3.1853 \(\int (A+B x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=452 \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (-5 a B e-A b e+6 b B d)}{15 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{13 e^7 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{11 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{7 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*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)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*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)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*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)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*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)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b
*x)) - (2*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x)) +
(2*b^5*B*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*e^7*(a + b*x))

________________________________________________________________________________________

Rubi [A]  time = 0.216829, antiderivative size = 452, 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} \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (-5 a B e-A b e+6 b B d)}{15 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{13 e^7 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{11 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{7 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*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)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*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)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*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)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*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)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b
*x)) - (2*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x)) +
(2*b^5*B*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*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 (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \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) (d+e x)^{3/2}}{e^6}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e) (d+e x)^{5/2}}{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)^{7/2}}{e^6}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^6}-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{11/2}}{e^6}+\frac{b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{13/2}}{e^6}+\frac{b^{10} B (d+e x)^{15/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{2 (b d-a e)^5 (B d-A e) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 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)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 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)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 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)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 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)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}-\frac{2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)}+\frac{2 b^5 B (d+e x)^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 e^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.285872, size = 239, normalized size = 0.53 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{5/2} \left (-51051 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+294525 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-696150 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+425425 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-109395 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)+153153 (b d-a e)^5 (B d-A e)+45045 b^5 B (d+e x)^6\right )}{765765 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(153153*(b*d - a*e)^5*(B*d - A*e) - 109395*(b*d - a*e)^4*(6*b*B*d - 5*A*b
*e - a*B*e)*(d + e*x) + 425425*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 696150*b^2*(b*d - a*e
)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^3 + 294525*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 - 5
1051*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^5 + 45045*b^5*B*(d + e*x)^6))/(765765*e^7*(a + b*x))

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Maple [A]  time = 0.01, size = 689, normalized size = 1.5 \begin{align*}{\frac{90090\,B{x}^{6}{b}^{5}{e}^{6}+102102\,A{x}^{5}{b}^{5}{e}^{6}+510510\,B{x}^{5}a{b}^{4}{e}^{6}-72072\,B{x}^{5}{b}^{5}d{e}^{5}+589050\,A{x}^{4}a{b}^{4}{e}^{6}-78540\,A{x}^{4}{b}^{5}d{e}^{5}+1178100\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}-392700\,B{x}^{4}a{b}^{4}d{e}^{5}+55440\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+1392300\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}-428400\,A{x}^{3}a{b}^{4}d{e}^{5}+57120\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+1392300\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}-856800\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+285600\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}-40320\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+1701700\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}-928200\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+285600\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-38080\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+850850\,B{x}^{2}{a}^{4}b{e}^{6}-928200\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+571200\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-190400\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+26880\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+1093950\,Ax{a}^{4}b{e}^{6}-972400\,Ax{a}^{3}{b}^{2}d{e}^{5}+530400\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}-163200\,Axa{b}^{4}{d}^{3}{e}^{3}+21760\,Ax{b}^{5}{d}^{4}{e}^{2}+218790\,Bx{a}^{5}{e}^{6}-486200\,Bx{a}^{4}bd{e}^{5}+530400\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}-326400\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+108800\,Bxa{b}^{4}{d}^{4}{e}^{2}-15360\,Bx{b}^{5}{d}^{5}e+306306\,A{a}^{5}{e}^{6}-437580\,Ad{e}^{5}{a}^{4}b+388960\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-212160\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+65280\,Aa{b}^{4}{d}^{4}{e}^{2}-8704\,A{b}^{5}{d}^{5}e-87516\,Bd{e}^{5}{a}^{5}+194480\,B{a}^{4}b{d}^{2}{e}^{4}-212160\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+130560\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-43520\,Ba{b}^{4}{d}^{5}e+6144\,B{b}^{5}{d}^{6}}{765765\,{e}^{7} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \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)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

2/765765*(e*x+d)^(5/2)*(45045*B*b^5*e^6*x^6+51051*A*b^5*e^6*x^5+255255*B*a*b^4*e^6*x^5-36036*B*b^5*d*e^5*x^5+2
94525*A*a*b^4*e^6*x^4-39270*A*b^5*d*e^5*x^4+589050*B*a^2*b^3*e^6*x^4-196350*B*a*b^4*d*e^5*x^4+27720*B*b^5*d^2*
e^4*x^4+696150*A*a^2*b^3*e^6*x^3-214200*A*a*b^4*d*e^5*x^3+28560*A*b^5*d^2*e^4*x^3+696150*B*a^3*b^2*e^6*x^3-428
400*B*a^2*b^3*d*e^5*x^3+142800*B*a*b^4*d^2*e^4*x^3-20160*B*b^5*d^3*e^3*x^3+850850*A*a^3*b^2*e^6*x^2-464100*A*a
^2*b^3*d*e^5*x^2+142800*A*a*b^4*d^2*e^4*x^2-19040*A*b^5*d^3*e^3*x^2+425425*B*a^4*b*e^6*x^2-464100*B*a^3*b^2*d*
e^5*x^2+285600*B*a^2*b^3*d^2*e^4*x^2-95200*B*a*b^4*d^3*e^3*x^2+13440*B*b^5*d^4*e^2*x^2+546975*A*a^4*b*e^6*x-48
6200*A*a^3*b^2*d*e^5*x+265200*A*a^2*b^3*d^2*e^4*x-81600*A*a*b^4*d^3*e^3*x+10880*A*b^5*d^4*e^2*x+109395*B*a^5*e
^6*x-243100*B*a^4*b*d*e^5*x+265200*B*a^3*b^2*d^2*e^4*x-163200*B*a^2*b^3*d^3*e^3*x+54400*B*a*b^4*d^4*e^2*x-7680
*B*b^5*d^5*e*x+153153*A*a^5*e^6-218790*A*a^4*b*d*e^5+194480*A*a^3*b^2*d^2*e^4-106080*A*a^2*b^3*d^3*e^3+32640*A
*a*b^4*d^4*e^2-4352*A*b^5*d^5*e-43758*B*a^5*d*e^5+97240*B*a^4*b*d^2*e^4-106080*B*a^3*b^2*d^3*e^3+65280*B*a^2*b
^3*d^4*e^2-21760*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 = 1.1949, size = 1243, normalized size = 2.75 \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)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3*d^5*e^2 + 11440*a^3*b^2*d^4*e^3 - 12
870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e
^6 + 650*a^2*b^3*e^7)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2*b^3*d*e^6 - 1430*a^3*b^2*e^7)*x^4
+ 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*
(32*b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 780*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e^5 - 17160*a^4*b*d*e^6 - 3003*a^
5*e^7)*x^2 + (128*b^5*d^6*e - 960*a*b^4*d^5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*e^4 + 6435*a^4*b*d^2
*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)*A/e^6 + 2/765765*(45045*b^5*e^8*x^8 + 3072*b^5*d^8 - 21760*a*b^4*d^7*
e + 65280*a^2*b^3*d^6*e^2 - 106080*a^3*b^2*d^5*e^3 + 97240*a^4*b*d^4*e^4 - 43758*a^5*d^3*e^5 + 3003*(18*b^5*d*
e^7 + 85*a*b^4*e^8)*x^7 + 231*(3*b^5*d^2*e^6 + 1360*a*b^4*d*e^7 + 2550*a^2*b^3*e^8)*x^6 - 63*(12*b^5*d^3*e^5 -
85*a*b^4*d^2*e^6 - 11900*a^2*b^3*d*e^7 - 11050*a^3*b^2*e^8)*x^5 + 35*(24*b^5*d^4*e^4 - 170*a*b^4*d^3*e^5 + 51
0*a^2*b^3*d^2*e^6 + 26520*a^3*b^2*d*e^7 + 12155*a^4*b*e^8)*x^4 - 5*(192*b^5*d^5*e^3 - 1360*a*b^4*d^4*e^4 + 408
0*a^2*b^3*d^3*e^5 - 6630*a^3*b^2*d^2*e^6 - 121550*a^4*b*d*e^7 - 21879*a^5*e^8)*x^3 + 3*(384*b^5*d^6*e^2 - 2720
*a*b^4*d^5*e^3 + 8160*a^2*b^3*d^4*e^4 - 13260*a^3*b^2*d^3*e^5 + 12155*a^4*b*d^2*e^6 + 58344*a^5*d*e^7)*x^2 - (
1536*b^5*d^7*e - 10880*a*b^4*d^6*e^2 + 32640*a^2*b^3*d^5*e^3 - 53040*a^3*b^2*d^4*e^4 + 48620*a^4*b*d^3*e^5 - 2
1879*a^5*d^2*e^6)*x)*sqrt(e*x + d)*B/e^7

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Fricas [B]  time = 1.4046, size = 1914, normalized size = 4.23 \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)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

2/765765*(45045*B*b^5*e^8*x^8 + 3072*B*b^5*d^8 + 153153*A*a^5*d^2*e^6 - 4352*(5*B*a*b^4 + A*b^5)*d^7*e + 32640
*(2*B*a^2*b^3 + A*a*b^4)*d^6*e^2 - 106080*(B*a^3*b^2 + A*a^2*b^3)*d^5*e^3 + 97240*(B*a^4*b + 2*A*a^3*b^2)*d^4*
e^4 - 43758*(B*a^5 + 5*A*a^4*b)*d^3*e^5 + 3003*(18*B*b^5*d*e^7 + 17*(5*B*a*b^4 + A*b^5)*e^8)*x^7 + 231*(3*B*b^
5*d^2*e^6 + 272*(5*B*a*b^4 + A*b^5)*d*e^7 + 1275*(2*B*a^2*b^3 + A*a*b^4)*e^8)*x^6 - 63*(12*B*b^5*d^3*e^5 - 17*
(5*B*a*b^4 + A*b^5)*d^2*e^6 - 5950*(2*B*a^2*b^3 + A*a*b^4)*d*e^7 - 11050*(B*a^3*b^2 + A*a^2*b^3)*e^8)*x^5 + 35
*(24*B*b^5*d^4*e^4 - 34*(5*B*a*b^4 + A*b^5)*d^3*e^5 + 255*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^6 + 26520*(B*a^3*b^2 +
A*a^2*b^3)*d*e^7 + 12155*(B*a^4*b + 2*A*a^3*b^2)*e^8)*x^4 - 5*(192*B*b^5*d^5*e^3 - 272*(5*B*a*b^4 + A*b^5)*d^
4*e^4 + 2040*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^5 - 6630*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^6 - 121550*(B*a^4*b + 2*A*a^
3*b^2)*d*e^7 - 21879*(B*a^5 + 5*A*a^4*b)*e^8)*x^3 + 3*(384*B*b^5*d^6*e^2 + 51051*A*a^5*e^8 - 544*(5*B*a*b^4 +
A*b^5)*d^5*e^3 + 4080*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^4 - 13260*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^5 + 12155*(B*a^4*b
+ 2*A*a^3*b^2)*d^2*e^6 + 58344*(B*a^5 + 5*A*a^4*b)*d*e^7)*x^2 - (1536*B*b^5*d^7*e - 306306*A*a^5*d*e^7 - 2176
*(5*B*a*b^4 + A*b^5)*d^6*e^2 + 16320*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^3 - 53040*(B*a^3*b^2 + A*a^2*b^3)*d^4*e^4 +
48620*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^5 - 21879*(B*a^5 + 5*A*a^4*b)*d^2*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)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.34621, size = 2279, normalized size = 5.04 \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)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

2/765765*(51051*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^5*d*e^(-1)*sgn(b*x + a) + 255255*(3*(x*e + d)^(5
/2) - 5*(x*e + d)^(3/2)*d)*A*a^4*b*d*e^(-1)*sgn(b*x + a) + 36465*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d +
35*(x*e + d)^(3/2)*d^2)*B*a^4*b*d*e^(-2)*sgn(b*x + a) + 72930*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*
(x*e + d)^(3/2)*d^2)*A*a^3*b^2*d*e^(-2)*sgn(b*x + a) + 24310*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189
*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a^3*b^2*d*e^(-3)*sgn(b*x + a) + 24310*(35*(x*e + d)^(9/2) -
135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*a^2*b^3*d*e^(-3)*sgn(b*x + a) + 2
210*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 115
5*(x*e + d)^(3/2)*d^4)*B*a^2*b^3*d*e^(-4)*sgn(b*x + a) + 1105*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d +
2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*a*b^4*d*e^(-4)*sgn(b*x + a)
+ 425*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3
+ 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*a*b^4*d*e^(-5)*sgn(b*x + a) + 85*(693*(x*e + d)^(13/
2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^
4 - 3003*(x*e + d)^(3/2)*d^5)*A*b^5*d*e^(-5)*sgn(b*x + a) + 17*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)
*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/
2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*B*b^5*d*e^(-6)*sgn(b*x + a) + 255255*(x*e + d)^(3/2)*A*a^5*d*sgn(b*x + a)
+ 7293*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^5*e^(-1)*sgn(b*x + a) + 36465*
(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^4*b*e^(-1)*sgn(b*x + a) + 12155*(35*(
x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a^4*b*e^(-2)*sgn
(b*x + a) + 24310*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*
d^3)*A*a^3*b^2*e^(-2)*sgn(b*x + a) + 2210*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2
)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*a^3*b^2*e^(-3)*sgn(b*x + a) + 2210*(315*(x*e +
d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2
)*d^4)*A*a^2*b^3*e^(-3)*sgn(b*x + a) + 850*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(
9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*a^2*b^3*e^(-4)*s
gn(b*x + a) + 425*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d
)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*A*a*b^4*e^(-4)*sgn(b*x + a) + 85*(3003*(x*e
+ d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e
+ d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*B*a*b^4*e^(-5)*sgn(b*x + a) + 17*(300
3*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 9652
5*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*A*b^5*e^(-5)*sgn(b*x + a) + 7*(
6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 +
425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*
d^7)*B*b^5*e^(-6)*sgn(b*x + a) + 51051*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^5*sgn(b*x + a))*e^(-1)