3.2110 \(\int (a+b x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=374 \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{19/2}}{19 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (b d-a e)}{17 e^7 (a+b x)}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)^2}{e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^3}{13 e^7 (a+b x)}+\frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^4}{11 e^7 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^5}{3 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^6}{7 e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (4*b*(b*d - a*e)^5*(d + e*
x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(11/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(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*(b*d - a*e)^2*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (
12*b^5*(b*d - a*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^6*(d + e*x)^(19/2
)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(19*e^7*(a + b*x))

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Rubi [A]  time = 0.16749, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {770, 21, 43} \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{19/2}}{19 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (b d-a e)}{17 e^7 (a+b x)}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)^2}{e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^3}{13 e^7 (a+b x)}+\frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^4}{11 e^7 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^5}{3 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^6}{7 e^7 (a+b x)} \]

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)^6*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (4*b*(b*d - a*e)^5*(d + e*
x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(11/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(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*(b*d - a*e)^2*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (
12*b^5*(b*d - a*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^6*(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 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^{5/2} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^{5/2} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^6 (d+e x)^{5/2}}{e^6}-\frac{6 b (b d-a e)^5 (d+e x)^{7/2}}{e^6}+\frac{15 b^2 (b d-a e)^4 (d+e x)^{9/2}}{e^6}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{11/2}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{13/2}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{15/2}}{e^6}+\frac{b^6 (d+e x)^{17/2}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 (b d-a e)^6 (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac{4 b (b d-a e)^5 (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac{30 b^2 (b d-a e)^4 (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac{40 b^3 (b d-a e)^3 (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 (b d-a e)^2 (d+e x)^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac{12 b^5 (b d-a 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^6 (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 = 0.229151, size = 163, normalized size = 0.44 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} \left (1322685 b^2 (d+e x)^2 (b d-a e)^4-1492260 b^3 (d+e x)^3 (b d-a e)^3+969969 b^4 (d+e x)^4 (b d-a e)^2-342342 b^5 (d+e x)^5 (b d-a e)-646646 b (d+e x) (b d-a e)^5+138567 (b d-a e)^6+51051 b^6 (d+e x)^6\right )}{969969 e^7 (a+b x)} \]

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)*(138567*(b*d - a*e)^6 - 646646*b*(b*d - a*e)^5*(d + e*x) + 1322685*b^2*(b
*d - a*e)^4*(d + e*x)^2 - 1492260*b^3*(b*d - a*e)^3*(d + e*x)^3 + 969969*b^4*(b*d - a*e)^2*(d + e*x)^4 - 34234
2*b^5*(b*d - a*e)*(d + e*x)^5 + 51051*b^6*(d + e*x)^6))/(969969*e^7*(a + b*x))

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Maple [A]  time = 0.007, size = 393, normalized size = 1.1 \begin{align*}{\frac{102102\,{x}^{6}{b}^{6}{e}^{6}+684684\,{x}^{5}a{b}^{5}{e}^{6}-72072\,{x}^{5}{b}^{6}d{e}^{5}+1939938\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-456456\,{x}^{4}a{b}^{5}d{e}^{5}+48048\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+2984520\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-1193808\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+280896\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-29568\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+2645370\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-1627920\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+651168\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-153216\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+16128\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+1293292\,x{a}^{5}b{e}^{6}-1175720\,x{a}^{4}{b}^{2}d{e}^{5}+723520\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-289408\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+68096\,xa{b}^{5}{d}^{4}{e}^{2}-7168\,x{b}^{6}{d}^{5}e+277134\,{a}^{6}{e}^{6}-369512\,d{e}^{5}{a}^{5}b+335920\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-206720\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+82688\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-19456\,a{b}^{5}{d}^{5}e+2048\,{b}^{6}{d}^{6}}{969969\,{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]

2/969969*(e*x+d)^(7/2)*(51051*b^6*e^6*x^6+342342*a*b^5*e^6*x^5-36036*b^6*d*e^5*x^5+969969*a^2*b^4*e^6*x^4-2282
28*a*b^5*d*e^5*x^4+24024*b^6*d^2*e^4*x^4+1492260*a^3*b^3*e^6*x^3-596904*a^2*b^4*d*e^5*x^3+140448*a*b^5*d^2*e^4
*x^3-14784*b^6*d^3*e^3*x^3+1322685*a^4*b^2*e^6*x^2-813960*a^3*b^3*d*e^5*x^2+325584*a^2*b^4*d^2*e^4*x^2-76608*a
*b^5*d^3*e^3*x^2+8064*b^6*d^4*e^2*x^2+646646*a^5*b*e^6*x-587860*a^4*b^2*d*e^5*x+361760*a^3*b^3*d^2*e^4*x-14470
4*a^2*b^4*d^3*e^3*x+34048*a*b^5*d^4*e^2*x-3584*b^6*d^5*e*x+138567*a^6*e^6-184756*a^5*b*d*e^5+167960*a^4*b^2*d^
2*e^4-103360*a^3*b^3*d^3*e^3+41344*a^2*b^4*d^4*e^2-9728*a*b^5*d^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a
)^5

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Maxima [B]  time = 1.26283, size = 1458, normalized size = 3.9 \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 = 1.14058, size = 1497, normalized size = 4. \begin{align*} \frac{2 \,{\left (51051 \, b^{6} e^{9} x^{9} + 1024 \, b^{6} d^{9} - 9728 \, a b^{5} d^{8} e + 41344 \, a^{2} b^{4} d^{7} e^{2} - 103360 \, a^{3} b^{3} d^{6} e^{3} + 167960 \, a^{4} b^{2} d^{5} e^{4} - 184756 \, a^{5} b d^{4} e^{5} + 138567 \, a^{6} d^{3} e^{6} + 9009 \,{\left (13 \, b^{6} d e^{8} + 38 \, a b^{5} e^{9}\right )} x^{8} + 3003 \,{\left (23 \, b^{6} d^{2} e^{7} + 266 \, a b^{5} d e^{8} + 323 \, a^{2} b^{4} e^{9}\right )} x^{7} + 231 \,{\left (b^{6} d^{3} e^{6} + 2090 \, a b^{5} d^{2} e^{7} + 10013 \, a^{2} b^{4} d e^{8} + 6460 \, a^{3} b^{3} e^{9}\right )} x^{6} - 63 \,{\left (4 \, b^{6} d^{4} e^{5} - 38 \, a b^{5} d^{3} e^{6} - 22933 \, a^{2} b^{4} d^{2} e^{7} - 58140 \, a^{3} b^{3} d e^{8} - 20995 \, a^{4} b^{2} e^{9}\right )} x^{5} + 7 \,{\left (40 \, b^{6} d^{5} e^{4} - 380 \, a b^{5} d^{4} e^{5} + 1615 \, a^{2} b^{4} d^{3} e^{6} + 342380 \, a^{3} b^{3} d^{2} e^{7} + 482885 \, a^{4} b^{2} d e^{8} + 92378 \, a^{5} b e^{9}\right )} x^{4} -{\left (320 \, b^{6} d^{6} e^{3} - 3040 \, a b^{5} d^{5} e^{4} + 12920 \, a^{2} b^{4} d^{4} e^{5} - 32300 \, a^{3} b^{3} d^{3} e^{6} - 2372435 \, a^{4} b^{2} d^{2} e^{7} - 1755182 \, a^{5} b d e^{8} - 138567 \, a^{6} e^{9}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{7} e^{2} - 1216 \, a b^{5} d^{6} e^{3} + 5168 \, a^{2} b^{4} d^{5} e^{4} - 12920 \, a^{3} b^{3} d^{4} e^{5} + 20995 \, a^{4} b^{2} d^{3} e^{6} + 461890 \, a^{5} b d^{2} e^{7} + 138567 \, a^{6} d e^{8}\right )} x^{2} -{\left (512 \, b^{6} d^{8} e - 4864 \, a b^{5} d^{7} e^{2} + 20672 \, a^{2} b^{4} d^{6} e^{3} - 51680 \, a^{3} b^{3} d^{5} e^{4} + 83980 \, a^{4} b^{2} d^{4} e^{5} - 92378 \, a^{5} b d^{3} e^{6} - 415701 \, a^{6} d^{2} e^{7}\right )} x\right )} \sqrt{e x + d}}{969969 \, e^{7}} \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]

2/969969*(51051*b^6*e^9*x^9 + 1024*b^6*d^9 - 9728*a*b^5*d^8*e + 41344*a^2*b^4*d^7*e^2 - 103360*a^3*b^3*d^6*e^3
+ 167960*a^4*b^2*d^5*e^4 - 184756*a^5*b*d^4*e^5 + 138567*a^6*d^3*e^6 + 9009*(13*b^6*d*e^8 + 38*a*b^5*e^9)*x^8
+ 3003*(23*b^6*d^2*e^7 + 266*a*b^5*d*e^8 + 323*a^2*b^4*e^9)*x^7 + 231*(b^6*d^3*e^6 + 2090*a*b^5*d^2*e^7 + 100
13*a^2*b^4*d*e^8 + 6460*a^3*b^3*e^9)*x^6 - 63*(4*b^6*d^4*e^5 - 38*a*b^5*d^3*e^6 - 22933*a^2*b^4*d^2*e^7 - 5814
0*a^3*b^3*d*e^8 - 20995*a^4*b^2*e^9)*x^5 + 7*(40*b^6*d^5*e^4 - 380*a*b^5*d^4*e^5 + 1615*a^2*b^4*d^3*e^6 + 3423
80*a^3*b^3*d^2*e^7 + 482885*a^4*b^2*d*e^8 + 92378*a^5*b*e^9)*x^4 - (320*b^6*d^6*e^3 - 3040*a*b^5*d^5*e^4 + 129
20*a^2*b^4*d^4*e^5 - 32300*a^3*b^3*d^3*e^6 - 2372435*a^4*b^2*d^2*e^7 - 1755182*a^5*b*d*e^8 - 138567*a^6*e^9)*x
^3 + 3*(128*b^6*d^7*e^2 - 1216*a*b^5*d^6*e^3 + 5168*a^2*b^4*d^5*e^4 - 12920*a^3*b^3*d^4*e^5 + 20995*a^4*b^2*d^
3*e^6 + 461890*a^5*b*d^2*e^7 + 138567*a^6*d*e^8)*x^2 - (512*b^6*d^8*e - 4864*a*b^5*d^7*e^2 + 20672*a^2*b^4*d^6
*e^3 - 51680*a^3*b^3*d^5*e^4 + 83980*a^4*b^2*d^4*e^5 - 92378*a^5*b*d^3*e^6 - 415701*a^6*d^2*e^7)*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 = 1.369, size = 2178, normalized size = 5.82 \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*(5819814*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^5*b*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^4*b^2*d^2*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^3*b^3*d^2*e^(-3)*sg
n(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)*a^2*b^4*d^2*e^(-4)*sgn(b*x + a) + 9690*(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 + 15
015*(x*e + d)^(3/2)*d^6)*b^6*d^2*e^(-6)*sgn(b*x + a) + 4849845*(x*e + d)^(3/2)*a^6*d^2*sgn(b*x + a) + 1662804*
(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^5*b*d*e^(-1)*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^4*b^2*d*e^(-2)
*sgn(b*x + a) + 167960*(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^3*b^3*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^2*b^4*d*e^(-4)*sgn(b*x + a) + 3876*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61
425*(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 - 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^6*d*e^(-6)*sgn(b*x + a) + 1939938*(3*(x
*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^6*d*sgn(b*x + a) + 277134*(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^5*b*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)*a^4*b
^2*e^(-2)*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 - 1
2870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a^3*b^3*e^(-3)*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 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*a^2*b^4*e^(-4)*sgn(b*x
+ a) + 798*(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)^(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^6*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^6*sgn(b*x + a))*e^(-
1)