3.1855 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=448 \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-5 a B e-A b e+6 b B d)}{11 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{9 e^7 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{7 e^7 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5 (B d-A e)}{e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (2*(b*d - a*e)^4*(
6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (2*b*(b*d - a*e)
^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (20*b^2*(b*d -
a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) + (10*b^3*(
b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) - (2*b
^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) + (2*b^5*B*(
d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x))

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Rubi [A]  time = 0.217931, antiderivative size = 448, normalized size of antiderivative = 1., 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 = {770, 77} \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-5 a B e-A b e+6 b B d)}{11 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{9 e^7 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{7 e^7 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5 (B d-A e)}{e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/Sqrt[d + e*x],x]

[Out]

(2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (2*(b*d - a*e)^4*(
6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (2*b*(b*d - a*e)
^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (20*b^2*(b*d -
a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) + (10*b^3*(
b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) - (2*b
^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) + (2*b^5*B*(
d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x))

Rule 770

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]

Rule 77

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]))))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{\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 )^5 (A+B x)}{\sqrt{d+e x}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^5 (b d-a e)^5 (-B d+A e)}{e^6 \sqrt{d+e x}}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e) \sqrt{d+e x}}{e^6}-\frac{5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e) (d+e x)^{3/2}}{e^6}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{5/2}}{e^6}-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{7/2}}{e^6}+\frac{b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{9/2}}{e^6}+\frac{b^{10} B (d+e x)^{11/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{2 (b d-a e)^5 (B d-A e) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac{2 (b d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac{2 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac{20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}+\frac{10 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}-\frac{2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac{2 b^5 B (d+e x)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.193495, size = 239, normalized size = 0.53 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (-819 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+5005 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-12870 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+9009 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-3003 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)+9009 (b d-a e)^5 (B d-A e)+693 b^5 B (d+e x)^6\right )}{9009 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(9009*(b*d - a*e)^5*(B*d - A*e) - 3003*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a
*B*e)*(d + e*x) + 9009*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 12870*b^2*(b*d - a*e)^2*(2*b*
B*d - A*b*e - a*B*e)*(d + e*x)^3 + 5005*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 - 819*b^4*(6*b
*B*d - A*b*e - 5*a*B*e)*(d + e*x)^5 + 693*b^5*B*(d + e*x)^6))/(9009*e^7*(a + b*x))

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Maple [A]  time = 0.009, size = 689, normalized size = 1.5 \begin{align*}{\frac{1386\,B{x}^{6}{b}^{5}{e}^{6}+1638\,A{x}^{5}{b}^{5}{e}^{6}+8190\,B{x}^{5}a{b}^{4}{e}^{6}-1512\,B{x}^{5}{b}^{5}d{e}^{5}+10010\,A{x}^{4}a{b}^{4}{e}^{6}-1820\,A{x}^{4}{b}^{5}d{e}^{5}+20020\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}-9100\,B{x}^{4}a{b}^{4}d{e}^{5}+1680\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+25740\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}-11440\,A{x}^{3}a{b}^{4}d{e}^{5}+2080\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+25740\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}-22880\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+10400\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}-1920\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+36036\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}-30888\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+13728\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-2496\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+18018\,B{x}^{2}{a}^{4}b{e}^{6}-30888\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+27456\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-12480\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+2304\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+30030\,Ax{a}^{4}b{e}^{6}-48048\,Ax{a}^{3}{b}^{2}d{e}^{5}+41184\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}-18304\,Axa{b}^{4}{d}^{3}{e}^{3}+3328\,Ax{b}^{5}{d}^{4}{e}^{2}+6006\,Bx{a}^{5}{e}^{6}-24024\,Bx{a}^{4}bd{e}^{5}+41184\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}-36608\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+16640\,Bxa{b}^{4}{d}^{4}{e}^{2}-3072\,Bx{b}^{5}{d}^{5}e+18018\,A{a}^{5}{e}^{6}-60060\,Ad{e}^{5}{a}^{4}b+96096\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-82368\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+36608\,Aa{b}^{4}{d}^{4}{e}^{2}-6656\,A{b}^{5}{d}^{5}e-12012\,Bd{e}^{5}{a}^{5}+48048\,B{a}^{4}b{d}^{2}{e}^{4}-82368\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+73216\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-33280\,Ba{b}^{4}{d}^{5}e+6144\,B{b}^{5}{d}^{6}}{9009\, \left ( bx+a \right ) ^{5}{e}^{7}}\sqrt{ex+d} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

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)^(5/2)/(e*x+d)^(1/2),x)

[Out]

2/9009*(e*x+d)^(1/2)*(693*B*b^5*e^6*x^6+819*A*b^5*e^6*x^5+4095*B*a*b^4*e^6*x^5-756*B*b^5*d*e^5*x^5+5005*A*a*b^
4*e^6*x^4-910*A*b^5*d*e^5*x^4+10010*B*a^2*b^3*e^6*x^4-4550*B*a*b^4*d*e^5*x^4+840*B*b^5*d^2*e^4*x^4+12870*A*a^2
*b^3*e^6*x^3-5720*A*a*b^4*d*e^5*x^3+1040*A*b^5*d^2*e^4*x^3+12870*B*a^3*b^2*e^6*x^3-11440*B*a^2*b^3*d*e^5*x^3+5
200*B*a*b^4*d^2*e^4*x^3-960*B*b^5*d^3*e^3*x^3+18018*A*a^3*b^2*e^6*x^2-15444*A*a^2*b^3*d*e^5*x^2+6864*A*a*b^4*d
^2*e^4*x^2-1248*A*b^5*d^3*e^3*x^2+9009*B*a^4*b*e^6*x^2-15444*B*a^3*b^2*d*e^5*x^2+13728*B*a^2*b^3*d^2*e^4*x^2-6
240*B*a*b^4*d^3*e^3*x^2+1152*B*b^5*d^4*e^2*x^2+15015*A*a^4*b*e^6*x-24024*A*a^3*b^2*d*e^5*x+20592*A*a^2*b^3*d^2
*e^4*x-9152*A*a*b^4*d^3*e^3*x+1664*A*b^5*d^4*e^2*x+3003*B*a^5*e^6*x-12012*B*a^4*b*d*e^5*x+20592*B*a^3*b^2*d^2*
e^4*x-18304*B*a^2*b^3*d^3*e^3*x+8320*B*a*b^4*d^4*e^2*x-1536*B*b^5*d^5*e*x+9009*A*a^5*e^6-30030*A*a^4*b*d*e^5+4
8048*A*a^3*b^2*d^2*e^4-41184*A*a^2*b^3*d^3*e^3+18304*A*a*b^4*d^4*e^2-3328*A*b^5*d^5*e-6006*B*a^5*d*e^5+24024*B
*a^4*b*d^2*e^4-41184*B*a^3*b^2*d^3*e^3+36608*B*a^2*b^3*d^4*e^2-16640*B*a*b^4*d^5*e+3072*B*b^5*d^6)*((b*x+a)^2)
^(5/2)/e^7/(b*x+a)^5

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Maxima [B]  time = 1.1389, size = 1023, normalized size = 2.28 \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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/693*(63*b^5*e^6*x^6 - 256*b^5*d^6 + 1408*a*b^4*d^5*e - 3168*a^2*b^3*d^4*e^2 + 3696*a^3*b^2*d^3*e^3 - 2310*a^
4*b*d^2*e^4 + 693*a^5*d*e^5 - 7*(b^5*d*e^5 - 55*a*b^4*e^6)*x^5 + 5*(2*b^5*d^2*e^4 - 11*a*b^4*d*e^5 + 198*a^2*b
^3*e^6)*x^4 - 2*(8*b^5*d^3*e^3 - 44*a*b^4*d^2*e^4 + 99*a^2*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (32*b^5*d^4*e^2
- 176*a*b^4*d^3*e^3 + 396*a^2*b^3*d^2*e^4 - 462*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (128*b^5*d^5*e - 704*a*b
^4*d^4*e^2 + 1584*a^2*b^3*d^3*e^3 - 1848*a^3*b^2*d^2*e^4 + 1155*a^4*b*d*e^5 - 693*a^5*e^6)*x)*A/(sqrt(e*x + d)
*e^6) + 2/9009*(693*b^5*e^7*x^7 + 3072*b^5*d^7 - 16640*a*b^4*d^6*e + 36608*a^2*b^3*d^5*e^2 - 41184*a^3*b^2*d^4
*e^3 + 24024*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 - 63*(b^5*d*e^6 - 65*a*b^4*e^7)*x^6 + 7*(12*b^5*d^2*e^5 - 65*a*b
^4*d*e^6 + 1430*a^2*b^3*e^7)*x^5 - 10*(12*b^5*d^3*e^4 - 65*a*b^4*d^2*e^5 + 143*a^2*b^3*d*e^6 - 1287*a^3*b^2*e^
7)*x^4 + (192*b^5*d^4*e^3 - 1040*a*b^4*d^3*e^4 + 2288*a^2*b^3*d^2*e^5 - 2574*a^3*b^2*d*e^6 + 9009*a^4*b*e^7)*x
^3 - (384*b^5*d^5*e^2 - 2080*a*b^4*d^4*e^3 + 4576*a^2*b^3*d^3*e^4 - 5148*a^3*b^2*d^2*e^5 + 3003*a^4*b*d*e^6 -
3003*a^5*e^7)*x^2 + (1536*b^5*d^6*e - 8320*a*b^4*d^5*e^2 + 18304*a^2*b^3*d^4*e^3 - 20592*a^3*b^2*d^3*e^4 + 120
12*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*B/(sqrt(e*x + d)*e^7)

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Fricas [A]  time = 1.43648, size = 1250, normalized size = 2.79 \begin{align*} \frac{2 \,{\left (693 \, B b^{5} e^{6} x^{6} + 3072 \, B b^{5} d^{6} + 9009 \, A a^{5} e^{6} - 3328 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 18304 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 41184 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 24024 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 6006 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 63 \,{\left (12 \, B b^{5} d e^{5} - 13 \,{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 35 \,{\left (24 \, B b^{5} d^{2} e^{4} - 26 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 143 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 10 \,{\left (96 \, B b^{5} d^{3} e^{3} - 104 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 572 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 1287 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 3 \,{\left (384 \, B b^{5} d^{4} e^{2} - 416 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 2288 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 5148 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 3003 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} -{\left (1536 \, B b^{5} d^{5} e - 1664 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 9152 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 20592 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 12012 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 3003 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt{e x + d}}{9009 \, e^{7}} \end{align*}

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)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/9009*(693*B*b^5*e^6*x^6 + 3072*B*b^5*d^6 + 9009*A*a^5*e^6 - 3328*(5*B*a*b^4 + A*b^5)*d^5*e + 18304*(2*B*a^2*
b^3 + A*a*b^4)*d^4*e^2 - 41184*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 24024*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - 6006*
(B*a^5 + 5*A*a^4*b)*d*e^5 - 63*(12*B*b^5*d*e^5 - 13*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 35*(24*B*b^5*d^2*e^4 - 26*(
5*B*a*b^4 + A*b^5)*d*e^5 + 143*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 10*(96*B*b^5*d^3*e^3 - 104*(5*B*a*b^4 + A*b^
5)*d^2*e^4 + 572*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 - 1287*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 3*(384*B*b^5*d^4*e^2
- 416*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 2288*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 5148*(B*a^3*b^2 + A*a^2*b^3)*d*e^5
+ 3003*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 - (1536*B*b^5*d^5*e - 1664*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 9152*(2*B*a^2
*b^3 + A*a*b^4)*d^3*e^3 - 20592*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 12012*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - 3003*(
B*a^5 + 5*A*a^4*b)*e^6)*x)*sqrt(e*x + d)/e^7

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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)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.24975, size = 1023, normalized size = 2.28 \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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/9009*(3003*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^5*e^(-1)*sgn(b*x + a) + 15015*((x*e + d)^(3/2) - 3*sqrt
(x*e + d)*d)*A*a^4*b*e^(-1)*sgn(b*x + a) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d
^2)*B*a^4*b*e^(-2)*sgn(b*x + a) + 6006*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^3
*b^2*e^(-2)*sgn(b*x + a) + 2574*(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^3*b^2*e^(-3)*sgn(b*x + a) + 2574*(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*a^2*b^3*e^(-3)*sgn(b*x + a) + 286*(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*a^2*b^3*e^(-4)*sgn(b*x + a) +
143*(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)*A*a*b^4*e^(-4)*sgn(b*x + a) + 65*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(
7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a*b^4*e^(-5)*sgn(b*x
+ a) + 13*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 +
1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*b^5*e^(-5)*sgn(b*x + a) + 3*(231*(x*e + d)^(13/2) - 1638*
(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*
e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*b^5*e^(-6)*sgn(b*x + a) + 9009*sqrt(x*e + d)*A*a^5*sgn(b*x + a))*
e^(-1)