3.19.47 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{\sqrt {d+e x}} \, dx\) [1847]

Optimal. Leaf size=306 \[ \frac {2 (b d-a e)^3 (B d-A e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)} \]

[Out]

-2/3*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b*d)*(e*x+d)^(3/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)+6/5*b*(-a*e+b*d)*(-A*b*
e-B*a*e+2*B*b*d)*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-2/7*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^(7/2)*((
b*x+a)^2)^(1/2)/e^5/(b*x+a)+2/9*b^3*B*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)+2*(-a*e+b*d)^3*(-A*e+B*d)*(e
*x+d)^(1/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {784, 78} \begin {gather*} -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3 (B d-A e)}{e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)) - (2*(b*d - a*e)^2*(
4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)) + (6*b*(b*d - a*e)
*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)) - (2*b^2*(4*b*B*d
- A*b*e - 3*a*B*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) + (2*b^3*B*(d + e*x)^(9/2)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 784

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3 (A+B x)}{\sqrt {d+e x}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (-B d+A e)}{e^4 \sqrt {d+e x}}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) \sqrt {d+e x}}{e^4}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{3/2}}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{5/2}}{e^4}+\frac {b^6 B (d+e x)^{7/2}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^3 (B d-A e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.16, size = 244, normalized size = 0.80 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \sqrt {d+e x} \left (105 a^3 e^3 (-2 B d+3 A e+B e x)+63 a^2 b e^2 \left (5 A e (-2 d+e x)+B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )-9 a b^2 e \left (-7 A e \left (8 d^2-4 d e x+3 e^2 x^2\right )+3 B \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )+b^3 \left (9 A e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+B \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )\right )}{315 e^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(105*a^3*e^3*(-2*B*d + 3*A*e + B*e*x) + 63*a^2*b*e^2*(5*A*e*(-2*d + e*x) +
B*(8*d^2 - 4*d*e*x + 3*e^2*x^2)) - 9*a*b^2*e*(-7*A*e*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 3*B*(16*d^3 - 8*d^2*e*x +
6*d*e^2*x^2 - 5*e^3*x^3)) + b^3*(9*A*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + B*(128*d^4 - 64*d^3*
e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4))))/(315*e^5*(a + b*x))

________________________________________________________________________________________

Maple [A]
time = 0.94, size = 317, normalized size = 1.04

method	result	size
gosper	\(\frac {2 \sqrt {e x +d}\, \left (35 B \,b^{3} x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 A \,a^{3} e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{315 e^{5} \left (b x +a \right )^{3}}\)	\(317\)
default	\(\frac {2 \sqrt {e x +d}\, \left (35 B \,b^{3} x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 A \,a^{3} e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{315 e^{5} \left (b x +a \right )^{3}}\)	\(317\)
risch	\(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (35 B \,b^{3} x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 A \,a^{3} e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \sqrt {e x +d}}{315 \left (b x +a \right ) e^{5}}\)	\(317\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/315*(e*x+d)^(1/2)*(35*B*b^3*e^4*x^4+45*A*b^3*e^4*x^3+135*B*a*b^2*e^4*x^3-40*B*b^3*d*e^3*x^3+189*A*a*b^2*e^4*
x^2-54*A*b^3*d*e^3*x^2+189*B*a^2*b*e^4*x^2-162*B*a*b^2*d*e^3*x^2+48*B*b^3*d^2*e^2*x^2+315*A*a^2*b*e^4*x-252*A*
a*b^2*d*e^3*x+72*A*b^3*d^2*e^2*x+105*B*a^3*e^4*x-252*B*a^2*b*d*e^3*x+216*B*a*b^2*d^2*e^2*x-64*B*b^3*d^3*e*x+31
5*A*a^3*e^4-630*A*a^2*b*d*e^3+504*A*a*b^2*d^2*e^2-144*A*b^3*d^3*e-210*B*a^3*d*e^3+504*B*a^2*b*d^2*e^2-432*B*a*
b^2*d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 360, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (5 \, b^{3} x^{4} e^{4} - 16 \, b^{3} d^{4} + 56 \, a b^{2} d^{3} e - 70 \, a^{2} b d^{2} e^{2} + 35 \, a^{3} d e^{3} - {\left (b^{3} d e^{3} - 21 \, a b^{2} e^{4}\right )} x^{3} + {\left (2 \, b^{3} d^{2} e^{2} - 7 \, a b^{2} d e^{3} + 35 \, a^{2} b e^{4}\right )} x^{2} - {\left (8 \, b^{3} d^{3} e - 28 \, a b^{2} d^{2} e^{2} + 35 \, a^{2} b d e^{3} - 35 \, a^{3} e^{4}\right )} x\right )} A e^{\left (-4\right )}}{35 \, \sqrt {x e + d}} + \frac {2 \, {\left (35 \, b^{3} x^{5} e^{5} + 128 \, b^{3} d^{5} - 432 \, a b^{2} d^{4} e + 504 \, a^{2} b d^{3} e^{2} - 210 \, a^{3} d^{2} e^{3} - 5 \, {\left (b^{3} d e^{4} - 27 \, a b^{2} e^{5}\right )} x^{4} + {\left (8 \, b^{3} d^{2} e^{3} - 27 \, a b^{2} d e^{4} + 189 \, a^{2} b e^{5}\right )} x^{3} - {\left (16 \, b^{3} d^{3} e^{2} - 54 \, a b^{2} d^{2} e^{3} + 63 \, a^{2} b d e^{4} - 105 \, a^{3} e^{5}\right )} x^{2} + {\left (64 \, b^{3} d^{4} e - 216 \, a b^{2} d^{3} e^{2} + 252 \, a^{2} b d^{2} e^{3} - 105 \, a^{3} d e^{4}\right )} x\right )} B e^{\left (-5\right )}}{315 \, \sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/35*(5*b^3*x^4*e^4 - 16*b^3*d^4 + 56*a*b^2*d^3*e - 70*a^2*b*d^2*e^2 + 35*a^3*d*e^3 - (b^3*d*e^3 - 21*a*b^2*e^
4)*x^3 + (2*b^3*d^2*e^2 - 7*a*b^2*d*e^3 + 35*a^2*b*e^4)*x^2 - (8*b^3*d^3*e - 28*a*b^2*d^2*e^2 + 35*a^2*b*d*e^3
- 35*a^3*e^4)*x)*A*e^(-4)/sqrt(x*e + d) + 2/315*(35*b^3*x^5*e^5 + 128*b^3*d^5 - 432*a*b^2*d^4*e + 504*a^2*b*d
^3*e^2 - 210*a^3*d^2*e^3 - 5*(b^3*d*e^4 - 27*a*b^2*e^5)*x^4 + (8*b^3*d^2*e^3 - 27*a*b^2*d*e^4 + 189*a^2*b*e^5)
*x^3 - (16*b^3*d^3*e^2 - 54*a*b^2*d^2*e^3 + 63*a^2*b*d*e^4 - 105*a^3*e^5)*x^2 + (64*b^3*d^4*e - 216*a*b^2*d^3*
e^2 + 252*a^2*b*d^2*e^3 - 105*a^3*d*e^4)*x)*B*e^(-5)/sqrt(x*e + d)

________________________________________________________________________________________

Fricas [A]
time = 1.44, size = 247, normalized size = 0.81 \begin {gather*} \frac {2}{315} \, {\left (128 \, B b^{3} d^{4} + {\left (35 \, B b^{3} x^{4} + 315 \, A a^{3} + 45 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 189 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} e^{4} - 2 \, {\left (20 \, B b^{3} d x^{3} + 27 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d x^{2} + 126 \, {\left (B a^{2} b + A a b^{2}\right )} d x + 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} e^{3} + 24 \, {\left (2 \, B b^{3} d^{2} x^{2} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} x + 21 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2}\right )} e^{2} - 16 \, {\left (4 \, B b^{3} d^{3} x + 9 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} e\right )} \sqrt {x e + d} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/315*(128*B*b^3*d^4 + (35*B*b^3*x^4 + 315*A*a^3 + 45*(3*B*a*b^2 + A*b^3)*x^3 + 189*(B*a^2*b + A*a*b^2)*x^2 +
105*(B*a^3 + 3*A*a^2*b)*x)*e^4 - 2*(20*B*b^3*d*x^3 + 27*(3*B*a*b^2 + A*b^3)*d*x^2 + 126*(B*a^2*b + A*a*b^2)*d*
x + 105*(B*a^3 + 3*A*a^2*b)*d)*e^3 + 24*(2*B*b^3*d^2*x^2 + 3*(3*B*a*b^2 + A*b^3)*d^2*x + 21*(B*a^2*b + A*a*b^2
)*d^2)*e^2 - 16*(4*B*b^3*d^3*x + 9*(3*B*a*b^2 + A*b^3)*d^3)*e)*sqrt(x*e + d)*e^(-5)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\sqrt {d + e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(1/2),x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/sqrt(d + e*x), x)

________________________________________________________________________________________

Giac [A]
time = 1.46, size = 393, normalized size = 1.28 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{3} e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 315 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{2} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 63 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a^{2} b e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 63 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a b^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 27 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a b^{2} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 9 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A b^{3} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B b^{3} e^{\left (-4\right )} \mathrm {sgn}\left (b x + a\right ) + 315 \, \sqrt {x e + d} A a^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/315*(105*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*e^(-1)*sgn(b*x + a) + 315*((x*e + d)^(3/2) - 3*sqrt(x*e
+ d)*d)*A*a^2*b*e^(-1)*sgn(b*x + a) + 63*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*
a^2*b*e^(-2)*sgn(b*x + a) + 63*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a*b^2*e^(-2
)*sgn(b*x + a) + 27*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)
*B*a*b^2*e^(-3)*sgn(b*x + a) + 9*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(
x*e + d)*d^3)*A*b^3*e^(-3)*sgn(b*x + a) + (35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^
2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*b^3*e^(-4)*sgn(b*x + a) + 315*sqrt(x*e + d)*A*a^3*sgn(b
*x + a))*e^(-1)

________________________________________________________________________________________

Mupad [B]
time = 2.74, size = 434, normalized size = 1.42 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {-420\,B\,a^3\,d^2\,e^3+630\,A\,a^3\,d\,e^4+1008\,B\,a^2\,b\,d^3\,e^2-1260\,A\,a^2\,b\,d^2\,e^3-864\,B\,a\,b^2\,d^4\,e+1008\,A\,a\,b^2\,d^3\,e^2+256\,B\,b^3\,d^5-288\,A\,b^3\,d^4\,e}{315\,b\,e^5}+\frac {2\,B\,b^2\,x^5}{9}+\frac {x^3\,\left (378\,B\,a^2\,b\,e^5-54\,B\,a\,b^2\,d\,e^4+378\,A\,a\,b^2\,e^5+16\,B\,b^3\,d^2\,e^3-18\,A\,b^3\,d\,e^4\right )}{315\,b\,e^5}+\frac {x\,\left (-210\,B\,a^3\,d\,e^4+630\,A\,a^3\,e^5+504\,B\,a^2\,b\,d^2\,e^3-630\,A\,a^2\,b\,d\,e^4-432\,B\,a\,b^2\,d^3\,e^2+504\,A\,a\,b^2\,d^2\,e^3+128\,B\,b^3\,d^4\,e-144\,A\,b^3\,d^3\,e^2\right )}{315\,b\,e^5}+\frac {x^2\,\left (210\,B\,a^3\,e^5-126\,B\,a^2\,b\,d\,e^4+630\,A\,a^2\,b\,e^5+108\,B\,a\,b^2\,d^2\,e^3-126\,A\,a\,b^2\,d\,e^4-32\,B\,b^3\,d^3\,e^2+36\,A\,b^3\,d^2\,e^3\right )}{315\,b\,e^5}+\frac {2\,b\,x^4\,\left (9\,A\,b\,e+27\,B\,a\,e-B\,b\,d\right )}{63\,e}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(d + e*x)^(1/2),x)

[Out]

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((256*B*b^3*d^5 + 630*A*a^3*d*e^4 - 288*A*b^3*d^4*e - 420*B*a^3*d^2*e^3 + 100
8*A*a*b^2*d^3*e^2 - 1260*A*a^2*b*d^2*e^3 + 1008*B*a^2*b*d^3*e^2 - 864*B*a*b^2*d^4*e)/(315*b*e^5) + (2*B*b^2*x^
5)/9 + (x^3*(378*A*a*b^2*e^5 + 378*B*a^2*b*e^5 - 18*A*b^3*d*e^4 + 16*B*b^3*d^2*e^3 - 54*B*a*b^2*d*e^4))/(315*b
*e^5) + (x*(630*A*a^3*e^5 - 210*B*a^3*d*e^4 + 128*B*b^3*d^4*e - 144*A*b^3*d^3*e^2 + 504*A*a*b^2*d^2*e^3 - 432*
B*a*b^2*d^3*e^2 + 504*B*a^2*b*d^2*e^3 - 630*A*a^2*b*d*e^4))/(315*b*e^5) + (x^2*(210*B*a^3*e^5 + 630*A*a^2*b*e^
5 + 36*A*b^3*d^2*e^3 - 32*B*b^3*d^3*e^2 + 108*B*a*b^2*d^2*e^3 - 126*A*a*b^2*d*e^4 - 126*B*a^2*b*d*e^4))/(315*b
*e^5) + (2*b*x^4*(9*A*b*e + 27*B*a*e - B*b*d))/(63*e)))/(x*(d + e*x)^(1/2) + (a*(d + e*x)^(1/2))/b)

________________________________________________________________________________________