\,b^3\,d\,e+126\,A\,b^4\,d^2\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\sqrt {a+b\,x}\,\left (90\,B\,a^4\,e^2-172\,B\,a^3\,b\,d\,e+100\,A\,a^3\,b\,e^2+18\,B\,a^2\,b^2\,d^2-288\,A\,a^2\,b^2\,d\,e+252\,A\,a\,b^3\,d^2\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}+\frac {4\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-9\,B\,a\,e+5\,B\,b\,d\right )}{315\,e^4\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,b^2\,x^3\,\left (a\,e-9\,b\,d\right )\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-9\,B\,a\,e+5\,B\,b\,d\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}\right )}{x^5+\frac {d^5}{e^5}+\frac {5\,d\,x^4}{e}+\frac {5\,d^4\,x}{e^4}+\frac {10\,d^2\,x^3}{e^2}+\frac {10\,d^3\,x^2}{e^3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(3/2))/(d + e*x)^(11/2),x)

[Out]

-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(70*A*a^4*e^2 + 20*B*a^4*d*e - 36*B*a^3*b*d^2 + 126*A*a^2*b^2*d^2 - 180*A*
a^3*b*d*e))/(315*e^5*(a*e - b*d)^3) + (x^2*(a + b*x)^(1/2)*(126*A*b^4*d^2 + 144*B*a*b^3*d^2 + 144*B*a^3*b*e^2
+ 6*A*a^2*b^2*e^2 - 36*A*a*b^3*d*e - 384*B*a^2*b^2*d*e))/(315*e^5*(a*e - b*d)^3) + (x*(a + b*x)^(1/2)*(90*B*a^
4*e^2 + 252*A*a*b^3*d^2 + 100*A*a^3*b*e^2 + 18*B*a^2*b^2*d^2 - 172*B*a^3*b*d*e - 288*A*a^2*b^2*d*e))/(315*e^5*
(a*e - b*d)^3) + (4*b^3*x^4*(a + b*x)^(1/2)*(4*A*b*e - 9*B*a*e + 5*B*b*d))/(315*e^4*(a*e - b*d)^3) - (2*b^2*x^
3*(a*e - 9*b*d)*(a + b*x)^(1/2)*(4*A*b*e - 9*B*a*e + 5*B*b*d))/(315*e^5*(a*e - b*d)^3)))/(x^5 + d^5/e^5 + (5*d
*x^4)/e + (5*d^4*x)/e^4 + (10*d^2*x^3)/e^2 + (10*d^3*x^2)/e^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(11/2),x)

[Out]

Timed out

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