3.1851 \(\int (A+B x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=452 \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{19/2} (-5 a B e-A b e+6 b B d)}{19 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{17 e^7 (a+b x)}-\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{3 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{13 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{11 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^5 (B d-A e)}{9 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{21/2}}{21 e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^5*(B*d - A*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*d - a*e)
^4*(6*b*B*d - 5*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*(b*d
- a*e)^3*(3*b*B*d - 2*A*b*e - 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)) - (4*
b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*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)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*e^7*(a +
b*x)) - (2*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(19/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(19*e^7*(a + b*x))
+ (2*b^5*B*(d + e*x)^(21/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(21*e^7*(a + b*x))

________________________________________________________________________________________

Rubi [A]  time = 0.283003, 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} \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{19/2} (-5 a B e-A b e+6 b B d)}{19 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{17 e^7 (a+b x)}-\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{3 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{13 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{11 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^5 (B d-A e)}{9 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{21/2}}{21 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(7/2)*(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)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*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)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*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)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*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)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*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)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*e^7*(a +
b*x)) - (2*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(19/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(19*e^7*(a + b*x))
+ (2*b^5*B*(d + e*x)^(21/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(21*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)^{7/2} \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)^{7/2} \, 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)^{7/2}}{e^6}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e) (d+e x)^{9/2}}{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)^{11/2}}{e^6}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{13/2}}{e^6}-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{15/2}}{e^6}+\frac{b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{17/2}}{e^6}+\frac{b^{10} B (d+e x)^{19/2}}{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)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 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)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 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)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 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)^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 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)^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 e^7 (a+b x)}-\frac{2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{19/2} \sqrt{a^2+2 a b x+b^2 x^2}}{19 e^7 (a+b x)}+\frac{2 b^5 B (d+e x)^{21/2} \sqrt{a^2+2 a b x+b^2 x^2}}{21 e^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.398825, size = 239, normalized size = 0.53 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{9/2} \left (-153153 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+855855 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-1939938 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+1119195 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-264537 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)+323323 (b d-a e)^5 (B d-A e)+138567 b^5 B (d+e x)^6\right )}{2909907 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(9/2)*(323323*(b*d - a*e)^5*(B*d - A*e) - 264537*(b*d - a*e)^4*(6*b*B*d - 5*A*b
*e - a*B*e)*(d + e*x) + 1119195*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 1939938*b^2*(b*d - a
*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^3 + 855855*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 -
153153*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^5 + 138567*b^5*B*(d + e*x)^6))/(2909907*e^7*(a + b*x))

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Maple [A]  time = 0.009, size = 689, normalized size = 1.5 \begin{align*}{\frac{277134\,B{x}^{6}{b}^{5}{e}^{6}+306306\,A{x}^{5}{b}^{5}{e}^{6}+1531530\,B{x}^{5}a{b}^{4}{e}^{6}-175032\,B{x}^{5}{b}^{5}d{e}^{5}+1711710\,A{x}^{4}a{b}^{4}{e}^{6}-180180\,A{x}^{4}{b}^{5}d{e}^{5}+3423420\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}-900900\,B{x}^{4}a{b}^{4}d{e}^{5}+102960\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+3879876\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}-912912\,A{x}^{3}a{b}^{4}d{e}^{5}+96096\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+3879876\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}-1825824\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+480480\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}-54912\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+4476780\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}-1790712\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+421344\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-44352\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+2238390\,B{x}^{2}{a}^{4}b{e}^{6}-1790712\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+842688\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-221760\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+25344\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+2645370\,Ax{a}^{4}b{e}^{6}-1627920\,Ax{a}^{3}{b}^{2}d{e}^{5}+651168\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}-153216\,Axa{b}^{4}{d}^{3}{e}^{3}+16128\,Ax{b}^{5}{d}^{4}{e}^{2}+529074\,Bx{a}^{5}{e}^{6}-813960\,Bx{a}^{4}bd{e}^{5}+651168\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}-306432\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+80640\,Bxa{b}^{4}{d}^{4}{e}^{2}-9216\,Bx{b}^{5}{d}^{5}e+646646\,A{a}^{5}{e}^{6}-587860\,Ad{e}^{5}{a}^{4}b+361760\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-144704\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+34048\,Aa{b}^{4}{d}^{4}{e}^{2}-3584\,A{b}^{5}{d}^{5}e-117572\,Bd{e}^{5}{a}^{5}+180880\,B{a}^{4}b{d}^{2}{e}^{4}-144704\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+68096\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-17920\,Ba{b}^{4}{d}^{5}e+2048\,B{b}^{5}{d}^{6}}{2909907\,{e}^{7} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{9}{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)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

2/2909907*(e*x+d)^(9/2)*(138567*B*b^5*e^6*x^6+153153*A*b^5*e^6*x^5+765765*B*a*b^4*e^6*x^5-87516*B*b^5*d*e^5*x^
5+855855*A*a*b^4*e^6*x^4-90090*A*b^5*d*e^5*x^4+1711710*B*a^2*b^3*e^6*x^4-450450*B*a*b^4*d*e^5*x^4+51480*B*b^5*
d^2*e^4*x^4+1939938*A*a^2*b^3*e^6*x^3-456456*A*a*b^4*d*e^5*x^3+48048*A*b^5*d^2*e^4*x^3+1939938*B*a^3*b^2*e^6*x
^3-912912*B*a^2*b^3*d*e^5*x^3+240240*B*a*b^4*d^2*e^4*x^3-27456*B*b^5*d^3*e^3*x^3+2238390*A*a^3*b^2*e^6*x^2-895
356*A*a^2*b^3*d*e^5*x^2+210672*A*a*b^4*d^2*e^4*x^2-22176*A*b^5*d^3*e^3*x^2+1119195*B*a^4*b*e^6*x^2-895356*B*a^
3*b^2*d*e^5*x^2+421344*B*a^2*b^3*d^2*e^4*x^2-110880*B*a*b^4*d^3*e^3*x^2+12672*B*b^5*d^4*e^2*x^2+1322685*A*a^4*
b*e^6*x-813960*A*a^3*b^2*d*e^5*x+325584*A*a^2*b^3*d^2*e^4*x-76608*A*a*b^4*d^3*e^3*x+8064*A*b^5*d^4*e^2*x+26453
7*B*a^5*e^6*x-406980*B*a^4*b*d*e^5*x+325584*B*a^3*b^2*d^2*e^4*x-153216*B*a^2*b^3*d^3*e^3*x+40320*B*a*b^4*d^4*e
^2*x-4608*B*b^5*d^5*e*x+323323*A*a^5*e^6-293930*A*a^4*b*d*e^5+180880*A*a^3*b^2*d^2*e^4-72352*A*a^2*b^3*d^3*e^3
+17024*A*a*b^4*d^4*e^2-1792*A*b^5*d^5*e-58786*B*a^5*d*e^5+90440*B*a^4*b*d^2*e^4-72352*B*a^3*b^2*d^3*e^3+34048*
B*a^2*b^3*d^4*e^2-8960*B*a*b^4*d^5*e+1024*B*b^5*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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Maxima [B]  time = 1.11306, size = 1675, normalized size = 3.71 \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)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

2/415701*(21879*b^5*e^9*x^9 - 256*b^5*d^9 + 2432*a*b^4*d^8*e - 10336*a^2*b^3*d^7*e^2 + 25840*a^3*b^2*d^6*e^3 -
41990*a^4*b*d^5*e^4 + 46189*a^5*d^4*e^5 + 1287*(58*b^5*d*e^8 + 95*a*b^4*e^9)*x^8 + 858*(101*b^5*d^2*e^7 + 494
*a*b^4*d*e^8 + 323*a^2*b^3*e^9)*x^7 + 66*(524*b^5*d^3*e^6 + 7619*a*b^4*d^2*e^7 + 14858*a^2*b^3*d*e^8 + 4845*a^
3*b^2*e^9)*x^6 + 9*(7*b^5*d^4*e^5 + 23028*a*b^4*d^3*e^6 + 133076*a^2*b^3*d^2*e^7 + 129200*a^3*b^2*d*e^8 + 2099
5*a^4*b*e^9)*x^5 - (70*b^5*d^5*e^4 - 665*a*b^4*d^4*e^5 - 516800*a^2*b^3*d^3*e^6 - 1479340*a^3*b^2*d^2*e^7 - 71
3830*a^4*b*d*e^8 - 46189*a^5*e^9)*x^4 + 2*(40*b^5*d^6*e^3 - 380*a*b^4*d^5*e^4 + 1615*a^2*b^3*d^4*e^5 + 342380*
a^3*b^2*d^3*e^6 + 482885*a^4*b*d^2*e^7 + 92378*a^5*d*e^8)*x^3 - 6*(16*b^5*d^7*e^2 - 152*a*b^4*d^6*e^3 + 646*a^
2*b^3*d^5*e^4 - 1615*a^3*b^2*d^4*e^5 - 83980*a^4*b*d^3*e^6 - 46189*a^5*d^2*e^7)*x^2 + (128*b^5*d^8*e - 1216*a*
b^4*d^7*e^2 + 5168*a^2*b^3*d^6*e^3 - 12920*a^3*b^2*d^5*e^4 + 20995*a^4*b*d^4*e^5 + 184756*a^5*d^3*e^6)*x)*sqrt
(e*x + d)*A/e^6 + 2/2909907*(138567*b^5*e^10*x^10 + 1024*b^5*d^10 - 8960*a*b^4*d^9*e + 34048*a^2*b^3*d^8*e^2 -
72352*a^3*b^2*d^7*e^3 + 90440*a^4*b*d^6*e^4 - 58786*a^5*d^5*e^5 + 7293*(64*b^5*d*e^9 + 105*a*b^4*e^10)*x^9 +
2574*(207*b^5*d^2*e^8 + 1015*a*b^4*d*e^9 + 665*a^2*b^3*e^10)*x^8 + 858*(242*b^5*d^3*e^7 + 3535*a*b^4*d^2*e^8 +
6916*a^2*b^3*d*e^9 + 2261*a^3*b^2*e^10)*x^7 + 231*(b^5*d^4*e^6 + 5240*a*b^4*d^3*e^7 + 30476*a^2*b^3*d^2*e^8 +
29716*a^3*b^2*d*e^9 + 4845*a^4*b*e^10)*x^6 - 63*(4*b^5*d^5*e^5 - 35*a*b^4*d^4*e^6 - 46056*a^2*b^3*d^3*e^7 - 1
33076*a^3*b^2*d^2*e^8 - 64600*a^4*b*d*e^9 - 4199*a^5*e^10)*x^5 + 14*(20*b^5*d^6*e^4 - 175*a*b^4*d^5*e^5 + 665*
a^2*b^3*d^4*e^6 + 258400*a^3*b^2*d^3*e^7 + 369835*a^4*b*d^2*e^8 + 71383*a^5*d*e^9)*x^4 - 2*(160*b^5*d^7*e^3 -
1400*a*b^4*d^6*e^4 + 5320*a^2*b^3*d^5*e^5 - 11305*a^3*b^2*d^4*e^6 - 1198330*a^4*b*d^3*e^7 - 676039*a^5*d^2*e^8
)*x^3 + 3*(128*b^5*d^8*e^2 - 1120*a*b^4*d^7*e^3 + 4256*a^2*b^3*d^6*e^4 - 9044*a^3*b^2*d^5*e^5 + 11305*a^4*b*d^
4*e^6 + 235144*a^5*d^3*e^7)*x^2 - (512*b^5*d^9*e - 4480*a*b^4*d^8*e^2 + 17024*a^2*b^3*d^7*e^3 - 36176*a^3*b^2*
d^6*e^4 + 45220*a^4*b*d^5*e^5 - 29393*a^5*d^4*e^6)*x)*sqrt(e*x + d)*B/e^7

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Fricas [B]  time = 1.38445, size = 2581, normalized size = 5.71 \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)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

2/2909907*(138567*B*b^5*e^10*x^10 + 1024*B*b^5*d^10 + 323323*A*a^5*d^4*e^6 - 1792*(5*B*a*b^4 + A*b^5)*d^9*e +
17024*(2*B*a^2*b^3 + A*a*b^4)*d^8*e^2 - 72352*(B*a^3*b^2 + A*a^2*b^3)*d^7*e^3 + 90440*(B*a^4*b + 2*A*a^3*b^2)*
d^6*e^4 - 58786*(B*a^5 + 5*A*a^4*b)*d^5*e^5 + 7293*(64*B*b^5*d*e^9 + 21*(5*B*a*b^4 + A*b^5)*e^10)*x^9 + 1287*(
414*B*b^5*d^2*e^8 + 406*(5*B*a*b^4 + A*b^5)*d*e^9 + 665*(2*B*a^2*b^3 + A*a*b^4)*e^10)*x^8 + 858*(242*B*b^5*d^3
*e^7 + 707*(5*B*a*b^4 + A*b^5)*d^2*e^8 + 3458*(2*B*a^2*b^3 + A*a*b^4)*d*e^9 + 2261*(B*a^3*b^2 + A*a^2*b^3)*e^1
0)*x^7 + 231*(B*b^5*d^4*e^6 + 1048*(5*B*a*b^4 + A*b^5)*d^3*e^7 + 15238*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^8 + 29716
*(B*a^3*b^2 + A*a^2*b^3)*d*e^9 + 4845*(B*a^4*b + 2*A*a^3*b^2)*e^10)*x^6 - 63*(4*B*b^5*d^5*e^5 - 7*(5*B*a*b^4 +
A*b^5)*d^4*e^6 - 23028*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^7 - 133076*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^8 - 64600*(B*a^
4*b + 2*A*a^3*b^2)*d*e^9 - 4199*(B*a^5 + 5*A*a^4*b)*e^10)*x^5 + 7*(40*B*b^5*d^6*e^4 + 46189*A*a^5*e^10 - 70*(5
*B*a*b^4 + A*b^5)*d^5*e^5 + 665*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^6 + 516800*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^7 + 739
670*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^8 + 142766*(B*a^5 + 5*A*a^4*b)*d*e^9)*x^4 - 2*(160*B*b^5*d^7*e^3 - 646646*A*
a^5*d*e^9 - 280*(5*B*a*b^4 + A*b^5)*d^6*e^4 + 2660*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^5 - 11305*(B*a^3*b^2 + A*a^2*
b^3)*d^4*e^6 - 1198330*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^7 - 676039*(B*a^5 + 5*A*a^4*b)*d^2*e^8)*x^3 + 3*(128*B*b^
5*d^8*e^2 + 646646*A*a^5*d^2*e^8 - 224*(5*B*a*b^4 + A*b^5)*d^7*e^3 + 2128*(2*B*a^2*b^3 + A*a*b^4)*d^6*e^4 - 90
44*(B*a^3*b^2 + A*a^2*b^3)*d^5*e^5 + 11305*(B*a^4*b + 2*A*a^3*b^2)*d^4*e^6 + 235144*(B*a^5 + 5*A*a^4*b)*d^3*e^
7)*x^2 - (512*B*b^5*d^9*e - 1293292*A*a^5*d^3*e^7 - 896*(5*B*a*b^4 + A*b^5)*d^8*e^2 + 8512*(2*B*a^2*b^3 + A*a*
b^4)*d^7*e^3 - 36176*(B*a^3*b^2 + A*a^2*b^3)*d^6*e^4 + 45220*(B*a^4*b + 2*A*a^3*b^2)*d^5*e^5 - 29393*(B*a^5 +
5*A*a^4*b)*d^4*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)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.70493, size = 5476, normalized size = 12.12 \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)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

2/14549535*(969969*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^5*d^3*e^(-1)*sgn(b*x + a) + 4849845*(3*(x*e +
d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^4*b*d^3*e^(-1)*sgn(b*x + a) + 692835*(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^3*e^(-2)*sgn(b*x + a) + 1385670*(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^3*e^(-2)*sgn(b*x + a) + 461890*(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^3*e^(-3)*sgn(b*x + a) + 461890*(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^3*e^
(-3)*sgn(b*x + a) + 41990*(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^2*b^3*d^3*e^(-4)*sgn(b*x + a) + 20995*(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^3*e^(-4)*sgn(b*x + a) + 8075*(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^3*e^(-5)*sgn(b*x
+ a) + 1615*(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^3*e^(-5)*sgn(b*x + a) + 323*(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^3*e^(-6)*sgn(b*x + a) + 4849845
*(x*e + d)^(3/2)*A*a^5*d^3*sgn(b*x + a) + 415701*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/
2)*d^2)*B*a^5*d^2*e^(-1)*sgn(b*x + a) + 2078505*(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*d^2*e^(-1)*sgn(b*x + a) + 692835*(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*d^2*e^(-2)*sgn(b*x + a) + 1385670*(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*d^2*e^(-2)*sgn(b*x + a) + 125970*(31
5*(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*d^2*e^(-3)*sgn(b*x + a) + 125970*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 29
70*(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*d^2*e^(-3)*sgn(b*x + a
) + 48450*(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*d^2*e^(-4)*sgn(b*x + a) + 24225*(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*d^2*e^(-4)*sgn(b*x + a) + 4845*(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 - 540
54*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*B*a*b^4*d^2*e^(-5)*sgn(b*x + a) + 969*(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)*A*b^5*d^2*e^(-5)*sgn(b*x + a) + 399*(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
*d^2*e^(-6)*sgn(b*x + a) + 2909907*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^5*d^2*sgn(b*x + a) + 138567*(
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^5*d*e^(-1)
*sgn(b*x + a) + 692835*(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^4*b*d*e^(-1)*sgn(b*x + a) + 62985*(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^4*b*d*e^(-2)*sgn(b*x + a) + 125970*(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^3*b^2*d*e^(-2)*sgn(b*x + a) + 48450*(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^3*
b^2*d*e^(-3)*sgn(b*x + a) + 48450*(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^2*b^3*d*e^(-3)*sgn(b*x
+ a) + 9690*(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^2*b^3*d*e^(
-4)*sgn(b*x + a) + 4845*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 10010
0*(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)*A*a
*b^4*d*e^(-4)*sgn(b*x + a) + 1995*(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*a*b^4*d*e^(-5)*sgn(b*x + a) + 399*(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 - 328
185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*A*b^5*d*e^(-5)*sgn(b*x + a)
+ 21*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 8953560*(x*e + d)^(
13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(7/2)*d^6 - 33256
08*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*B*b^5*d*e^(-6)*sgn(b*x + a) + 415701*(15*(x*e + d)^(7/2)
- 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^5*d*sgn(b*x + a) + 4199*(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^5*e^(-1)*s
gn(b*x + a) + 20995*(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^4*b*e^(-1)*sgn(b*x + a) + 8075*(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^4*b*e^(-2)*sgn(b*x + a) + 16150*(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^3*b^2*e^
(-2)*sgn(b*x + a) + 3230*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 1001
00*(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^3*b^2*e^(-3)*sgn(b*x + a) + 3230*(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)*A*a^2*b^3*e^(-3)*sgn(b*x + a) + 1330*(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 + 15315
3*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*B*a^2*b^3*e^(-4)*sgn(b*x + a) + 665*(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)*A*a*b^4*e^(-4)*sg
n(b*x + a) + 35*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 8953560*
(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(7/2)*
d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*B*a*b^4*e^(-5)*sgn(b*x + a) + 7*(109395*(x*e +
d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 8953560*(x*e + d)^(13/2)*d^3 + 1322685
0*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*
d^7 + 692835*(x*e + d)^(3/2)*d^8)*A*b^5*e^(-5)*sgn(b*x + a) + 3*(230945*(x*e + d)^(21/2) - 2297295*(x*e + d)^(
19/2)*d + 10270260*(x*e + d)^(17/2)*d^2 - 27159132*(x*e + d)^(15/2)*d^3 + 47006190*(x*e + d)^(13/2)*d^4 - 5555
2770*(x*e + d)^(11/2)*d^5 + 45265220*(x*e + d)^(9/2)*d^6 - 24942060*(x*e + d)^(7/2)*d^7 + 8729721*(x*e + d)^(5
/2)*d^8 - 1616615*(x*e + d)^(3/2)*d^9)*B*b^5*e^(-6)*sgn(b*x + a) + 46189*(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^5*sgn(b*x + a))*e^(-1)