3.1852 \(\int (A+B x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=452 \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (-5 a B e-A b e+6 b B d)}{17 e^7 (a+b x)}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{13 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{11 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{9 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5 (B d-A e)}{7 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{19/2}}{19 e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^5*(B*d - A*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*d - a*e)
^4*(6*b*B*d - 5*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)) + (10*b*(b*d -
a*e)^3*(3*b*B*d - 2*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)) - (20*b
^2*(b*d - a*e)^2*(2*b*B*d - 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))
+ (2*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*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)) - (2*b^4*(6*b*B*d - A*b*e - 5*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^5*B*(d + e*x)^(19/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(19*e^7*(a + b*x))

________________________________________________________________________________________

Rubi [A]  time = 0.215834, 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} \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (-5 a B e-A b e+6 b B d)}{17 e^7 (a+b x)}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{13 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{11 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{9 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5 (B d-A e)}{7 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{19/2}}{19 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.310466, size = 239, normalized size = 0.53 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} \left (-171171 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+969969 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-2238390 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+1322685 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-323323 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)+415701 (b d-a e)^5 (B d-A e)+153153 b^5 B (d+e x)^6\right )}{2909907 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(415701*(b*d - a*e)^5*(B*d - A*e) - 323323*(b*d - a*e)^4*(6*b*B*d - 5*A*b
*e - a*B*e)*(d + e*x) + 1322685*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 2238390*b^2*(b*d - a
*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^3 + 969969*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 -
171171*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^5 + 153153*b^5*B*(d + e*x)^6))/(2909907*e^7*(a + b*x))

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Maple [A]  time = 0.007, size = 689, normalized size = 1.5 \begin{align*}{\frac{306306\,B{x}^{6}{b}^{5}{e}^{6}+342342\,A{x}^{5}{b}^{5}{e}^{6}+1711710\,B{x}^{5}a{b}^{4}{e}^{6}-216216\,B{x}^{5}{b}^{5}d{e}^{5}+1939938\,A{x}^{4}a{b}^{4}{e}^{6}-228228\,A{x}^{4}{b}^{5}d{e}^{5}+3879876\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}-1141140\,B{x}^{4}a{b}^{4}d{e}^{5}+144144\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+4476780\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}-1193808\,A{x}^{3}a{b}^{4}d{e}^{5}+140448\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+4476780\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}-2387616\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+702240\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}-88704\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+5290740\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}-2441880\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+651168\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-76608\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+2645370\,B{x}^{2}{a}^{4}b{e}^{6}-2441880\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+1302336\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-383040\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+48384\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+3233230\,Ax{a}^{4}b{e}^{6}-2351440\,Ax{a}^{3}{b}^{2}d{e}^{5}+1085280\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}-289408\,Axa{b}^{4}{d}^{3}{e}^{3}+34048\,Ax{b}^{5}{d}^{4}{e}^{2}+646646\,Bx{a}^{5}{e}^{6}-1175720\,Bx{a}^{4}bd{e}^{5}+1085280\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}-578816\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+170240\,Bxa{b}^{4}{d}^{4}{e}^{2}-21504\,Bx{b}^{5}{d}^{5}e+831402\,A{a}^{5}{e}^{6}-923780\,Ad{e}^{5}{a}^{4}b+671840\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-310080\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+82688\,Aa{b}^{4}{d}^{4}{e}^{2}-9728\,A{b}^{5}{d}^{5}e-184756\,Bd{e}^{5}{a}^{5}+335920\,B{a}^{4}b{d}^{2}{e}^{4}-310080\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+165376\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-48640\,Ba{b}^{4}{d}^{5}e+6144\,B{b}^{5}{d}^{6}}{2909907\,{e}^{7} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{7}{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)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

2/2909907*(e*x+d)^(7/2)*(153153*B*b^5*e^6*x^6+171171*A*b^5*e^6*x^5+855855*B*a*b^4*e^6*x^5-108108*B*b^5*d*e^5*x
^5+969969*A*a*b^4*e^6*x^4-114114*A*b^5*d*e^5*x^4+1939938*B*a^2*b^3*e^6*x^4-570570*B*a*b^4*d*e^5*x^4+72072*B*b^
5*d^2*e^4*x^4+2238390*A*a^2*b^3*e^6*x^3-596904*A*a*b^4*d*e^5*x^3+70224*A*b^5*d^2*e^4*x^3+2238390*B*a^3*b^2*e^6
*x^3-1193808*B*a^2*b^3*d*e^5*x^3+351120*B*a*b^4*d^2*e^4*x^3-44352*B*b^5*d^3*e^3*x^3+2645370*A*a^3*b^2*e^6*x^2-
1220940*A*a^2*b^3*d*e^5*x^2+325584*A*a*b^4*d^2*e^4*x^2-38304*A*b^5*d^3*e^3*x^2+1322685*B*a^4*b*e^6*x^2-1220940
*B*a^3*b^2*d*e^5*x^2+651168*B*a^2*b^3*d^2*e^4*x^2-191520*B*a*b^4*d^3*e^3*x^2+24192*B*b^5*d^4*e^2*x^2+1616615*A
*a^4*b*e^6*x-1175720*A*a^3*b^2*d*e^5*x+542640*A*a^2*b^3*d^2*e^4*x-144704*A*a*b^4*d^3*e^3*x+17024*A*b^5*d^4*e^2
*x+323323*B*a^5*e^6*x-587860*B*a^4*b*d*e^5*x+542640*B*a^3*b^2*d^2*e^4*x-289408*B*a^2*b^3*d^3*e^3*x+85120*B*a*b
^4*d^4*e^2*x-10752*B*b^5*d^5*e*x+415701*A*a^5*e^6-461890*A*a^4*b*d*e^5+335920*A*a^3*b^2*d^2*e^4-155040*A*a^2*b
^3*d^3*e^3+41344*A*a*b^4*d^4*e^2-4864*A*b^5*d^5*e-92378*B*a^5*d*e^5+167960*B*a^4*b*d^2*e^4-155040*B*a^3*b^2*d^
3*e^3+82688*B*a^2*b^3*d^4*e^2-24320*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 = 1.12537, size = 1458, normalized size = 3.23 \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)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

2/153153*(9009*b^5*e^8*x^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3*d^6*e^2 + 17680*a^3*b^2*d^5*e^3 - 2
4310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b
^4*d*e^7 + 510*a^2*b^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 4590*a^2*b^3*d*e^7 + 2210*a^3*b^2*e^8
)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3*e^5 - 5406*a^2*b^3*d^2*e^6 - 10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^
4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4 + 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^2*d^2*e^6 + 230945*a^4*b*d*e^7 +
21879*a^5*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*a^3*b^2*d^3*e^5 - 60
775*a^4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e - 1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a
^3*b^2*d^4*e^4 + 12155*a^4*b*d^3*e^5 + 65637*a^5*d^2*e^6)*x)*sqrt(e*x + d)*A/e^6 + 2/2909907*(153153*b^5*e^9*x
^9 + 3072*b^5*d^9 - 24320*a*b^4*d^8*e + 82688*a^2*b^3*d^7*e^2 - 155040*a^3*b^2*d^6*e^3 + 167960*a^4*b*d^5*e^4
- 92378*a^5*d^4*e^5 + 9009*(39*b^5*d*e^8 + 95*a*b^4*e^9)*x^8 + 3003*(69*b^5*d^2*e^7 + 665*a*b^4*d*e^8 + 646*a^
2*b^3*e^9)*x^7 + 231*(3*b^5*d^3*e^6 + 5225*a*b^4*d^2*e^7 + 20026*a^2*b^3*d*e^8 + 9690*a^3*b^2*e^9)*x^6 - 63*(1
2*b^5*d^4*e^5 - 95*a*b^4*d^3*e^6 - 45866*a^2*b^3*d^2*e^7 - 87210*a^3*b^2*d*e^8 - 20995*a^4*b*e^9)*x^5 + 7*(120
*b^5*d^5*e^4 - 950*a*b^4*d^4*e^5 + 3230*a^2*b^3*d^3*e^6 + 513570*a^3*b^2*d^2*e^7 + 482885*a^4*b*d*e^8 + 46189*
a^5*e^9)*x^4 - (960*b^5*d^6*e^3 - 7600*a*b^4*d^5*e^4 + 25840*a^2*b^3*d^4*e^5 - 48450*a^3*b^2*d^3*e^6 - 2372435
*a^4*b*d^2*e^7 - 877591*a^5*d*e^8)*x^3 + 3*(384*b^5*d^7*e^2 - 3040*a*b^4*d^6*e^3 + 10336*a^2*b^3*d^5*e^4 - 193
80*a^3*b^2*d^4*e^5 + 20995*a^4*b*d^3*e^6 + 230945*a^5*d^2*e^7)*x^2 - (1536*b^5*d^8*e - 12160*a*b^4*d^7*e^2 + 4
1344*a^2*b^3*d^6*e^3 - 77520*a^3*b^2*d^5*e^4 + 83980*a^4*b*d^4*e^5 - 46189*a^5*d^3*e^6)*x)*sqrt(e*x + d)*B/e^7

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Fricas [B]  time = 1.4342, size = 2260, normalized size = 5. \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)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

2/2909907*(153153*B*b^5*e^9*x^9 + 3072*B*b^5*d^9 + 415701*A*a^5*d^3*e^6 - 4864*(5*B*a*b^4 + A*b^5)*d^8*e + 413
44*(2*B*a^2*b^3 + A*a*b^4)*d^7*e^2 - 155040*(B*a^3*b^2 + A*a^2*b^3)*d^6*e^3 + 167960*(B*a^4*b + 2*A*a^3*b^2)*d
^5*e^4 - 92378*(B*a^5 + 5*A*a^4*b)*d^4*e^5 + 9009*(39*B*b^5*d*e^8 + 19*(5*B*a*b^4 + A*b^5)*e^9)*x^8 + 3003*(69
*B*b^5*d^2*e^7 + 133*(5*B*a*b^4 + A*b^5)*d*e^8 + 323*(2*B*a^2*b^3 + A*a*b^4)*e^9)*x^7 + 231*(3*B*b^5*d^3*e^6 +
1045*(5*B*a*b^4 + A*b^5)*d^2*e^7 + 10013*(2*B*a^2*b^3 + A*a*b^4)*d*e^8 + 9690*(B*a^3*b^2 + A*a^2*b^3)*e^9)*x^
6 - 63*(12*B*b^5*d^4*e^5 - 19*(5*B*a*b^4 + A*b^5)*d^3*e^6 - 22933*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^7 - 87210*(B*a
^3*b^2 + A*a^2*b^3)*d*e^8 - 20995*(B*a^4*b + 2*A*a^3*b^2)*e^9)*x^5 + 7*(120*B*b^5*d^5*e^4 - 190*(5*B*a*b^4 + A
*b^5)*d^4*e^5 + 1615*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^6 + 513570*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^7 + 482885*(B*a^4*
b + 2*A*a^3*b^2)*d*e^8 + 46189*(B*a^5 + 5*A*a^4*b)*e^9)*x^4 - (960*B*b^5*d^6*e^3 - 415701*A*a^5*e^9 - 1520*(5*
B*a*b^4 + A*b^5)*d^5*e^4 + 12920*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^5 - 48450*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^6 - 237
2435*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^7 - 877591*(B*a^5 + 5*A*a^4*b)*d*e^8)*x^3 + 3*(384*B*b^5*d^7*e^2 + 415701*A
*a^5*d*e^8 - 608*(5*B*a*b^4 + A*b^5)*d^6*e^3 + 5168*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^4 - 19380*(B*a^3*b^2 + A*a^2
*b^3)*d^4*e^5 + 20995*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^6 + 230945*(B*a^5 + 5*A*a^4*b)*d^2*e^7)*x^2 - (1536*B*b^5*
d^8*e - 1247103*A*a^5*d^2*e^7 - 2432*(5*B*a*b^4 + A*b^5)*d^7*e^2 + 20672*(2*B*a^2*b^3 + A*a*b^4)*d^6*e^3 - 775
20*(B*a^3*b^2 + A*a^2*b^3)*d^5*e^4 + 83980*(B*a^4*b + 2*A*a^3*b^2)*d^4*e^5 - 46189*(B*a^5 + 5*A*a^4*b)*d^3*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)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.49747, size = 3772, normalized size = 8.35 \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)^(5/2)*(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^2*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^2*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^2*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^2*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^2*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^2*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^2*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^2*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^2*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^2*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^2*e^(-6)*sgn(b*x + a) + 4849845
*(x*e + d)^(3/2)*A*a^5*d^2*sgn(b*x + a) + 277134*(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*e^(-1)*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^4*b*d*e^(-1)*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^4*b*d*e^(-2)*sgn(b*x + a) + 923780*(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*e^(-2)*sgn(b*x + a) + 83980*(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*d*e^(-3)*sgn(b*x + a) + 83980*(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*d*e^(-3)*sgn(b*x + a) + 32300*(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*e^(-4)*sgn(b*x + a) + 16150*(693*(x*e + d)^(13/2) - 40
95*(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 - 300
3*(x*e + d)^(3/2)*d^5)*A*a*b^4*d*e^(-4)*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)*B*a*b^4*d*e^(-5)*sgn(b*x + a) + 646*(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*e^(-5)*sgn(b*x + a) + 266*(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)*B*b^5*d*e^(-6)*sgn(b*x + a)
+ 1939938*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^5*d*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)*B*a^5*e^(-1)*sgn(b*x + a) + 230945*(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*e^(-1)*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)*B*a^4*b*e^(-2)*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)*A*a^3*b^2*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)*B*a^3*b^2*e^(-3)*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^2*b^3*e^(-3)*sgn(b*x + a) + 3230*(3003*(x*e + d)^(15/2) - 2079
0*(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 - 5
4054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*B*a^2*b^3*e^(-4)*sgn(b*x + a) + 1615*(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*b^4*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)*B*a*b^
4*e^(-5)*sgn(b*x + a) + 133*(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*b^5*e^(-5)*sgn(b*x + a) + 7*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(1
7/2)*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 - 1293292
0*(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*b^5*e^(-6)*sgn(b*x + a) + 138567*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^
5*sgn(b*x + a))*e^(-1)