3.23.8 \(\int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx\) [2208]

Optimal. Leaf size=198 \[ -\frac {2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{21 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {8 b (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{315 e (b d-a e)^4 (d+e x)^{3/2}} \]

[Out]

-2/9*(-A*e+B*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)/(e*x+d)^(9/2)+2/21*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*
d)^2/(e*x+d)^(7/2)+8/105*b*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^3/(e*x+d)^(5/2)+16/315*b^2*(2*A*
b*e-3*B*a*e+B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^4/(e*x+d)^(3/2)

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Rubi [A]
time = 0.07, antiderivative size = 198, normalized size of antiderivative = 1.00, 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 = {79, 47, 37} \begin {gather*} \frac {16 b^2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac {8 b (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac {2 (a+b x)^{3/2} (-3 a B e+2 A b e+b B d)}{21 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^
(3/2))/(21*e*(b*d - a*e)^2*(d + e*x)^(7/2)) + (8*b*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(3/2))/(105*e*(b*d -
a*e)^3*(d + e*x)^(5/2)) + (16*b^2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(3/2))/(315*e*(b*d - a*e)^4*(d + e*x)^
(3/2))

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 47

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 79

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] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {(b B d+2 A b e-3 a B e) \int \frac {\sqrt {a+b x}}{(d+e x)^{9/2}} \, dx}{3 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{21 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {(4 b (b B d+2 A b e-3 a B e)) \int \frac {\sqrt {a+b x}}{(d+e x)^{7/2}} \, dx}{21 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{21 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {8 b (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac {\left (8 b^2 (b B d+2 A b e-3 a B e)\right ) \int \frac {\sqrt {a+b x}}{(d+e x)^{5/2}} \, dx}{105 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{21 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {8 b (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{315 e (b d-a e)^4 (d+e x)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 199, normalized size = 1.01 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (35 B d e^2 (a+b x)^3-35 A e^3 (a+b x)^3-90 b B d e (a+b x)^2 (d+e x)+135 A b e^2 (a+b x)^2 (d+e x)-45 a B e^2 (a+b x)^2 (d+e x)+63 b^2 B d (a+b x) (d+e x)^2-189 A b^2 e (a+b x) (d+e x)^2+126 a b B e (a+b x) (d+e x)^2+105 A b^3 (d+e x)^3-105 a b^2 B (d+e x)^3\right )}{315 (b d-a e)^4 (d+e x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^(3/2)*(35*B*d*e^2*(a + b*x)^3 - 35*A*e^3*(a + b*x)^3 - 90*b*B*d*e*(a + b*x)^2*(d + e*x) + 135*A*b
*e^2*(a + b*x)^2*(d + e*x) - 45*a*B*e^2*(a + b*x)^2*(d + e*x) + 63*b^2*B*d*(a + b*x)*(d + e*x)^2 - 189*A*b^2*e
*(a + b*x)*(d + e*x)^2 + 126*a*b*B*e*(a + b*x)*(d + e*x)^2 + 105*A*b^3*(d + e*x)^3 - 105*a*b^2*B*(d + e*x)^3))
/(315*(b*d - a*e)^4*(d + e*x)^(9/2))

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Maple [A]
time = 0.09, size = 281, normalized size = 1.42

method	result	size
default	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-16 A \,b^{3} e^{3} x^{3}+24 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}-72 A \,b^{3} d \,e^{2} x^{2}-36 B \,a^{2} b \,e^{3} x^{2}+120 B a \,b^{2} d \,e^{2} x^{2}-36 B \,b^{3} d^{2} e \,x^{2}-30 A \,a^{2} b \,e^{3} x +108 A a \,b^{2} d \,e^{2} x -126 A \,b^{3} d^{2} e x +45 B \,a^{3} e^{3} x -177 B \,a^{2} b d \,e^{2} x +243 B a \,b^{2} d^{2} e x -63 B \,b^{3} d^{3} x +35 a^{3} A \,e^{3}-135 A \,a^{2} b d \,e^{2}+189 A a \,b^{2} d^{2} e -105 A \,b^{3} d^{3}+10 B \,a^{3} d \,e^{2}-36 B \,a^{2} b \,d^{2} e +42 B a \,b^{2} d^{3}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (a e -b d \right )^{4}}\)	\(281\)
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-16 A \,b^{3} e^{3} x^{3}+24 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}-72 A \,b^{3} d \,e^{2} x^{2}-36 B \,a^{2} b \,e^{3} x^{2}+120 B a \,b^{2} d \,e^{2} x^{2}-36 B \,b^{3} d^{2} e \,x^{2}-30 A \,a^{2} b \,e^{3} x +108 A a \,b^{2} d \,e^{2} x -126 A \,b^{3} d^{2} e x +45 B \,a^{3} e^{3} x -177 B \,a^{2} b d \,e^{2} x +243 B a \,b^{2} d^{2} e x -63 B \,b^{3} d^{3} x +35 a^{3} A \,e^{3}-135 A \,a^{2} b d \,e^{2}+189 A a \,b^{2} d^{2} e -105 A \,b^{3} d^{3}+10 B \,a^{3} d \,e^{2}-36 B \,a^{2} b \,d^{2} e +42 B a \,b^{2} d^{3}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\)	\(322\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(11/2),x,method=_RETURNVERBOSE)

[Out]

-2/315*(b*x+a)^(3/2)/(e*x+d)^(9/2)*(-16*A*b^3*e^3*x^3+24*B*a*b^2*e^3*x^3-8*B*b^3*d*e^2*x^3+24*A*a*b^2*e^3*x^2-
72*A*b^3*d*e^2*x^2-36*B*a^2*b*e^3*x^2+120*B*a*b^2*d*e^2*x^2-36*B*b^3*d^2*e*x^2-30*A*a^2*b*e^3*x+108*A*a*b^2*d*
e^2*x-126*A*b^3*d^2*e*x+45*B*a^3*e^3*x-177*B*a^2*b*d*e^2*x+243*B*a*b^2*d^2*e*x-63*B*b^3*d^3*x+35*A*a^3*e^3-135
*A*a^2*b*d*e^2+189*A*a*b^2*d^2*e-105*A*b^3*d^3+10*B*a^3*d*e^2-36*B*a^2*b*d^2*e+42*B*a*b^2*d^3)/(a*e-b*d)^4

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/(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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (185) = 370\).
time = 63.27, size = 704, normalized size = 3.56 \begin {gather*} \frac {2 \, {\left (63 \, B b^{4} d^{3} x^{2} + 21 \, {\left (B a b^{3} + 5 \, A b^{4}\right )} d^{3} x - 21 \, {\left (2 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} d^{3} - {\left (35 \, A a^{4} + 8 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} - 4 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 3 \, {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{4} + A a^{3} b\right )} x\right )} e^{3} + {\left (8 \, B b^{4} d x^{4} - 8 \, {\left (14 \, B a b^{3} - 9 \, A b^{4}\right )} d x^{3} + 3 \, {\left (19 \, B a^{2} b^{2} - 12 \, A a b^{3}\right )} d x^{2} + {\left (167 \, B a^{3} b + 27 \, A a^{2} b^{2}\right )} d x - 5 \, {\left (2 \, B a^{4} - 27 \, A a^{3} b\right )} d\right )} e^{2} + 9 \, {\left (4 \, B b^{4} d^{2} x^{3} - {\left (23 \, B a b^{3} - 14 \, A b^{4}\right )} d^{2} x^{2} - {\left (23 \, B a^{2} b^{2} + 7 \, A a b^{3}\right )} d^{2} x + {\left (4 \, B a^{3} b - 21 \, A a^{2} b^{2}\right )} d^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{315 \, {\left (b^{4} d^{9} + a^{4} x^{5} e^{9} - {\left (4 \, a^{3} b d x^{5} - 5 \, a^{4} d x^{4}\right )} e^{8} + 2 \, {\left (3 \, a^{2} b^{2} d^{2} x^{5} - 10 \, a^{3} b d^{2} x^{4} + 5 \, a^{4} d^{2} x^{3}\right )} e^{7} - 2 \, {\left (2 \, a b^{3} d^{3} x^{5} - 15 \, a^{2} b^{2} d^{3} x^{4} + 20 \, a^{3} b d^{3} x^{3} - 5 \, a^{4} d^{3} x^{2}\right )} e^{6} + {\left (b^{4} d^{4} x^{5} - 20 \, a b^{3} d^{4} x^{4} + 60 \, a^{2} b^{2} d^{4} x^{3} - 40 \, a^{3} b d^{4} x^{2} + 5 \, a^{4} d^{4} x\right )} e^{5} + {\left (5 \, b^{4} d^{5} x^{4} - 40 \, a b^{3} d^{5} x^{3} + 60 \, a^{2} b^{2} d^{5} x^{2} - 20 \, a^{3} b d^{5} x + a^{4} d^{5}\right )} e^{4} + 2 \, {\left (5 \, b^{4} d^{6} x^{3} - 20 \, a b^{3} d^{6} x^{2} + 15 \, a^{2} b^{2} d^{6} x - 2 \, a^{3} b d^{6}\right )} e^{3} + 2 \, {\left (5 \, b^{4} d^{7} x^{2} - 10 \, a b^{3} d^{7} x + 3 \, a^{2} b^{2} d^{7}\right )} e^{2} + {\left (5 \, b^{4} d^{8} x - 4 \, a b^{3} d^{8}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

2/315*(63*B*b^4*d^3*x^2 + 21*(B*a*b^3 + 5*A*b^4)*d^3*x - 21*(2*B*a^2*b^2 - 5*A*a*b^3)*d^3 - (35*A*a^4 + 8*(3*B
*a*b^3 - 2*A*b^4)*x^4 - 4*(3*B*a^2*b^2 - 2*A*a*b^3)*x^3 + 3*(3*B*a^3*b - 2*A*a^2*b^2)*x^2 + 5*(9*B*a^4 + A*a^3
*b)*x)*e^3 + (8*B*b^4*d*x^4 - 8*(14*B*a*b^3 - 9*A*b^4)*d*x^3 + 3*(19*B*a^2*b^2 - 12*A*a*b^3)*d*x^2 + (167*B*a^
3*b + 27*A*a^2*b^2)*d*x - 5*(2*B*a^4 - 27*A*a^3*b)*d)*e^2 + 9*(4*B*b^4*d^2*x^3 - (23*B*a*b^3 - 14*A*b^4)*d^2*x
^2 - (23*B*a^2*b^2 + 7*A*a*b^3)*d^2*x + (4*B*a^3*b - 21*A*a^2*b^2)*d^2)*e)*sqrt(b*x + a)*sqrt(x*e + d)/(b^4*d^
9 + a^4*x^5*e^9 - (4*a^3*b*d*x^5 - 5*a^4*d*x^4)*e^8 + 2*(3*a^2*b^2*d^2*x^5 - 10*a^3*b*d^2*x^4 + 5*a^4*d^2*x^3)
*e^7 - 2*(2*a*b^3*d^3*x^5 - 15*a^2*b^2*d^3*x^4 + 20*a^3*b*d^3*x^3 - 5*a^4*d^3*x^2)*e^6 + (b^4*d^4*x^5 - 20*a*b
^3*d^4*x^4 + 60*a^2*b^2*d^4*x^3 - 40*a^3*b*d^4*x^2 + 5*a^4*d^4*x)*e^5 + (5*b^4*d^5*x^4 - 40*a*b^3*d^5*x^3 + 60
*a^2*b^2*d^5*x^2 - 20*a^3*b*d^5*x + a^4*d^5)*e^4 + 2*(5*b^4*d^6*x^3 - 20*a*b^3*d^6*x^2 + 15*a^2*b^2*d^6*x - 2*
a^3*b*d^6)*e^3 + 2*(5*b^4*d^7*x^2 - 10*a*b^3*d^7*x + 3*a^2*b^2*d^7)*e^2 + (5*b^4*d^8*x - 4*a*b^3*d^8)*e)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (185) = 370\).
time = 2.76, size = 579, normalized size = 2.92 \begin {gather*} \frac {2 \, {\left ({\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (B b^{10} d {\left | b \right |} e^{6} - 3 \, B a b^{9} {\left | b \right |} e^{7} + 2 \, 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 (B b^{11} d^{2} {\left | b \right |} e^{5} - 4 \, B a b^{10} d {\left | b \right |} e^{6} + 2 \, A b^{11} d {\left | b \right |} e^{6} + 3 \, B a^{2} b^{9} {\left | b \right |} e^{7} - 2 \, A a 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 b^{12} d^{3} {\left | b \right |} e^{4} - 5 \, B a b^{11} d^{2} {\left | b \right |} e^{5} + 2 \, A b^{12} d^{2} {\left | b \right |} e^{5} + 7 \, B a^{2} b^{10} d {\left | b \right |} e^{6} - 4 \, A a b^{11} d {\left | b \right |} e^{6} - 3 \, B a^{3} b^{9} {\left | b \right |} e^{7} + 2 \, 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 )} {\left (b x + a\right )} - \frac {105 \, {\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 {3}{2}}}{315 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

2/315*((4*(b*x + a)*(2*(B*b^10*d*abs(b)*e^6 - 3*B*a*b^9*abs(b)*e^7 + 2*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*(B*b^11*d^2*abs(b)*e^5 - 4*B*a*b
^10*d*abs(b)*e^6 + 2*A*b^11*d*abs(b)*e^6 + 3*B*a^2*b^9*abs(b)*e^7 - 2*A*a*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*b^12*d^3*abs(b)*e^4 - 5*B*a*b^11*d^2
*abs(b)*e^5 + 2*A*b^12*d^2*abs(b)*e^5 + 7*B*a^2*b^10*d*abs(b)*e^6 - 4*A*a*b^11*d*abs(b)*e^6 - 3*B*a^3*b^9*abs(
b)*e^7 + 2*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))*(b*x + a) - 105*(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)
^(3/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(9/2)

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Mupad [B]
time = 2.34, size = 428, normalized size = 2.16 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {x\,\sqrt {a+b\,x}\,\left (-90\,B\,a^4\,e^3+334\,B\,a^3\,b\,d\,e^2-10\,A\,a^3\,b\,e^3-414\,B\,a^2\,b^2\,d^2\,e+54\,A\,a^2\,b^2\,d\,e^2+42\,B\,a\,b^3\,d^3-126\,A\,a\,b^3\,d^2\,e+210\,A\,b^4\,d^3\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^4}-\frac {\sqrt {a+b\,x}\,\left (20\,B\,a^4\,d\,e^2+70\,A\,a^4\,e^3-72\,B\,a^3\,b\,d^2\,e-270\,A\,a^3\,b\,d\,e^2+84\,B\,a^2\,b^2\,d^3+378\,A\,a^2\,b^2\,d^2\,e-210\,A\,a\,b^3\,d^3\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{315\,e^3\,{\left (a\,e-b\,d\right )}^4}-\frac {8\,b^2\,x^3\,\left (a\,e-9\,b\,d\right )\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{315\,e^4\,{\left (a\,e-b\,d\right )}^4}+\frac {2\,b\,x^2\,\sqrt {a+b\,x}\,\left (a^2\,e^2-6\,a\,b\,d\,e+21\,b^2\,d^2\right )\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{105\,e^5\,{\left (a\,e-b\,d\right )}^4}\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(1/2))/(d + e*x)^(11/2),x)

[Out]

((d + e*x)^(1/2)*((x*(a + b*x)^(1/2)*(210*A*b^4*d^3 - 90*B*a^4*e^3 - 10*A*a^3*b*e^3 + 42*B*a*b^3*d^3 + 54*A*a^
2*b^2*d*e^2 - 414*B*a^2*b^2*d^2*e - 126*A*a*b^3*d^2*e + 334*B*a^3*b*d*e^2))/(315*e^5*(a*e - b*d)^4) - ((a + b*
x)^(1/2)*(70*A*a^4*e^3 - 210*A*a*b^3*d^3 + 20*B*a^4*d*e^2 + 84*B*a^2*b^2*d^3 + 378*A*a^2*b^2*d^2*e - 270*A*a^3
*b*d*e^2 - 72*B*a^3*b*d^2*e))/(315*e^5*(a*e - b*d)^4) + (16*b^3*x^4*(a + b*x)^(1/2)*(2*A*b*e - 3*B*a*e + B*b*d
))/(315*e^3*(a*e - b*d)^4) - (8*b^2*x^3*(a*e - 9*b*d)*(a + b*x)^(1/2)*(2*A*b*e - 3*B*a*e + B*b*d))/(315*e^4*(a
*e - b*d)^4) + (2*b*x^2*(a + b*x)^(1/2)*(a^2*e^2 + 21*b^2*d^2 - 6*a*b*d*e)*(2*A*b*e - 3*B*a*e + B*b*d))/(105*e
^5*(a*e - b*d)^4)))/(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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