3.18.42 \(\int (A+B x) (d+e x)^2 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [1742]

Optimal. Leaf size=198 \[ \frac {(A b-a B) (b d-a e)^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac {B e^2 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^4} \]

[Out]

1/6*(A*b-B*a)*(-a*e+b*d)^2*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^4+1/7*(-a*e+b*d)*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^6*((
b*x+a)^2)^(1/2)/b^4+1/8*e*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^7*((b*x+a)^2)^(1/2)/b^4+1/9*B*e^2*(b*x+a)^8*((b*x+a)
^2)^(1/2)/b^4

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Rubi [A]
time = 0.22, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {784, 78} \begin {gather*} \frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (-3 a B e+A b e+2 b B d)}{8 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e) (-3 a B e+2 A b e+b B d)}{7 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^2}{6 b^4}+\frac {B e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*(b*d - a*e)^2*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^4) + ((b*d - a*e)*(b*B*d + 2*A*b*e
- 3*a*B*e)*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^4) + (e*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^7*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(8*b^4) + (B*e^2*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^4)

Rule 78

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]))))

Rule 784

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]

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^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)^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 {(A b-a B) (b d-a e)^2 \left (a b+b^2 x\right )^5}{b^3}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) \left (a b+b^2 x\right )^6}{b^4}+\frac {e (2 b B d+A b e-3 a B e) \left (a b+b^2 x\right )^7}{b^5}+\frac {B e^2 \left (a b+b^2 x\right )^8}{b^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (b d-a e)^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac {B e^2 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 347, normalized size = 1.75 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (42 a^5 \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+42 a^4 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+84 a^3 b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+12 a^2 b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )+3 a b^4 x^4 \left (8 A \left (21 d^2+35 d e x+15 e^2 x^2\right )+5 B x \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )+b^5 x^5 \left (3 A \left (28 d^2+48 d e x+21 e^2 x^2\right )+2 B x \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )\right )}{504 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(42*a^5*(4*A*(3*d^2 + 3*d*e*x + e^2*x^2) + B*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)) + 42*a^4*b*
x*(5*A*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 2*B*x*(10*d^2 + 15*d*e*x + 6*e^2*x^2)) + 84*a^3*b^2*x^2*(2*A*(10*d^2 +
15*d*e*x + 6*e^2*x^2) + B*x*(15*d^2 + 24*d*e*x + 10*e^2*x^2)) + 12*a^2*b^3*x^3*(7*A*(15*d^2 + 24*d*e*x + 10*e^
2*x^2) + 4*B*x*(21*d^2 + 35*d*e*x + 15*e^2*x^2)) + 3*a*b^4*x^4*(8*A*(21*d^2 + 35*d*e*x + 15*e^2*x^2) + 5*B*x*(
28*d^2 + 48*d*e*x + 21*e^2*x^2)) + b^5*x^5*(3*A*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 2*B*x*(36*d^2 + 63*d*e*x +
28*e^2*x^2))))/(504*(a + b*x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(479\) vs. \(2(146)=292\).
time = 0.80, size = 480, normalized size = 2.42

method	result	size
gosper	\(\frac {x \left (56 B \,e^{2} b^{5} x^{8}+63 x^{7} A \,b^{5} e^{2}+315 x^{7} B \,e^{2} a \,b^{4}+126 x^{7} B \,b^{5} d e +360 x^{6} A a \,b^{4} e^{2}+144 x^{6} A \,b^{5} d e +720 x^{6} B \,e^{2} a^{2} b^{3}+720 x^{6} B a \,b^{4} d e +72 x^{6} B \,b^{5} d^{2}+840 x^{5} A \,a^{2} b^{3} e^{2}+840 x^{5} A a \,b^{4} d e +84 x^{5} A \,d^{2} b^{5}+840 x^{5} B \,e^{2} a^{3} b^{2}+1680 x^{5} B \,a^{2} b^{3} d e +420 x^{5} B a \,b^{4} d^{2}+1008 A \,a^{3} b^{2} e^{2} x^{4}+2016 A \,a^{2} b^{3} d e \,x^{4}+504 A a \,b^{4} d^{2} x^{4}+504 B \,a^{4} b \,e^{2} x^{4}+2016 B \,a^{3} b^{2} d e \,x^{4}+1008 B \,a^{2} b^{3} d^{2} x^{4}+630 x^{3} A \,a^{4} b \,e^{2}+2520 x^{3} A \,a^{3} b^{2} d e +1260 x^{3} A \,d^{2} a^{2} b^{3}+126 x^{3} B \,e^{2} a^{5}+1260 x^{3} B \,a^{4} b d e +1260 x^{3} B \,a^{3} b^{2} d^{2}+168 x^{2} A \,a^{5} e^{2}+1680 x^{2} A \,a^{4} b d e +1680 x^{2} A \,d^{2} a^{3} b^{2}+336 x^{2} B \,a^{5} d e +840 x^{2} B \,a^{4} b \,d^{2}+504 x A \,a^{5} d e +1260 x A \,d^{2} a^{4} b +252 x B \,a^{5} d^{2}+504 A \,d^{2} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5}}\)	\(480\)
default	\(\frac {x \left (56 B \,e^{2} b^{5} x^{8}+63 x^{7} A \,b^{5} e^{2}+315 x^{7} B \,e^{2} a \,b^{4}+126 x^{7} B \,b^{5} d e +360 x^{6} A a \,b^{4} e^{2}+144 x^{6} A \,b^{5} d e +720 x^{6} B \,e^{2} a^{2} b^{3}+720 x^{6} B a \,b^{4} d e +72 x^{6} B \,b^{5} d^{2}+840 x^{5} A \,a^{2} b^{3} e^{2}+840 x^{5} A a \,b^{4} d e +84 x^{5} A \,d^{2} b^{5}+840 x^{5} B \,e^{2} a^{3} b^{2}+1680 x^{5} B \,a^{2} b^{3} d e +420 x^{5} B a \,b^{4} d^{2}+1008 A \,a^{3} b^{2} e^{2} x^{4}+2016 A \,a^{2} b^{3} d e \,x^{4}+504 A a \,b^{4} d^{2} x^{4}+504 B \,a^{4} b \,e^{2} x^{4}+2016 B \,a^{3} b^{2} d e \,x^{4}+1008 B \,a^{2} b^{3} d^{2} x^{4}+630 x^{3} A \,a^{4} b \,e^{2}+2520 x^{3} A \,a^{3} b^{2} d e +1260 x^{3} A \,d^{2} a^{2} b^{3}+126 x^{3} B \,e^{2} a^{5}+1260 x^{3} B \,a^{4} b d e +1260 x^{3} B \,a^{3} b^{2} d^{2}+168 x^{2} A \,a^{5} e^{2}+1680 x^{2} A \,a^{4} b d e +1680 x^{2} A \,d^{2} a^{3} b^{2}+336 x^{2} B \,a^{5} d e +840 x^{2} B \,a^{4} b \,d^{2}+504 x A \,a^{5} d e +1260 x A \,d^{2} a^{4} b +252 x B \,a^{5} d^{2}+504 A \,d^{2} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5}}\)	\(480\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, B \,e^{2} b^{5} x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (A \,e^{2}+2 B d e \right ) b^{5}+5 B \,e^{2} a \,b^{4}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (2 A d e +B \,d^{2}\right ) b^{5}+5 \left (A \,e^{2}+2 B d e \right ) a \,b^{4}+10 B \,e^{2} a^{2} b^{3}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A \,d^{2} b^{5}+5 \left (2 A d e +B \,d^{2}\right ) a \,b^{4}+10 \left (A \,e^{2}+2 B d e \right ) a^{2} b^{3}+10 B \,e^{2} a^{3} b^{2}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,d^{2} a \,b^{4}+10 \left (2 A d e +B \,d^{2}\right ) a^{2} b^{3}+10 \left (A \,e^{2}+2 B d e \right ) a^{3} b^{2}+5 B \,e^{2} a^{4} b \right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,d^{2} a^{2} b^{3}+10 \left (2 A d e +B \,d^{2}\right ) a^{3} b^{2}+5 \left (A \,e^{2}+2 B d e \right ) a^{4} b +B \,e^{2} a^{5}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,d^{2} a^{3} b^{2}+5 \left (2 A d e +B \,d^{2}\right ) a^{4} b +\left (A \,e^{2}+2 B d e \right ) a^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,d^{2} a^{4} b +\left (2 A d e +B \,d^{2}\right ) a^{5}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A \,d^{2} a^{5} x}{b x +a}\)	\(517\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^2*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/504*x*(56*B*b^5*e^2*x^8+63*A*b^5*e^2*x^7+315*B*a*b^4*e^2*x^7+126*B*b^5*d*e*x^7+360*A*a*b^4*e^2*x^6+144*A*b^5
*d*e*x^6+720*B*a^2*b^3*e^2*x^6+720*B*a*b^4*d*e*x^6+72*B*b^5*d^2*x^6+840*A*a^2*b^3*e^2*x^5+840*A*a*b^4*d*e*x^5+
84*A*b^5*d^2*x^5+840*B*a^3*b^2*e^2*x^5+1680*B*a^2*b^3*d*e*x^5+420*B*a*b^4*d^2*x^5+1008*A*a^3*b^2*e^2*x^4+2016*
A*a^2*b^3*d*e*x^4+504*A*a*b^4*d^2*x^4+504*B*a^4*b*e^2*x^4+2016*B*a^3*b^2*d*e*x^4+1008*B*a^2*b^3*d^2*x^4+630*A*
a^4*b*e^2*x^3+2520*A*a^3*b^2*d*e*x^3+1260*A*a^2*b^3*d^2*x^3+126*B*a^5*e^2*x^3+1260*B*a^4*b*d*e*x^3+1260*B*a^3*
b^2*d^2*x^3+168*A*a^5*e^2*x^2+1680*A*a^4*b*d*e*x^2+1680*A*a^3*b^2*d^2*x^2+336*B*a^5*d*e*x^2+840*B*a^4*b*d^2*x^
2+504*A*a^5*d*e*x+1260*A*a^4*b*d^2*x+252*B*a^5*d^2*x+504*A*a^5*d^2)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (152) = 304\).
time = 0.28, size = 454, normalized size = 2.29 \begin {gather*} \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A d^{2} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a d^{2}}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{3} x e^{2}}{6 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B x^{2} e^{2}}{9 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, B d e + A e^{2}\right )} a^{2} x}{6 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (B d^{2} + 2 \, A d e\right )} a x}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{4} e^{2}}{6 \, b^{4}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a x e^{2}}{72 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, B d e + A e^{2}\right )} a^{3}}{6 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (B d^{2} + 2 \, A d e\right )} a^{2}}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, B d e + A e^{2}\right )} x}{8 \, b^{2}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{2} e^{2}}{504 \, b^{4}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, B d e + A e^{2}\right )} a}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (B d^{2} + 2 \, A d e\right )}}{7 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*d^2*x + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*a*d^2/b - 1/6*(b^2*x^2 + 2
*a*b*x + a^2)^(5/2)*B*a^3*x*e^2/b^3 + 1/9*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*x^2*e^2/b^2 + 1/6*(b^2*x^2 + 2*a*b
*x + a^2)^(5/2)*(2*B*d*e + A*e^2)*a^2*x/b^2 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*d^2 + 2*A*d*e)*a*x/b - 1/
6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^4*e^2/b^4 - 11/72*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a*x*e^2/b^3 + 1/6*(b
^2*x^2 + 2*a*b*x + a^2)^(5/2)*(2*B*d*e + A*e^2)*a^3/b^3 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*d^2 + 2*A*d*e
)*a^2/b^2 + 1/8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(2*B*d*e + A*e^2)*x/b^2 + 83/504*(b^2*x^2 + 2*a*b*x + a^2)^(7/
2)*B*a^2*e^2/b^4 - 9/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(2*B*d*e + A*e^2)*a/b^3 + 1/7*(b^2*x^2 + 2*a*b*x + a^2
)^(7/2)*(B*d^2 + 2*A*d*e)/b^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (152) = 304\).
time = 1.07, size = 389, normalized size = 1.96 \begin {gather*} \frac {1}{7} \, B b^{5} d^{2} x^{7} + A a^{5} d^{2} x + \frac {1}{6} \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} x^{6} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} x^{5} + \frac {5}{2} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} x^{4} + \frac {5}{3} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} x^{2} + \frac {1}{504} \, {\left (56 \, B b^{5} x^{9} + 168 \, A a^{5} x^{3} + 63 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + 360 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 504 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + 126 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4}\right )} e^{2} + \frac {1}{84} \, {\left (21 \, B b^{5} d x^{8} + 84 \, A a^{5} d x^{2} + 24 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d x^{7} + 140 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d x^{6} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d x^{5} + 210 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d x^{4} + 56 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/7*B*b^5*d^2*x^7 + A*a^5*d^2*x + 1/6*(5*B*a*b^4 + A*b^5)*d^2*x^6 + (2*B*a^2*b^3 + A*a*b^4)*d^2*x^5 + 5/2*(B*a
^3*b^2 + A*a^2*b^3)*d^2*x^4 + 5/3*(B*a^4*b + 2*A*a^3*b^2)*d^2*x^3 + 1/2*(B*a^5 + 5*A*a^4*b)*d^2*x^2 + 1/504*(5
6*B*b^5*x^9 + 168*A*a^5*x^3 + 63*(5*B*a*b^4 + A*b^5)*x^8 + 360*(2*B*a^2*b^3 + A*a*b^4)*x^7 + 840*(B*a^3*b^2 +
A*a^2*b^3)*x^6 + 504*(B*a^4*b + 2*A*a^3*b^2)*x^5 + 126*(B*a^5 + 5*A*a^4*b)*x^4)*e^2 + 1/84*(21*B*b^5*d*x^8 + 8
4*A*a^5*d*x^2 + 24*(5*B*a*b^4 + A*b^5)*d*x^7 + 140*(2*B*a^2*b^3 + A*a*b^4)*d*x^6 + 336*(B*a^3*b^2 + A*a^2*b^3)
*d*x^5 + 210*(B*a^4*b + 2*A*a^3*b^2)*d*x^4 + 56*(B*a^5 + 5*A*a^4*b)*d*x^3)*e

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + B x\right ) \left (d + e x\right )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**2*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**2*((a + b*x)**2)**(5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (152) = 304\).
time = 1.13, size = 679, normalized size = 3.43 \begin {gather*} \frac {1}{9} \, B b^{5} x^{9} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, B b^{5} d x^{8} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, B b^{5} d^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, B a b^{4} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{8} \, A b^{5} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, B a b^{4} d x^{7} e \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, A b^{5} d x^{7} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, B a b^{4} d^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A b^{5} d^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, B a^{2} b^{3} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, A a b^{4} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, B a^{2} b^{3} d x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, A a b^{4} d x^{6} e \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{2} b^{3} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + A a b^{4} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{3} b^{2} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, A a^{2} b^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, B a^{3} b^{2} d x^{5} e \mathrm {sgn}\left (b x + a\right ) + 4 \, A a^{2} b^{3} d x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, B a^{3} b^{2} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{2} b^{3} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + B a^{4} b x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} b^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, B a^{4} b d x^{4} e \mathrm {sgn}\left (b x + a\right ) + 5 \, A a^{3} b^{2} d x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{4} b d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, A a^{3} b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, B a^{5} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, A a^{4} b x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B a^{5} d x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, A a^{4} b d x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{5} d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{4} b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, A a^{5} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + A a^{5} d x^{2} e \mathrm {sgn}\left (b x + a\right ) + A a^{5} d^{2} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/9*B*b^5*x^9*e^2*sgn(b*x + a) + 1/4*B*b^5*d*x^8*e*sgn(b*x + a) + 1/7*B*b^5*d^2*x^7*sgn(b*x + a) + 5/8*B*a*b^4
*x^8*e^2*sgn(b*x + a) + 1/8*A*b^5*x^8*e^2*sgn(b*x + a) + 10/7*B*a*b^4*d*x^7*e*sgn(b*x + a) + 2/7*A*b^5*d*x^7*e
*sgn(b*x + a) + 5/6*B*a*b^4*d^2*x^6*sgn(b*x + a) + 1/6*A*b^5*d^2*x^6*sgn(b*x + a) + 10/7*B*a^2*b^3*x^7*e^2*sgn
(b*x + a) + 5/7*A*a*b^4*x^7*e^2*sgn(b*x + a) + 10/3*B*a^2*b^3*d*x^6*e*sgn(b*x + a) + 5/3*A*a*b^4*d*x^6*e*sgn(b
*x + a) + 2*B*a^2*b^3*d^2*x^5*sgn(b*x + a) + A*a*b^4*d^2*x^5*sgn(b*x + a) + 5/3*B*a^3*b^2*x^6*e^2*sgn(b*x + a)
+ 5/3*A*a^2*b^3*x^6*e^2*sgn(b*x + a) + 4*B*a^3*b^2*d*x^5*e*sgn(b*x + a) + 4*A*a^2*b^3*d*x^5*e*sgn(b*x + a) +
5/2*B*a^3*b^2*d^2*x^4*sgn(b*x + a) + 5/2*A*a^2*b^3*d^2*x^4*sgn(b*x + a) + B*a^4*b*x^5*e^2*sgn(b*x + a) + 2*A*a
^3*b^2*x^5*e^2*sgn(b*x + a) + 5/2*B*a^4*b*d*x^4*e*sgn(b*x + a) + 5*A*a^3*b^2*d*x^4*e*sgn(b*x + a) + 5/3*B*a^4*
b*d^2*x^3*sgn(b*x + a) + 10/3*A*a^3*b^2*d^2*x^3*sgn(b*x + a) + 1/4*B*a^5*x^4*e^2*sgn(b*x + a) + 5/4*A*a^4*b*x^
4*e^2*sgn(b*x + a) + 2/3*B*a^5*d*x^3*e*sgn(b*x + a) + 10/3*A*a^4*b*d*x^3*e*sgn(b*x + a) + 1/2*B*a^5*d^2*x^2*sg
n(b*x + a) + 5/2*A*a^4*b*d^2*x^2*sgn(b*x + a) + 1/3*A*a^5*x^3*e^2*sgn(b*x + a) + A*a^5*d*x^2*e*sgn(b*x + a) +
A*a^5*d^2*x*sgn(b*x + a)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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