3.2113 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=370 \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^7 (a+b x)}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^7 (a+b x)}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^7 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^6}{e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (4*b*(b*d - a*e)^5*(d + e*x)^(
3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (6*b^2*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a +
b*x)) + (10*b^4*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (12*b^5*(b*d
- a*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(13/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x))

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Rubi [A]  time = 0.14003, antiderivative size = 370, 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} \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^7 (a+b x)}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^7 (a+b x)}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^7 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^6}{e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (4*b*(b*d - a*e)^5*(d + e*x)^(
3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (6*b^2*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a +
b*x)) + (10*b^4*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (12*b^5*(b*d
- a*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(13/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(13*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 \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{(a+b x) \left (a b+b^2 x\right )^5}{\sqrt{d+e x}} \, 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 \frac{(a+b x)^6}{\sqrt{d+e x}} \, 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}{e^6 \sqrt{d+e x}}-\frac{6 b (b d-a e)^5 \sqrt{d+e x}}{e^6}+\frac{15 b^2 (b d-a e)^4 (d+e x)^{3/2}}{e^6}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{5/2}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{7/2}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{9/2}}{e^6}+\frac{b^6 (d+e x)^{11/2}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 (b d-a e)^6 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac{4 b (b d-a e)^5 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac{6 b^2 (b d-a e)^4 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac{40 b^3 (b d-a e)^3 (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^4 (b d-a e)^2 (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac{12 b^5 (b d-a e) (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac{2 b^6 (d+e x)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.10393, size = 163, normalized size = 0.44 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (9009 b^2 (d+e x)^2 (b d-a e)^4-8580 b^3 (d+e x)^3 (b d-a e)^3+5005 b^4 (d+e x)^4 (b d-a e)^2-1638 b^5 (d+e x)^5 (b d-a e)-6006 b (d+e x) (b d-a e)^5+3003 (b d-a e)^6+231 b^6 (d+e x)^6\right )}{3003 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(3003*(b*d - a*e)^6 - 6006*b*(b*d - a*e)^5*(d + e*x) + 9009*b^2*(b*d - a*e)
^4*(d + e*x)^2 - 8580*b^3*(b*d - a*e)^3*(d + e*x)^3 + 5005*b^4*(b*d - a*e)^2*(d + e*x)^4 - 1638*b^5*(b*d - a*e
)*(d + e*x)^5 + 231*b^6*(d + e*x)^6))/(3003*e^7*(a + b*x))

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Maple [A]  time = 0.007, size = 393, normalized size = 1.1 \begin{align*}{\frac{462\,{x}^{6}{b}^{6}{e}^{6}+3276\,{x}^{5}a{b}^{5}{e}^{6}-504\,{x}^{5}{b}^{6}d{e}^{5}+10010\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-3640\,{x}^{4}a{b}^{5}d{e}^{5}+560\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+17160\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-11440\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+4160\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-640\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+18018\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-20592\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+13728\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-4992\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+768\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+12012\,x{a}^{5}b{e}^{6}-24024\,x{a}^{4}{b}^{2}d{e}^{5}+27456\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-18304\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+6656\,xa{b}^{5}{d}^{4}{e}^{2}-1024\,x{b}^{6}{d}^{5}e+6006\,{a}^{6}{e}^{6}-24024\,d{e}^{5}{a}^{5}b+48048\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-54912\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+36608\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-13312\,a{b}^{5}{d}^{5}e+2048\,{b}^{6}{d}^{6}}{3003\, \left ( bx+a \right ) ^{5}{e}^{7}}\sqrt{ex+d} \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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x)

[Out]

2/3003*(e*x+d)^(1/2)*(231*b^6*e^6*x^6+1638*a*b^5*e^6*x^5-252*b^6*d*e^5*x^5+5005*a^2*b^4*e^6*x^4-1820*a*b^5*d*e
^5*x^4+280*b^6*d^2*e^4*x^4+8580*a^3*b^3*e^6*x^3-5720*a^2*b^4*d*e^5*x^3+2080*a*b^5*d^2*e^4*x^3-320*b^6*d^3*e^3*
x^3+9009*a^4*b^2*e^6*x^2-10296*a^3*b^3*d*e^5*x^2+6864*a^2*b^4*d^2*e^4*x^2-2496*a*b^5*d^3*e^3*x^2+384*b^6*d^4*e
^2*x^2+6006*a^5*b*e^6*x-12012*a^4*b^2*d*e^5*x+13728*a^3*b^3*d^2*e^4*x-9152*a^2*b^4*d^3*e^3*x+3328*a*b^5*d^4*e^
2*x-512*b^6*d^5*e*x+3003*a^6*e^6-12012*a^5*b*d*e^5+24024*a^4*b^2*d^2*e^4-27456*a^3*b^3*d^3*e^3+18304*a^2*b^4*d
^4*e^2-6656*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 = 1.34959, size = 1023, normalized size = 2.76 \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)^(1/2),x, algorithm="maxima")

[Out]

2/693*(63*b^5*e^6*x^6 - 256*b^5*d^6 + 1408*a*b^4*d^5*e - 3168*a^2*b^3*d^4*e^2 + 3696*a^3*b^2*d^3*e^3 - 2310*a^
4*b*d^2*e^4 + 693*a^5*d*e^5 - 7*(b^5*d*e^5 - 55*a*b^4*e^6)*x^5 + 5*(2*b^5*d^2*e^4 - 11*a*b^4*d*e^5 + 198*a^2*b
^3*e^6)*x^4 - 2*(8*b^5*d^3*e^3 - 44*a*b^4*d^2*e^4 + 99*a^2*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (32*b^5*d^4*e^2
- 176*a*b^4*d^3*e^3 + 396*a^2*b^3*d^2*e^4 - 462*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (128*b^5*d^5*e - 704*a*b
^4*d^4*e^2 + 1584*a^2*b^3*d^3*e^3 - 1848*a^3*b^2*d^2*e^4 + 1155*a^4*b*d*e^5 - 693*a^5*e^6)*x)*a/(sqrt(e*x + d)
*e^6) + 2/9009*(693*b^5*e^7*x^7 + 3072*b^5*d^7 - 16640*a*b^4*d^6*e + 36608*a^2*b^3*d^5*e^2 - 41184*a^3*b^2*d^4
*e^3 + 24024*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 - 63*(b^5*d*e^6 - 65*a*b^4*e^7)*x^6 + 7*(12*b^5*d^2*e^5 - 65*a*b
^4*d*e^6 + 1430*a^2*b^3*e^7)*x^5 - 10*(12*b^5*d^3*e^4 - 65*a*b^4*d^2*e^5 + 143*a^2*b^3*d*e^6 - 1287*a^3*b^2*e^
7)*x^4 + (192*b^5*d^4*e^3 - 1040*a*b^4*d^3*e^4 + 2288*a^2*b^3*d^2*e^5 - 2574*a^3*b^2*d*e^6 + 9009*a^4*b*e^7)*x
^3 - (384*b^5*d^5*e^2 - 2080*a*b^4*d^4*e^3 + 4576*a^2*b^3*d^3*e^4 - 5148*a^3*b^2*d^2*e^5 + 3003*a^4*b*d*e^6 -
3003*a^5*e^7)*x^2 + (1536*b^5*d^6*e - 8320*a*b^4*d^5*e^2 + 18304*a^2*b^3*d^4*e^3 - 20592*a^3*b^2*d^3*e^4 + 120
12*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*b/(sqrt(e*x + d)*e^7)

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Fricas [A]  time = 1.02207, size = 815, normalized size = 2.2 \begin{align*} \frac{2 \,{\left (231 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 6656 \, a b^{5} d^{5} e + 18304 \, a^{2} b^{4} d^{4} e^{2} - 27456 \, a^{3} b^{3} d^{3} e^{3} + 24024 \, a^{4} b^{2} d^{2} e^{4} - 12012 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} - 126 \,{\left (2 \, b^{6} d e^{5} - 13 \, a b^{5} e^{6}\right )} x^{5} + 35 \,{\left (8 \, b^{6} d^{2} e^{4} - 52 \, a b^{5} d e^{5} + 143 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (16 \, b^{6} d^{3} e^{3} - 104 \, a b^{5} d^{2} e^{4} + 286 \, a^{2} b^{4} d e^{5} - 429 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{4} e^{2} - 832 \, a b^{5} d^{3} e^{3} + 2288 \, a^{2} b^{4} d^{2} e^{4} - 3432 \, a^{3} b^{3} d e^{5} + 3003 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \,{\left (256 \, b^{6} d^{5} e - 1664 \, a b^{5} d^{4} e^{2} + 4576 \, a^{2} b^{4} d^{3} e^{3} - 6864 \, a^{3} b^{3} d^{2} e^{4} + 6006 \, a^{4} b^{2} d e^{5} - 3003 \, a^{5} b e^{6}\right )} x\right )} \sqrt{e x + d}}{3003 \, 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)^(1/2),x, algorithm="fricas")

[Out]

2/3003*(231*b^6*e^6*x^6 + 1024*b^6*d^6 - 6656*a*b^5*d^5*e + 18304*a^2*b^4*d^4*e^2 - 27456*a^3*b^3*d^3*e^3 + 24
024*a^4*b^2*d^2*e^4 - 12012*a^5*b*d*e^5 + 3003*a^6*e^6 - 126*(2*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 + 35*(8*b^6*d^2*
e^4 - 52*a*b^5*d*e^5 + 143*a^2*b^4*e^6)*x^4 - 20*(16*b^6*d^3*e^3 - 104*a*b^5*d^2*e^4 + 286*a^2*b^4*d*e^5 - 429
*a^3*b^3*e^6)*x^3 + 3*(128*b^6*d^4*e^2 - 832*a*b^5*d^3*e^3 + 2288*a^2*b^4*d^2*e^4 - 3432*a^3*b^3*d*e^5 + 3003*
a^4*b^2*e^6)*x^2 - 2*(256*b^6*d^5*e - 1664*a*b^5*d^4*e^2 + 4576*a^2*b^4*d^3*e^3 - 6864*a^3*b^3*d^2*e^4 + 6006*
a^4*b^2*d*e^5 - 3003*a^5*b*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)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.23485, size = 590, normalized size = 1.59 \begin{align*} \frac{2}{3003} \,{\left (6006 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{5} b e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) + 3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} a^{4} b^{2} e^{\left (-2\right )} \mathrm{sgn}\left (b x + a\right ) + 1716 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} a^{3} b^{3} e^{\left (-3\right )} \mathrm{sgn}\left (b x + a\right ) + 143 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} a^{2} b^{4} e^{\left (-4\right )} \mathrm{sgn}\left (b x + a\right ) + 26 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} - 693 \, \sqrt{x e + d} d^{5}\right )} a b^{5} e^{\left (-5\right )} \mathrm{sgn}\left (b x + a\right ) +{\left (231 \,{\left (x e + d\right )}^{\frac{13}{2}} - 1638 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 5005 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 8580 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 6006 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} + 3003 \, \sqrt{x e + d} d^{6}\right )} b^{6} e^{\left (-6\right )} \mathrm{sgn}\left (b x + a\right ) + 3003 \, \sqrt{x e + d} a^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\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)^(1/2),x, algorithm="giac")

[Out]

2/3003*(6006*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*e^(-1)*sgn(b*x + a) + 3003*(3*(x*e + d)^(5/2) - 10*(x
*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4*b^2*e^(-2)*sgn(b*x + a) + 1716*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(
5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^3*b^3*e^(-3)*sgn(b*x + a) + 143*(35*(x*e + d)^(9/2)
- 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^2*b^4*e
^(-4)*sgn(b*x + a) + 26*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d
)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a*b^5*e^(-5)*sgn(b*x + a) + (231*(x*e + d)^(13
/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4
- 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^6*e^(-6)*sgn(b*x + a) + 3003*sqrt(x*e + d)*a^6*sgn(b*x
+ a))*e^(-1)