3.10.21 \(\int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^7} \, dx\) [921]

Optimal. Leaf size=310 \[ \frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^5 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}-\frac {\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{11/2}} \]

[Out]

-1/6*A*(c*x^2+b*x+a)^(3/2)/a/x^6+1/20*(3*A*b-4*B*a)*(c*x^2+b*x+a)^(3/2)/a^2/x^5-1/160*(-20*A*a*c+21*A*b^2-28*B
*a*b)*(c*x^2+b*x+a)^(3/2)/a^3/x^4+1/960*(-196*A*a*b*c+105*A*b^3+128*B*a^2*c-140*B*a*b^2)*(c*x^2+b*x+a)^(3/2)/a
^4/x^3-1/1024*(-4*a*c+b^2)*(4*a*b*B*(-12*a*c+7*b^2)-A*(16*a^2*c^2-56*a*b^2*c+21*b^4))*arctanh(1/2*(b*x+2*a)/a^
(1/2)/(c*x^2+b*x+a)^(1/2))/a^(11/2)+1/512*(4*a*b*B*(-12*a*c+7*b^2)-A*(16*a^2*c^2-56*a*b^2*c+21*b^4))*(b*x+2*a)
*(c*x^2+b*x+a)^(1/2)/a^5/x^2

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Rubi [A]
time = 0.25, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {848, 820, 734, 738, 212} \begin {gather*} -\frac {\left (a+b x+c x^2\right )^{3/2} \left (-20 a A c-28 a b B+21 A b^2\right )}{160 a^3 x^4}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{11/2}}+\frac {(2 a+b x) \sqrt {a+b x+c x^2} \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right )}{512 a^5 x^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (128 a^2 B c-196 a A b c-140 a b^2 B+105 A b^3\right )}{960 a^4 x^3}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^7,x]

[Out]

((4*a*b*B*(7*b^2 - 12*a*c) - A*(21*b^4 - 56*a*b^2*c + 16*a^2*c^2))*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(512*a^5
*x^2) - (A*(a + b*x + c*x^2)^(3/2))/(6*a*x^6) + ((3*A*b - 4*a*B)*(a + b*x + c*x^2)^(3/2))/(20*a^2*x^5) - ((21*
A*b^2 - 28*a*b*B - 20*a*A*c)*(a + b*x + c*x^2)^(3/2))/(160*a^3*x^4) + ((105*A*b^3 - 140*a*b^2*B - 196*a*A*b*c
+ 128*a^2*B*c)*(a + b*x + c*x^2)^(3/2))/(960*a^4*x^3) - ((b^2 - 4*a*c)*(4*a*b*B*(7*b^2 - 12*a*c) - A*(21*b^4 -
56*a*b^2*c + 16*a^2*c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(1024*a^(11/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^7} \, dx &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}-\frac {\int \frac {\left (\frac {3}{2} (3 A b-4 a B)+3 A c x\right ) \sqrt {a+b x+c x^2}}{x^6} \, dx}{6 a}\\ &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}+\frac {\int \frac {\left (\frac {3}{4} \left (21 A b^2-28 a b B-20 a A c\right )+3 (3 A b-4 a B) c x\right ) \sqrt {a+b x+c x^2}}{x^5} \, dx}{30 a^2}\\ &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}-\frac {\int \frac {\left (\frac {3}{8} \left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right )+\frac {3}{4} c \left (21 A b^2-28 a b B-20 a A c\right ) x\right ) \sqrt {a+b x+c x^2}}{x^4} \, dx}{120 a^3}\\ &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}-\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx}{128 a^4}\\ &=\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^5 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}+\frac {\left (\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{1024 a^5}\\ &=\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^5 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}-\frac {\left (\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{512 a^5}\\ &=\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^5 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}-\frac {\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 2.42, size = 329, normalized size = 1.06 \begin {gather*} \frac {-\sqrt {a} \sqrt {a+x (b+c x)} \left (315 A b^5 x^5+256 a^5 (5 A+6 B x)-210 a b^3 x^4 (2 b B x+A (b+8 c x))+64 a^4 x (A (2 b+5 c x)+B x (3 b+8 c x))-16 a^3 x^2 \left (A \left (9 b^2+34 b c x+30 c^2 x^2\right )+2 B x \left (7 b^2+29 b c x+32 c^2 x^2\right )\right )+8 a^2 b x^3 \left (5 b B x (7 b+46 c x)+A \left (21 b^2+112 b c x+226 c^2 x^2\right )\right )\right )-315 A b^6 x^6 \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-60 a \left (7 b^5 B+35 A b^4 c-40 a b^3 B c-60 a A b^2 c^2+48 a^2 b B c^2+16 a^2 A c^3\right ) x^6 \tanh ^{-1}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{7680 a^{11/2} x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^7,x]

[Out]

(-(Sqrt[a]*Sqrt[a + x*(b + c*x)]*(315*A*b^5*x^5 + 256*a^5*(5*A + 6*B*x) - 210*a*b^3*x^4*(2*b*B*x + A*(b + 8*c*
x)) + 64*a^4*x*(A*(2*b + 5*c*x) + B*x*(3*b + 8*c*x)) - 16*a^3*x^2*(A*(9*b^2 + 34*b*c*x + 30*c^2*x^2) + 2*B*x*(
7*b^2 + 29*b*c*x + 32*c^2*x^2)) + 8*a^2*b*x^3*(5*b*B*x*(7*b + 46*c*x) + A*(21*b^2 + 112*b*c*x + 226*c^2*x^2)))
) - 315*A*b^6*x^6*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] - 60*a*(7*b^5*B + 35*A*b^4*c - 40*a*b^3
*B*c - 60*a*A*b^2*c^2 + 48*a^2*b*B*c^2 + 16*a^2*A*c^3)*x^6*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt
[a]])/(7680*a^(11/2)*x^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2626\) vs. \(2(280)=560\).
time = 0.79, size = 2627, normalized size = 8.47

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (1808 A \,a^{2} b \,c^{2} x^{5}-1680 A a \,b^{3} c \,x^{5}+315 A \,b^{5} x^{5}-1024 B \,a^{3} c^{2} x^{5}+1840 B \,a^{2} b^{2} c \,x^{5}-420 B a \,b^{4} x^{5}-480 A \,a^{3} c^{2} x^{4}+896 A \,a^{2} b^{2} c \,x^{4}-210 A a \,b^{4} x^{4}-928 B \,a^{3} b c \,x^{4}+280 B \,a^{2} b^{3} x^{4}-544 A \,a^{3} b c \,x^{3}+168 A \,a^{2} b^{3} x^{3}+512 B \,a^{4} c \,x^{3}-224 B \,a^{3} b^{2} x^{3}+320 A \,a^{4} c \,x^{2}-144 A \,a^{3} b^{2} x^{2}+192 B \,a^{4} b \,x^{2}+128 a^{4} b A x +1536 a^{5} B x +1280 a^{5} A \right )}{7680 x^{6} a^{5}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,c^{3}}{16 a^{\frac {5}{2}}}+\frac {15 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,b^{2} c^{2}}{64 a^{\frac {7}{2}}}-\frac {35 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,b^{4} c}{256 a^{\frac {9}{2}}}+\frac {21 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,b^{6}}{1024 a^{\frac {11}{2}}}-\frac {3 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) B b \,c^{2}}{16 a^{\frac {5}{2}}}+\frac {5 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) B \,b^{3} c}{32 a^{\frac {7}{2}}}-\frac {7 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) B \,b^{5}}{256 a^{\frac {9}{2}}}\)	\(519\)
default	\(\text {Expression too large to display}\)	\(2627\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

B*(-1/5/a/x^5*(c*x^2+b*x+a)^(3/2)-7/10*b/a*(-1/4/a/x^4*(c*x^2+b*x+a)^(3/2)-5/8*b/a*(-1/3/a/x^3*(c*x^2+b*x+a)^(
3/2)-1/2*b/a*(-1/2/a/x^2*(c*x^2+b*x+a)^(3/2)-1/4*b/a*(-1/a/x*(c*x^2+b*x+a)^(3/2)+1/2*b/a*((c*x^2+b*x+a)^(1/2)+
1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x
))+2*c/a*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))))+1/2*c/a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b
*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))))-1/4*c/a*(-1/2/a/x^2*(c*x^2+b*x+a)^(3/2)-1/4*b/a*(-1/a/x*(c*x^2+b*x+a)^
(3/2)+1/2*b/a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b
*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))+2*c/a*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+1/2*c/a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))))-2/5*c/a*(-1/3/a/x^3*(c*x^2+b*x+a)^(
3/2)-1/2*b/a*(-1/2/a/x^2*(c*x^2+b*x+a)^(3/2)-1/4*b/a*(-1/a/x*(c*x^2+b*x+a)^(3/2)+1/2*b/a*((c*x^2+b*x+a)^(1/2)+
1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x
))+2*c/a*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))))+1/2*c/a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b
*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)))))+A*(-1/6/a/x^6*(c*x^2+b*x+a)^(3/2)-3/4*b/a*(-1/5/a/x^5*(c*x^2+b*x+a)^(
3/2)-7/10*b/a*(-1/4/a/x^4*(c*x^2+b*x+a)^(3/2)-5/8*b/a*(-1/3/a/x^3*(c*x^2+b*x+a)^(3/2)-1/2*b/a*(-1/2/a/x^2*(c*x
^2+b*x+a)^(3/2)-1/4*b/a*(-1/a/x*(c*x^2+b*x+a)^(3/2)+1/2*b/a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+
(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))+2*c/a*(1/4*(2*c*x+b)*(c*x^
2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+1/2*c/a*((c*x^2+b*x+a)^
(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1
/2))/x))))-1/4*c/a*(-1/2/a/x^2*(c*x^2+b*x+a)^(3/2)-1/4*b/a*(-1/a/x*(c*x^2+b*x+a)^(3/2)+1/2*b/a*((c*x^2+b*x+a)^
(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1
/2))/x))+2*c/a*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))))+1/2*c/a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln(
(2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))))-2/5*c/a*(-1/3/a/x^3*(c*x^2+b*x+a)^(3/2)-1/2*b/a*(-1/2/a/x^2*(c*x
^2+b*x+a)^(3/2)-1/4*b/a*(-1/a/x*(c*x^2+b*x+a)^(3/2)+1/2*b/a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+
(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))+2*c/a*(1/4*(2*c*x+b)*(c*x^
2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+1/2*c/a*((c*x^2+b*x+a)^
(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1
/2))/x)))))-1/2*c/a*(-1/4/a/x^4*(c*x^2+b*x+a)^(3/2)-5/8*b/a*(-1/3/a/x^3*(c*x^2+b*x+a)^(3/2)-1/2*b/a*(-1/2/a/x^
2*(c*x^2+b*x+a)^(3/2)-1/4*b/a*(-1/a/x*(c*x^2+b*x+a)^(3/2)+1/2*b/a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))+2*c/a*(1/4*(2*c*x+b)
*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+1/2*c/a*((c*x^2+b
*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x
+a)^(1/2))/x))))-1/4*c/a*(-1/2/a/x^2*(c*x^2+b*x+a)^(3/2)-1/4*b/a*(-1/a/x*(c*x^2+b*x+a)^(3/2)+1/2*b/a*((c*x^2+b
*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x
+a)^(1/2))/x))+2*c/a*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))))+1/2*c/a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/
2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)))))

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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)*(c*x^2+b*x+a)^(1/2)/x^7,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(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [A]
time = 7.64, size = 709, normalized size = 2.29 \begin {gather*} \left [\frac {15 \, {\left (28 \, B a b^{5} - 21 \, A b^{6} + 64 \, A a^{3} c^{3} + 48 \, {\left (4 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} c^{2} - 20 \, {\left (8 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} c\right )} \sqrt {a} x^{6} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, {\left (1280 \, A a^{6} - {\left (420 \, B a^{2} b^{4} - 315 \, A a b^{5} + 16 \, {\left (64 \, B a^{4} - 113 \, A a^{3} b\right )} c^{2} - 80 \, {\left (23 \, B a^{3} b^{2} - 21 \, A a^{2} b^{3}\right )} c\right )} x^{5} + 2 \, {\left (140 \, B a^{3} b^{3} - 105 \, A a^{2} b^{4} - 240 \, A a^{4} c^{2} - 16 \, {\left (29 \, B a^{4} b - 28 \, A a^{3} b^{2}\right )} c\right )} x^{4} - 8 \, {\left (28 \, B a^{4} b^{2} - 21 \, A a^{3} b^{3} - 4 \, {\left (16 \, B a^{5} - 17 \, A a^{4} b\right )} c\right )} x^{3} + 16 \, {\left (12 \, B a^{5} b - 9 \, A a^{4} b^{2} + 20 \, A a^{5} c\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, a^{6} x^{6}}, \frac {15 \, {\left (28 \, B a b^{5} - 21 \, A b^{6} + 64 \, A a^{3} c^{3} + 48 \, {\left (4 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} c^{2} - 20 \, {\left (8 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} c\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \, {\left (1280 \, A a^{6} - {\left (420 \, B a^{2} b^{4} - 315 \, A a b^{5} + 16 \, {\left (64 \, B a^{4} - 113 \, A a^{3} b\right )} c^{2} - 80 \, {\left (23 \, B a^{3} b^{2} - 21 \, A a^{2} b^{3}\right )} c\right )} x^{5} + 2 \, {\left (140 \, B a^{3} b^{3} - 105 \, A a^{2} b^{4} - 240 \, A a^{4} c^{2} - 16 \, {\left (29 \, B a^{4} b - 28 \, A a^{3} b^{2}\right )} c\right )} x^{4} - 8 \, {\left (28 \, B a^{4} b^{2} - 21 \, A a^{3} b^{3} - 4 \, {\left (16 \, B a^{5} - 17 \, A a^{4} b\right )} c\right )} x^{3} + 16 \, {\left (12 \, B a^{5} b - 9 \, A a^{4} b^{2} + 20 \, A a^{5} c\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, a^{6} x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^7,x, algorithm="fricas")

[Out]

[1/30720*(15*(28*B*a*b^5 - 21*A*b^6 + 64*A*a^3*c^3 + 48*(4*B*a^3*b - 5*A*a^2*b^2)*c^2 - 20*(8*B*a^2*b^3 - 7*A*
a*b^4)*c)*sqrt(a)*x^6*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)
/x^2) - 4*(1280*A*a^6 - (420*B*a^2*b^4 - 315*A*a*b^5 + 16*(64*B*a^4 - 113*A*a^3*b)*c^2 - 80*(23*B*a^3*b^2 - 21
*A*a^2*b^3)*c)*x^5 + 2*(140*B*a^3*b^3 - 105*A*a^2*b^4 - 240*A*a^4*c^2 - 16*(29*B*a^4*b - 28*A*a^3*b^2)*c)*x^4
- 8*(28*B*a^4*b^2 - 21*A*a^3*b^3 - 4*(16*B*a^5 - 17*A*a^4*b)*c)*x^3 + 16*(12*B*a^5*b - 9*A*a^4*b^2 + 20*A*a^5*
c)*x^2 + 128*(12*B*a^6 + A*a^5*b)*x)*sqrt(c*x^2 + b*x + a))/(a^6*x^6), 1/15360*(15*(28*B*a*b^5 - 21*A*b^6 + 64
*A*a^3*c^3 + 48*(4*B*a^3*b - 5*A*a^2*b^2)*c^2 - 20*(8*B*a^2*b^3 - 7*A*a*b^4)*c)*sqrt(-a)*x^6*arctan(1/2*sqrt(c
*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(a*c*x^2 + a*b*x + a^2)) - 2*(1280*A*a^6 - (420*B*a^2*b^4 - 315*A*a*b^5 +
16*(64*B*a^4 - 113*A*a^3*b)*c^2 - 80*(23*B*a^3*b^2 - 21*A*a^2*b^3)*c)*x^5 + 2*(140*B*a^3*b^3 - 105*A*a^2*b^4
- 240*A*a^4*c^2 - 16*(29*B*a^4*b - 28*A*a^3*b^2)*c)*x^4 - 8*(28*B*a^4*b^2 - 21*A*a^3*b^3 - 4*(16*B*a^5 - 17*A*
a^4*b)*c)*x^3 + 16*(12*B*a^5*b - 9*A*a^4*b^2 + 20*A*a^5*c)*x^2 + 128*(12*B*a^6 + A*a^5*b)*x)*sqrt(c*x^2 + b*x
+ a))/(a^6*x^6)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**7,x)

[Out]

Integral((A + B*x)*sqrt(a + b*x + c*x**2)/x**7, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1955 vs. \(2 (280) = 560\).
time = 1.64, size = 1955, normalized size = 6.31 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^7,x, algorithm="giac")

[Out]

1/512*(28*B*a*b^5 - 21*A*b^6 - 160*B*a^2*b^3*c + 140*A*a*b^4*c + 192*B*a^3*b*c^2 - 240*A*a^2*b^2*c^2 + 64*A*a^
3*c^3)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^5) - 1/7680*(420*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^11*B*a*b^5 - 315*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*b^6 - 2400*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^11*B*a^2*b^3*c + 2100*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a*b^4*c + 2880*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^11*B*a^3*b*c^2 - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^2*b^2*c^2 + 960*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^11*A*a^3*c^3 - 2380*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^2*b^5 + 1785*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^9*A*a*b^6 + 13600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^3*b^3*c - 11900*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^9*A*a^2*b^4*c - 16320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^4*b*c^2 + 20400*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^3*b^2*c^2 - 5440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^4*c^3 - 30720*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^5*c^(5/2) + 5544*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^3*b^5 - 4
158*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*b^6 - 31680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^4*b^3*c
+ 27720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^3*b^4*c - 48000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^5*
b*c^2 - 47520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^4*b^2*c^2 - 36480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
7*A*a^5*c^3 - 97280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^5*b^2*c^(3/2) + 20480*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^6*B*a^6*c^(5/2) - 163840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^5*b*c^(5/2) - 6744*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^5*B*a^4*b^5 + 5058*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*b^6 - 16320*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^5*B*a^5*b^3*c - 33720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^4*b^4*c + 13440*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^5*B*a^6*b*c^2 - 170400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^5*b^2*c^2 - 3648
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^6*c^3 - 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^5*b^4*sqrt
(c) + 76800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^6*b^2*c^(3/2) - 168960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^4*A*a^5*b^3*c^(3/2) - 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^6*b*c^(5/2) + 2740*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^3*B*a^5*b^5 - 3335*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^4*b^6 + 23840*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^3*B*a^6*b^3*c - 47740*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^5*b^4*c + 45120*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*B*a^7*b*c^2 - 102480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^6*b^2*c^2 - 5440*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^7*c^3 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^6*b^4*sqrt(c) -
15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^5*b^5*sqrt(c) + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B
*a^7*b^2*c^(3/2) - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^6*b^3*c^(3/2) + 12288*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2*B*a^8*c^(5/2) - 24576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^7*b*c^(5/2) + 420*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*B*a^6*b^5 - 315*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^5*b^6 + 12960*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*B*a^7*b^3*c - 13260*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^6*b^4*c + 2880*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*B*a^8*b*c^2 - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^7*b^2*c^2 + 960*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*A*a^8*c^3 + 5120*B*a^8*b^2*c^(3/2) - 5120*A*a^7*b^3*c^(3/2) - 2048*B*a^9*c^(5/2) + 4096
*A*a^8*b*c^(5/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^6*a^5)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^7,x)

[Out]

int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^7, x)

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