3.18.49 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^5} \, dx\) [1749]

Optimal. Leaf size=421 \[ -\frac {b^4 (5 b B d-A b e-5 a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}-\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {10 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]

[Out]

-b^4*(-A*b*e-5*B*a*e+5*B*b*d)*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+1/2*b^5*B*x^2*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-1/4*
(-a*e+b*d)^5*(-A*e+B*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^4+1/3*(-a*e+b*d)^4*(-5*A*b*e-B*a*e+6*B*b*d)*((b*
x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^3-5/2*b*(-a*e+b*d)^3*(-2*A*b*e-B*a*e+3*B*b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/
(e*x+d)^2+10*b^2*(-a*e+b*d)^2*(-A*b*e-B*a*e+2*B*b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)+5*b^3*(-a*e+b*d)*(-
A*b*e-2*B*a*e+3*B*b*d)*ln(e*x+d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)

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Rubi [A]
time = 0.51, antiderivative size = 421, 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 {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{4 e^7 (a+b x) (d+e x)^4}-\frac {b^4 x \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+5 b B d)}{e^6 (a+b x)}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x)}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^5} \, 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^9 (-5 b B d+A b e+5 a B e)}{e^6}+\frac {b^{10} B x}{e^5}-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^5}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^4}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^3}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^2}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {b^4 (5 b B d-A b e-5 a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}-\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {10 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 497, normalized size = 1.18 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^5 e^5 (3 A e+B (d+4 e x))+5 a^4 b e^4 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+10 a^2 b^3 e^2 \left (3 A e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-B d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )-5 a b^4 e \left (A d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-B \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )+b^5 \left (A e \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )-3 B \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )\right )-60 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^4 \log (d+e x)\right )}{12 e^7 (a+b x) (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(Sqrt[(a + b*x)^2]*(a^5*e^5*(3*A*e + B*(d + 4*e*x)) + 5*a^4*b*e^4*(A*e*(d + 4*e*x) + B*(d^2 + 4*d*e*x +
6*e^2*x^2)) + 10*a^3*b^2*e^3*(A*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + 3*B*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3
)) + 10*a^2*b^3*e^2*(3*A*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - B*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*
x^2 + 48*e^3*x^3)) - 5*a*b^4*e*(A*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) - B*(77*d^5 + 248*d^4
*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5)) + b^5*(A*e*(77*d^5 + 248*d^4*e*x + 252*d
^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) - 3*B*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^
3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*x^6)) - 60*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d +
e*x)^4*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1162\) vs. \(2(336)=672\).
time = 0.84, size = 1163, normalized size = 2.76

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} \left (\frac {1}{2} B b e \,x^{2}+A b e x +5 B a e x -5 B b d x \right )}{\left (b x +a \right ) e^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-10 A \,a^{2} b^{3} e^{5}+20 A a \,b^{4} d \,e^{4}-10 A \,b^{5} d^{2} e^{3}-10 B \,a^{3} b^{2} e^{5}+40 B \,a^{2} b^{3} d \,e^{4}-50 B a \,b^{4} d^{2} e^{3}+20 b^{5} B \,d^{3} e^{2}\right ) x^{3}-\frac {5 e b \left (2 A \,a^{3} b \,e^{4}+6 A \,a^{2} b^{2} d \,e^{3}-18 A a \,b^{3} d^{2} e^{2}+10 A \,b^{4} d^{3} e +B \,a^{4} e^{4}+6 B \,a^{3} b d \,e^{3}-36 B \,a^{2} b^{2} d^{2} e^{2}+50 B a \,b^{3} d^{3} e -21 B \,b^{4} d^{4}\right ) x^{2}}{2}+\left (-\frac {5}{3} A \,a^{4} b \,e^{5}-\frac {10}{3} A \,a^{3} b^{2} d \,e^{4}-10 A \,a^{2} b^{3} d^{2} e^{3}+\frac {110}{3} A a \,b^{4} d^{3} e^{2}-\frac {65}{3} A \,b^{5} d^{4} e -\frac {1}{3} B \,a^{5} e^{5}-\frac {5}{3} B \,a^{4} b d \,e^{4}-10 B \,a^{3} b^{2} d^{2} e^{3}+\frac {220}{3} B \,a^{2} b^{3} d^{3} e^{2}-\frac {325}{3} B a \,b^{4} d^{4} e +47 b^{5} B \,d^{5}\right ) x -\frac {3 a^{5} A \,e^{6}+5 A \,a^{4} b d \,e^{5}+10 A \,a^{3} b^{2} d^{2} e^{4}+30 A \,a^{2} b^{3} d^{3} e^{3}-125 A a \,b^{4} d^{4} e^{2}+77 A \,b^{5} d^{5} e +B \,a^{5} d \,e^{5}+5 B \,a^{4} b \,d^{2} e^{4}+30 B \,a^{3} b^{2} d^{3} e^{3}-250 B \,a^{2} b^{3} d^{4} e^{2}+385 B a \,b^{4} d^{5} e -171 b^{5} B \,d^{6}}{12 e}\right )}{\left (b x +a \right ) e^{6} \left (e x +d \right )^{4}}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, b^{3} \left (A a b \,e^{2}-A \,b^{2} d e +2 a^{2} B \,e^{2}-5 B a b d e +3 b^{2} B \,d^{2}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\)	\(633\)
default	\(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (171 b^{5} B \,d^{6}-48 A \,b^{5} d^{2} e^{4} x^{3}-120 B \,a^{3} b^{2} e^{6} x^{3}-96 B \,b^{5} d^{3} e^{3} x^{3}-60 A \,a^{3} b^{2} e^{6} x^{2}-252 A \,b^{5} d^{3} e^{3} x^{2}-3 a^{5} A \,e^{6}+60 B a \,b^{4} e^{6} x^{5}-10 A \,a^{3} b^{2} d^{2} e^{4}+125 A a \,b^{4} d^{4} e^{2}-5 A \,a^{4} b d \,e^{5}+360 A \ln \left (e x +d \right ) a \,b^{4} d^{2} e^{4} x^{2}-120 A \,a^{2} b^{3} d^{2} e^{4} x +440 A a \,b^{4} d^{3} e^{3} x -20 B \,a^{4} b d \,e^{5} x -120 B \,a^{3} b^{2} d^{2} e^{4} x +880 B \,a^{2} b^{3} d^{3} e^{3} x +180 B \ln \left (e x +d \right ) b^{5} d^{6}-240 B a \,b^{4} d^{2} e^{4} x^{3}-180 A \,a^{2} b^{3} d \,e^{5} x^{2}+540 A a \,b^{4} d^{2} e^{4} x^{2}-180 B \,a^{3} b^{2} d \,e^{5} x^{2}+1080 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-1260 B a \,b^{4} d^{3} e^{3} x^{2}-40 A \,a^{3} b^{2} d \,e^{5} x -248 A \,b^{5} d^{4} e^{2} x -300 B \ln \left (e x +d \right ) a \,b^{4} d \,e^{5} x^{4}-1200 B \ln \left (e x +d \right ) a \,b^{4} d^{4} e^{2} x +6 B \,b^{5} e^{6} x^{6}+12 A \,b^{5} e^{6} x^{5}-4 B \,a^{5} e^{6} x +250 B \,a^{2} b^{3} d^{4} e^{2}-30 B \,a^{3} b^{2} d^{3} e^{3}-5 B \,a^{4} b \,d^{2} e^{4}-385 B a \,b^{4} d^{5} e -77 A \,b^{5} d^{5} e -B \,a^{5} d \,e^{5}+60 A \ln \left (e x +d \right ) a \,b^{4} e^{6} x^{4}-60 A \ln \left (e x +d \right ) b^{5} d \,e^{5} x^{4}+120 B \ln \left (e x +d \right ) a^{2} b^{3} e^{6} x^{4}+180 B \ln \left (e x +d \right ) b^{5} d^{2} e^{4} x^{4}-240 A \ln \left (e x +d \right ) b^{5} d^{4} e^{2} x +720 B \ln \left (e x +d \right ) b^{5} d^{5} e x -240 A \ln \left (e x +d \right ) b^{5} d^{2} e^{4} x^{3}-360 A \ln \left (e x +d \right ) b^{5} d^{3} e^{3} x^{2}-30 A \,a^{2} b^{3} d^{3} e^{3}-60 A \ln \left (e x +d \right ) b^{5} d^{5} e +240 A \ln \left (e x +d \right ) a \,b^{4} d^{3} e^{3} x +480 B \ln \left (e x +d \right ) a^{2} b^{3} d^{3} e^{3} x -204 B \,b^{5} d^{2} e^{4} x^{4}-120 A \,a^{2} b^{3} e^{6} x^{3}+480 B \ln \left (e x +d \right ) a^{2} b^{3} d \,e^{5} x^{3}-1200 B \ln \left (e x +d \right ) a \,b^{4} d^{2} e^{4} x^{3}+240 A a \,b^{4} d \,e^{5} x^{3}+480 B \,a^{2} b^{3} d \,e^{5} x^{3}+240 A \ln \left (e x +d \right ) a \,b^{4} d \,e^{5} x^{3}+60 A \ln \left (e x +d \right ) a \,b^{4} d^{4} e^{2}+120 B \ln \left (e x +d \right ) a^{2} b^{3} d^{4} e^{2}-300 B \ln \left (e x +d \right ) a \,b^{4} d^{5} e -36 B \,b^{5} d \,e^{5} x^{5}+48 A \,b^{5} d \,e^{5} x^{4}+504 B \,b^{5} d^{5} e x +720 B \ln \left (e x +d \right ) a^{2} b^{3} d^{2} e^{4} x^{2}-1800 B \ln \left (e x +d \right ) a \,b^{4} d^{3} e^{3} x^{2}-30 B \,a^{4} b \,e^{6} x^{2}+396 B \,b^{5} d^{4} e^{2} x^{2}-20 A \,a^{4} b \,e^{6} x +1080 B \ln \left (e x +d \right ) b^{5} d^{4} e^{2} x^{2}+720 B \ln \left (e x +d \right ) b^{5} d^{3} e^{3} x^{3}-1240 B a \,b^{4} d^{4} e^{2} x +240 B a \,b^{4} d \,e^{5} x^{4}\right )}{12 \left (b x +a \right )^{5} e^{7} \left (e x +d \right )^{4}}\)	\(1163\)

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)^5,x,method=_RETURNVERBOSE)

[Out]

1/12*((b*x+a)^2)^(5/2)*(171*b^5*B*d^6-48*A*b^5*d^2*e^4*x^3-120*B*a^3*b^2*e^6*x^3-96*B*b^5*d^3*e^3*x^3-60*A*a^3
*b^2*e^6*x^2-252*A*b^5*d^3*e^3*x^2-3*a^5*A*e^6+60*B*a*b^4*e^6*x^5-10*A*a^3*b^2*d^2*e^4+125*A*a*b^4*d^4*e^2-5*A
*a^4*b*d*e^5+360*A*ln(e*x+d)*a*b^4*d^2*e^4*x^2-120*A*a^2*b^3*d^2*e^4*x+440*A*a*b^4*d^3*e^3*x-20*B*a^4*b*d*e^5*
x-120*B*a^3*b^2*d^2*e^4*x+880*B*a^2*b^3*d^3*e^3*x+180*B*ln(e*x+d)*b^5*d^6-240*B*a*b^4*d^2*e^4*x^3-180*A*a^2*b^
3*d*e^5*x^2+540*A*a*b^4*d^2*e^4*x^2-180*B*a^3*b^2*d*e^5*x^2+1080*B*a^2*b^3*d^2*e^4*x^2-1260*B*a*b^4*d^3*e^3*x^
2-40*A*a^3*b^2*d*e^5*x-248*A*b^5*d^4*e^2*x-300*B*ln(e*x+d)*a*b^4*d*e^5*x^4-1200*B*ln(e*x+d)*a*b^4*d^4*e^2*x+6*
B*b^5*e^6*x^6+12*A*b^5*e^6*x^5-4*B*a^5*e^6*x+250*B*a^2*b^3*d^4*e^2-30*B*a^3*b^2*d^3*e^3-5*B*a^4*b*d^2*e^4-385*
B*a*b^4*d^5*e-77*A*b^5*d^5*e-B*a^5*d*e^5+60*A*ln(e*x+d)*a*b^4*e^6*x^4-60*A*ln(e*x+d)*b^5*d*e^5*x^4+120*B*ln(e*
x+d)*a^2*b^3*e^6*x^4+180*B*ln(e*x+d)*b^5*d^2*e^4*x^4-240*A*ln(e*x+d)*b^5*d^4*e^2*x+720*B*ln(e*x+d)*b^5*d^5*e*x
-240*A*ln(e*x+d)*b^5*d^2*e^4*x^3-360*A*ln(e*x+d)*b^5*d^3*e^3*x^2-30*A*a^2*b^3*d^3*e^3-60*A*ln(e*x+d)*b^5*d^5*e
+240*A*ln(e*x+d)*a*b^4*d^3*e^3*x+480*B*ln(e*x+d)*a^2*b^3*d^3*e^3*x-204*B*b^5*d^2*e^4*x^4-120*A*a^2*b^3*e^6*x^3
+480*B*ln(e*x+d)*a^2*b^3*d*e^5*x^3-1200*B*ln(e*x+d)*a*b^4*d^2*e^4*x^3+240*A*a*b^4*d*e^5*x^3+480*B*a^2*b^3*d*e^
5*x^3+240*A*ln(e*x+d)*a*b^4*d*e^5*x^3+60*A*ln(e*x+d)*a*b^4*d^4*e^2+120*B*ln(e*x+d)*a^2*b^3*d^4*e^2-300*B*ln(e*
x+d)*a*b^4*d^5*e-36*B*b^5*d*e^5*x^5+48*A*b^5*d*e^5*x^4+504*B*b^5*d^5*e*x+720*B*ln(e*x+d)*a^2*b^3*d^2*e^4*x^2-1
800*B*ln(e*x+d)*a*b^4*d^3*e^3*x^2-30*B*a^4*b*e^6*x^2+396*B*b^5*d^4*e^2*x^2-20*A*a^4*b*e^6*x+1080*B*ln(e*x+d)*b
^5*d^4*e^2*x^2+720*B*ln(e*x+d)*b^5*d^3*e^3*x^3-1240*B*a*b^4*d^4*e^2*x+240*B*a*b^4*d*e^5*x^4)/(b*x+a)^5/e^7/(e*
x+d)^4

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 834 vs. \(2 (350) = 700\).
time = 1.11, size = 834, normalized size = 1.98 \begin {gather*} \frac {171 \, B b^{5} d^{6} + {\left (6 \, B b^{5} x^{6} - 3 \, A a^{5} + 12 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} - 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x\right )} e^{6} - {\left (36 \, B b^{5} d x^{5} - 48 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d x^{4} - 240 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d x^{3} + 180 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d x^{2} + 20 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d x + {\left (B a^{5} + 5 \, A a^{4} b\right )} d\right )} e^{5} - {\left (204 \, B b^{5} d^{2} x^{4} + 48 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} x^{3} - 540 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} x^{2} + 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} x + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2}\right )} e^{4} - 2 \, {\left (48 \, B b^{5} d^{3} x^{3} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} x^{2} - 220 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} x + 15 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3}\right )} e^{3} + {\left (396 \, B b^{5} d^{4} x^{2} - 248 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} x + 125 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4}\right )} e^{2} + 7 \, {\left (72 \, B b^{5} d^{5} x - 11 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5}\right )} e + 60 \, {\left (3 \, B b^{5} d^{6} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} e^{6} - {\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d x^{4} - 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d x^{3}\right )} e^{5} + {\left (3 \, B b^{5} d^{2} x^{4} - 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} x^{3} + 6 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} x^{2}\right )} e^{4} + 2 \, {\left (6 \, B b^{5} d^{3} x^{3} - 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} x^{2} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} x\right )} e^{3} + {\left (18 \, B b^{5} d^{4} x^{2} - 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} x + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4}\right )} e^{2} + {\left (12 \, B b^{5} d^{5} x - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5}\right )} e\right )} \log \left (x e + d\right )}{12 \, {\left (x^{4} e^{11} + 4 \, d x^{3} e^{10} + 6 \, d^{2} x^{2} e^{9} + 4 \, d^{3} x e^{8} + d^{4} e^{7}\right )}} \end {gather*}

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)^5,x, algorithm="fricas")

[Out]

1/12*(171*B*b^5*d^6 + (6*B*b^5*x^6 - 3*A*a^5 + 12*(5*B*a*b^4 + A*b^5)*x^5 - 120*(B*a^3*b^2 + A*a^2*b^3)*x^3 -
30*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 4*(B*a^5 + 5*A*a^4*b)*x)*e^6 - (36*B*b^5*d*x^5 - 48*(5*B*a*b^4 + A*b^5)*d*x^4
- 240*(2*B*a^2*b^3 + A*a*b^4)*d*x^3 + 180*(B*a^3*b^2 + A*a^2*b^3)*d*x^2 + 20*(B*a^4*b + 2*A*a^3*b^2)*d*x + (B
*a^5 + 5*A*a^4*b)*d)*e^5 - (204*B*b^5*d^2*x^4 + 48*(5*B*a*b^4 + A*b^5)*d^2*x^3 - 540*(2*B*a^2*b^3 + A*a*b^4)*d
^2*x^2 + 120*(B*a^3*b^2 + A*a^2*b^3)*d^2*x + 5*(B*a^4*b + 2*A*a^3*b^2)*d^2)*e^4 - 2*(48*B*b^5*d^3*x^3 + 126*(5
*B*a*b^4 + A*b^5)*d^3*x^2 - 220*(2*B*a^2*b^3 + A*a*b^4)*d^3*x + 15*(B*a^3*b^2 + A*a^2*b^3)*d^3)*e^3 + (396*B*b
^5*d^4*x^2 - 248*(5*B*a*b^4 + A*b^5)*d^4*x + 125*(2*B*a^2*b^3 + A*a*b^4)*d^4)*e^2 + 7*(72*B*b^5*d^5*x - 11*(5*
B*a*b^4 + A*b^5)*d^5)*e + 60*(3*B*b^5*d^6 + (2*B*a^2*b^3 + A*a*b^4)*x^4*e^6 - ((5*B*a*b^4 + A*b^5)*d*x^4 - 4*(
2*B*a^2*b^3 + A*a*b^4)*d*x^3)*e^5 + (3*B*b^5*d^2*x^4 - 4*(5*B*a*b^4 + A*b^5)*d^2*x^3 + 6*(2*B*a^2*b^3 + A*a*b^
4)*d^2*x^2)*e^4 + 2*(6*B*b^5*d^3*x^3 - 3*(5*B*a*b^4 + A*b^5)*d^3*x^2 + 2*(2*B*a^2*b^3 + A*a*b^4)*d^3*x)*e^3 +
(18*B*b^5*d^4*x^2 - 4*(5*B*a*b^4 + A*b^5)*d^4*x + (2*B*a^2*b^3 + A*a*b^4)*d^4)*e^2 + (12*B*b^5*d^5*x - (5*B*a*
b^4 + A*b^5)*d^5)*e)*log(x*e + d))/(x^4*e^11 + 4*d*x^3*e^10 + 6*d^2*x^2*e^9 + 4*d^3*x*e^8 + d^4*e^7)

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

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)**5,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 870 vs. \(2 (350) = 700\).
time = 1.11, size = 870, normalized size = 2.07 \begin {gather*} 5 \, {\left (3 \, B b^{5} d^{2} \mathrm {sgn}\left (b x + a\right ) - 5 \, B a b^{4} d e \mathrm {sgn}\left (b x + a\right ) - A b^{5} d e \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{2} b^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + A a b^{4} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (B b^{5} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) - 10 \, B b^{5} d x e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, B a b^{4} x e^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, A b^{5} x e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-10\right )} + \frac {{\left (171 \, B b^{5} d^{6} \mathrm {sgn}\left (b x + a\right ) - 385 \, B a b^{4} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 77 \, A b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 250 \, B a^{2} b^{3} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 125 \, A a b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 30 \, B a^{3} b^{2} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 30 \, A a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 5 \, B a^{4} b d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 10 \, A a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - B a^{5} d e^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, A a^{4} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - 3 \, A a^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 120 \, {\left (2 \, B b^{5} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 5 \, B a b^{4} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - A b^{5} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 4 \, B a^{2} b^{3} d e^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a b^{4} d e^{5} \mathrm {sgn}\left (b x + a\right ) - B a^{3} b^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) - A a^{2} b^{3} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 30 \, {\left (21 \, B b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 50 \, B a b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 10 \, A b^{5} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, B a^{2} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 18 \, A a b^{4} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, B a^{3} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) - 6 \, A a^{2} b^{3} d e^{5} \mathrm {sgn}\left (b x + a\right ) - B a^{4} b e^{6} \mathrm {sgn}\left (b x + a\right ) - 2 \, A a^{3} b^{2} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 4 \, {\left (141 \, B b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 325 \, B a b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 65 \, A b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 220 \, B a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 110 \, A a b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 30 \, B a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 30 \, A a^{2} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 5 \, B a^{4} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - 10 \, A a^{3} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) - B a^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) - 5 \, A a^{4} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{12 \, {\left (x e + d\right )}^{4}} \end {gather*}

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)^5,x, algorithm="giac")

[Out]

5*(3*B*b^5*d^2*sgn(b*x + a) - 5*B*a*b^4*d*e*sgn(b*x + a) - A*b^5*d*e*sgn(b*x + a) + 2*B*a^2*b^3*e^2*sgn(b*x +
a) + A*a*b^4*e^2*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) + 1/2*(B*b^5*x^2*e^5*sgn(b*x + a) - 10*B*b^5*d*x*e^4*s
gn(b*x + a) + 10*B*a*b^4*x*e^5*sgn(b*x + a) + 2*A*b^5*x*e^5*sgn(b*x + a))*e^(-10) + 1/12*(171*B*b^5*d^6*sgn(b*
x + a) - 385*B*a*b^4*d^5*e*sgn(b*x + a) - 77*A*b^5*d^5*e*sgn(b*x + a) + 250*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 1
25*A*a*b^4*d^4*e^2*sgn(b*x + a) - 30*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 30*A*a^2*b^3*d^3*e^3*sgn(b*x + a) - 5*B*
a^4*b*d^2*e^4*sgn(b*x + a) - 10*A*a^3*b^2*d^2*e^4*sgn(b*x + a) - B*a^5*d*e^5*sgn(b*x + a) - 5*A*a^4*b*d*e^5*sg
n(b*x + a) - 3*A*a^5*e^6*sgn(b*x + a) + 120*(2*B*b^5*d^3*e^3*sgn(b*x + a) - 5*B*a*b^4*d^2*e^4*sgn(b*x + a) - A
*b^5*d^2*e^4*sgn(b*x + a) + 4*B*a^2*b^3*d*e^5*sgn(b*x + a) + 2*A*a*b^4*d*e^5*sgn(b*x + a) - B*a^3*b^2*e^6*sgn(
b*x + a) - A*a^2*b^3*e^6*sgn(b*x + a))*x^3 + 30*(21*B*b^5*d^4*e^2*sgn(b*x + a) - 50*B*a*b^4*d^3*e^3*sgn(b*x +
a) - 10*A*b^5*d^3*e^3*sgn(b*x + a) + 36*B*a^2*b^3*d^2*e^4*sgn(b*x + a) + 18*A*a*b^4*d^2*e^4*sgn(b*x + a) - 6*B
*a^3*b^2*d*e^5*sgn(b*x + a) - 6*A*a^2*b^3*d*e^5*sgn(b*x + a) - B*a^4*b*e^6*sgn(b*x + a) - 2*A*a^3*b^2*e^6*sgn(
b*x + a))*x^2 + 4*(141*B*b^5*d^5*e*sgn(b*x + a) - 325*B*a*b^4*d^4*e^2*sgn(b*x + a) - 65*A*b^5*d^4*e^2*sgn(b*x
+ a) + 220*B*a^2*b^3*d^3*e^3*sgn(b*x + a) + 110*A*a*b^4*d^3*e^3*sgn(b*x + a) - 30*B*a^3*b^2*d^2*e^4*sgn(b*x +
a) - 30*A*a^2*b^3*d^2*e^4*sgn(b*x + a) - 5*B*a^4*b*d*e^5*sgn(b*x + a) - 10*A*a^3*b^2*d*e^5*sgn(b*x + a) - B*a^
5*e^6*sgn(b*x + a) - 5*A*a^4*b*e^6*sgn(b*x + a))*x)*e^(-7)/(x*e + d)^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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