3.1856 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=446 \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-5 a B e-A b e+6 b B d)}{9 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (a+b x)}-\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{e^7 (a+b x) \sqrt{d+e x}}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^7 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.216317, antiderivative size = 446, 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} \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-5 a B e-A b e+6 b B d)}{9 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (a+b x)}-\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{e^7 (a+b x) \sqrt{d+e x}}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(3/2),x]

[Out]

(-2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) - (2*(b*d - a*e)^4*
(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (10*b*(b*d - a*e)^3
*(3*b*B*d - 2*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)) - (4*b^2*(b*d -
a*e)^2*(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])/(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)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (2*b^4*
(6*b*B*d - A*b*e - 5*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^5*B*(d + e
*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*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}}{(d+e x)^{3/2}} \, 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)}{(d+e x)^{3/2}} \, 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 (d+e x)^{3/2}}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 \sqrt{d+e x}}-\frac{5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e) \sqrt{d+e x}}{e^6}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{3/2}}{e^6}-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{5/2}}{e^6}+\frac{b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{7/2}}{e^6}+\frac{b^{10} B (d+e x)^{9/2}}{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{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt{d+e x}}-\frac{2 (b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac{10 b (b d-a e)^3 (3 b B d-2 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{4 b^2 (b d-a e)^2 (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}}{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)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac{2 b^4 (6 b B d-A b e-5 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^5 B (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.186555, size = 239, normalized size = 0.54 \[ \frac{2 \sqrt{(a+b x)^2} \left (-77 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+495 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-1386 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+1155 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-693 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)-693 (b d-a e)^5 (B d-A e)+63 b^5 B (d+e x)^6\right )}{693 e^7 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(3/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-693*(b*d - a*e)^5*(B*d - A*e) - 693*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)
+ 1155*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 1386*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*
B*e)*(d + e*x)^3 + 495*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 - 77*b^4*(6*b*B*d - A*b*e - 5*a
*B*e)*(d + e*x)^5 + 63*b^5*B*(d + e*x)^6))/(693*e^7*(a + b*x)*Sqrt[d + e*x])

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Maple [A]  time = 0.009, size = 689, normalized size = 1.5 \begin{align*} -{\frac{-126\,B{x}^{6}{b}^{5}{e}^{6}-154\,A{x}^{5}{b}^{5}{e}^{6}-770\,B{x}^{5}a{b}^{4}{e}^{6}+168\,B{x}^{5}{b}^{5}d{e}^{5}-990\,A{x}^{4}a{b}^{4}{e}^{6}+220\,A{x}^{4}{b}^{5}d{e}^{5}-1980\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+1100\,B{x}^{4}a{b}^{4}d{e}^{5}-240\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}-2772\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+1584\,A{x}^{3}a{b}^{4}d{e}^{5}-352\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}-2772\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+3168\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}-1760\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+384\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}-4620\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+5544\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}-3168\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+704\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}-2310\,B{x}^{2}{a}^{4}b{e}^{6}+5544\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}-6336\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+3520\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}-768\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}-6930\,Ax{a}^{4}b{e}^{6}+18480\,Ax{a}^{3}{b}^{2}d{e}^{5}-22176\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+12672\,Axa{b}^{4}{d}^{3}{e}^{3}-2816\,Ax{b}^{5}{d}^{4}{e}^{2}-1386\,Bx{a}^{5}{e}^{6}+9240\,Bx{a}^{4}bd{e}^{5}-22176\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+25344\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}-14080\,Bxa{b}^{4}{d}^{4}{e}^{2}+3072\,Bx{b}^{5}{d}^{5}e+1386\,A{a}^{5}{e}^{6}-13860\,Ad{e}^{5}{a}^{4}b+36960\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-44352\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+25344\,Aa{b}^{4}{d}^{4}{e}^{2}-5632\,A{b}^{5}{d}^{5}e-2772\,Bd{e}^{5}{a}^{5}+18480\,B{a}^{4}b{d}^{2}{e}^{4}-44352\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+50688\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-28160\,Ba{b}^{4}{d}^{5}e+6144\,B{b}^{5}{d}^{6}}{693\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{ex+d}}}} \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)^(3/2),x)

[Out]

-2/693/(e*x+d)^(1/2)*(-63*B*b^5*e^6*x^6-77*A*b^5*e^6*x^5-385*B*a*b^4*e^6*x^5+84*B*b^5*d*e^5*x^5-495*A*a*b^4*e^
6*x^4+110*A*b^5*d*e^5*x^4-990*B*a^2*b^3*e^6*x^4+550*B*a*b^4*d*e^5*x^4-120*B*b^5*d^2*e^4*x^4-1386*A*a^2*b^3*e^6
*x^3+792*A*a*b^4*d*e^5*x^3-176*A*b^5*d^2*e^4*x^3-1386*B*a^3*b^2*e^6*x^3+1584*B*a^2*b^3*d*e^5*x^3-880*B*a*b^4*d
^2*e^4*x^3+192*B*b^5*d^3*e^3*x^3-2310*A*a^3*b^2*e^6*x^2+2772*A*a^2*b^3*d*e^5*x^2-1584*A*a*b^4*d^2*e^4*x^2+352*
A*b^5*d^3*e^3*x^2-1155*B*a^4*b*e^6*x^2+2772*B*a^3*b^2*d*e^5*x^2-3168*B*a^2*b^3*d^2*e^4*x^2+1760*B*a*b^4*d^3*e^
3*x^2-384*B*b^5*d^4*e^2*x^2-3465*A*a^4*b*e^6*x+9240*A*a^3*b^2*d*e^5*x-11088*A*a^2*b^3*d^2*e^4*x+6336*A*a*b^4*d
^3*e^3*x-1408*A*b^5*d^4*e^2*x-693*B*a^5*e^6*x+4620*B*a^4*b*d*e^5*x-11088*B*a^3*b^2*d^2*e^4*x+12672*B*a^2*b^3*d
^3*e^3*x-7040*B*a*b^4*d^4*e^2*x+1536*B*b^5*d^5*e*x+693*A*a^5*e^6-6930*A*a^4*b*d*e^5+18480*A*a^3*b^2*d^2*e^4-22
176*A*a^2*b^3*d^3*e^3+12672*A*a*b^4*d^4*e^2-2816*A*b^5*d^5*e-1386*B*a^5*d*e^5+9240*B*a^4*b*d^2*e^4-22176*B*a^3
*b^2*d^3*e^3+25344*B*a^2*b^3*d^4*e^2-14080*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 [A]  time = 1.0979, size = 814, normalized size = 1.83 \begin{align*} \frac{2 \,{\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \,{\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} +{\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} A}{63 \, \sqrt{e x + d} e^{6}} + \frac{2 \,{\left (63 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 14080 \, a b^{4} d^{5} e - 25344 \, a^{2} b^{3} d^{4} e^{2} + 22176 \, a^{3} b^{2} d^{3} e^{3} - 9240 \, a^{4} b d^{2} e^{4} + 1386 \, a^{5} d e^{5} - 7 \,{\left (12 \, b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 10 \,{\left (12 \, b^{5} d^{2} e^{4} - 55 \, a b^{4} d e^{5} + 99 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \,{\left (96 \, b^{5} d^{3} e^{3} - 440 \, a b^{4} d^{2} e^{4} + 792 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} +{\left (384 \, b^{5} d^{4} e^{2} - 1760 \, a b^{4} d^{3} e^{3} + 3168 \, a^{2} b^{3} d^{2} e^{4} - 2772 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} -{\left (1536 \, b^{5} d^{5} e - 7040 \, a b^{4} d^{4} e^{2} + 12672 \, a^{2} b^{3} d^{3} e^{3} - 11088 \, a^{3} b^{2} d^{2} e^{4} + 4620 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )} B}{693 \, \sqrt{e x + d} 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)^(3/2),x, algorithm="maxima")

[Out]

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^2 - 1680*a^3*b^2*d^2*e^3 + 630*a^4*b
*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*
x^3 - 2*(16*b^5*d^3*e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e - 576*a
*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 840*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*A/(sqrt(e*x + d)*e^6) + 2/693*(63*
b^5*e^6*x^6 - 3072*b^5*d^6 + 14080*a*b^4*d^5*e - 25344*a^2*b^3*d^4*e^2 + 22176*a^3*b^2*d^3*e^3 - 9240*a^4*b*d^
2*e^4 + 1386*a^5*d*e^5 - 7*(12*b^5*d*e^5 - 55*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2*e^4 - 55*a*b^4*d*e^5 + 99*a^2*b^
3*e^6)*x^4 - 2*(96*b^5*d^3*e^3 - 440*a*b^4*d^2*e^4 + 792*a^2*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (384*b^5*d^4*e
^2 - 1760*a*b^4*d^3*e^3 + 3168*a^2*b^3*d^2*e^4 - 2772*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (1536*b^5*d^5*e -
7040*a*b^4*d^4*e^2 + 12672*a^2*b^3*d^3*e^3 - 11088*a^3*b^2*d^2*e^4 + 4620*a^4*b*d*e^5 - 693*a^5*e^6)*x)*B/(sqr
t(e*x + d)*e^7)

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Fricas [A]  time = 1.30827, size = 1247, normalized size = 2.8 \begin{align*} \frac{2 \,{\left (63 \, B b^{5} e^{6} x^{6} - 3072 \, B b^{5} d^{6} - 693 \, A a^{5} e^{6} + 2816 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 12672 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 22176 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 9240 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 1386 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 7 \,{\left (12 \, B b^{5} d e^{5} - 11 \,{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \,{\left (24 \, B b^{5} d^{2} e^{4} - 22 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 99 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 2 \,{\left (96 \, B b^{5} d^{3} e^{3} - 88 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 396 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 693 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} +{\left (384 \, B b^{5} d^{4} e^{2} - 352 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 1584 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 2772 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 1155 \,{\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 - 1408 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 6336 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 11088 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 4620 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 693 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt{e x + d}}{693 \,{\left (e^{8} x + d e^{7}\right )}} \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)^(3/2),x, algorithm="fricas")

[Out]

2/693*(63*B*b^5*e^6*x^6 - 3072*B*b^5*d^6 - 693*A*a^5*e^6 + 2816*(5*B*a*b^4 + A*b^5)*d^5*e - 12672*(2*B*a^2*b^3
+ A*a*b^4)*d^4*e^2 + 22176*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 - 9240*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 1386*(B*a
^5 + 5*A*a^4*b)*d*e^5 - 7*(12*B*b^5*d*e^5 - 11*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(24*B*b^5*d^2*e^4 - 22*(5*B*a*
b^4 + A*b^5)*d*e^5 + 99*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 2*(96*B*b^5*d^3*e^3 - 88*(5*B*a*b^4 + A*b^5)*d^2*e^
4 + 396*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 - 693*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + (384*B*b^5*d^4*e^2 - 352*(5*B*a
*b^4 + A*b^5)*d^3*e^3 + 1584*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 2772*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 1155*(B*a^
4*b + 2*A*a^3*b^2)*e^6)*x^2 - (1536*B*b^5*d^5*e - 1408*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 6336*(2*B*a^2*b^3 + A*a*b
^4)*d^3*e^3 - 11088*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 4620*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - 693*(B*a^5 + 5*A*a^
4*b)*e^6)*x)*sqrt(e*x + d)/(e^8*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)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.23194, size = 1519, normalized size = 3.41 \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)^(3/2),x, algorithm="giac")

[Out]

2/693*(63*(x*e + d)^(11/2)*B*b^5*e^70*sgn(b*x + a) - 462*(x*e + d)^(9/2)*B*b^5*d*e^70*sgn(b*x + a) + 1485*(x*e
+ d)^(7/2)*B*b^5*d^2*e^70*sgn(b*x + a) - 2772*(x*e + d)^(5/2)*B*b^5*d^3*e^70*sgn(b*x + a) + 3465*(x*e + d)^(3
/2)*B*b^5*d^4*e^70*sgn(b*x + a) - 4158*sqrt(x*e + d)*B*b^5*d^5*e^70*sgn(b*x + a) + 385*(x*e + d)^(9/2)*B*a*b^4
*e^71*sgn(b*x + a) + 77*(x*e + d)^(9/2)*A*b^5*e^71*sgn(b*x + a) - 2475*(x*e + d)^(7/2)*B*a*b^4*d*e^71*sgn(b*x
+ a) - 495*(x*e + d)^(7/2)*A*b^5*d*e^71*sgn(b*x + a) + 6930*(x*e + d)^(5/2)*B*a*b^4*d^2*e^71*sgn(b*x + a) + 13
86*(x*e + d)^(5/2)*A*b^5*d^2*e^71*sgn(b*x + a) - 11550*(x*e + d)^(3/2)*B*a*b^4*d^3*e^71*sgn(b*x + a) - 2310*(x
*e + d)^(3/2)*A*b^5*d^3*e^71*sgn(b*x + a) + 17325*sqrt(x*e + d)*B*a*b^4*d^4*e^71*sgn(b*x + a) + 3465*sqrt(x*e
+ d)*A*b^5*d^4*e^71*sgn(b*x + a) + 990*(x*e + d)^(7/2)*B*a^2*b^3*e^72*sgn(b*x + a) + 495*(x*e + d)^(7/2)*A*a*b
^4*e^72*sgn(b*x + a) - 5544*(x*e + d)^(5/2)*B*a^2*b^3*d*e^72*sgn(b*x + a) - 2772*(x*e + d)^(5/2)*A*a*b^4*d*e^7
2*sgn(b*x + a) + 13860*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^72*sgn(b*x + a) + 6930*(x*e + d)^(3/2)*A*a*b^4*d^2*e^72
*sgn(b*x + a) - 27720*sqrt(x*e + d)*B*a^2*b^3*d^3*e^72*sgn(b*x + a) - 13860*sqrt(x*e + d)*A*a*b^4*d^3*e^72*sgn
(b*x + a) + 1386*(x*e + d)^(5/2)*B*a^3*b^2*e^73*sgn(b*x + a) + 1386*(x*e + d)^(5/2)*A*a^2*b^3*e^73*sgn(b*x + a
) - 6930*(x*e + d)^(3/2)*B*a^3*b^2*d*e^73*sgn(b*x + a) - 6930*(x*e + d)^(3/2)*A*a^2*b^3*d*e^73*sgn(b*x + a) +
20790*sqrt(x*e + d)*B*a^3*b^2*d^2*e^73*sgn(b*x + a) + 20790*sqrt(x*e + d)*A*a^2*b^3*d^2*e^73*sgn(b*x + a) + 11
55*(x*e + d)^(3/2)*B*a^4*b*e^74*sgn(b*x + a) + 2310*(x*e + d)^(3/2)*A*a^3*b^2*e^74*sgn(b*x + a) - 6930*sqrt(x*
e + d)*B*a^4*b*d*e^74*sgn(b*x + a) - 13860*sqrt(x*e + d)*A*a^3*b^2*d*e^74*sgn(b*x + a) + 693*sqrt(x*e + d)*B*a
^5*e^75*sgn(b*x + a) + 3465*sqrt(x*e + d)*A*a^4*b*e^75*sgn(b*x + a))*e^(-77) - 2*(B*b^5*d^6*sgn(b*x + a) - 5*B
*a*b^4*d^5*e*sgn(b*x + a) - A*b^5*d^5*e*sgn(b*x + a) + 10*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 5*A*a*b^4*d^4*e^2*s
gn(b*x + a) - 10*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 10*A*a^2*b^3*d^3*e^3*sgn(b*x + a) + 5*B*a^4*b*d^2*e^4*sgn(b*
x + a) + 10*A*a^3*b^2*d^2*e^4*sgn(b*x + a) - B*a^5*d*e^5*sgn(b*x + a) - 5*A*a^4*b*d*e^5*sgn(b*x + a) + A*a^5*e
^6*sgn(b*x + a))*e^(-7)/sqrt(x*e + d)