3.13.1 \(\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx\) [1201]

Optimal. Leaf size=175 \[ -\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{48 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{32 c \left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2}+\frac {\tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{64 c^{3/2} \left (b^2-4 a c\right )^{5/2} d^7} \]

[Out]

1/64*arctan(2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^(3/2)/(-4*a*c+b^2)^(5/2)/d^7-1/12*(c*x^2+b*x+a
)^(1/2)/c/d^7/(2*c*x+b)^6+1/48*(c*x^2+b*x+a)^(1/2)/c/(-4*a*c+b^2)/d^7/(2*c*x+b)^4+1/32*(c*x^2+b*x+a)^(1/2)/c/(
-4*a*c+b^2)^2/d^7/(2*c*x+b)^2

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Rubi [A]
time = 0.17, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {698, 707, 702, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{64 c^{3/2} d^7 \left (b^2-4 a c\right )^{5/2}}+\frac {\sqrt {a+b x+c x^2}}{32 c d^7 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac {\sqrt {a+b x+c x^2}}{48 c d^7 \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*Sqrt[a + b*x + c*x^2]/(c*d^7*(b + 2*c*x)^6) + Sqrt[a + b*x + c*x^2]/(48*c*(b^2 - 4*a*c)*d^7*(b + 2*c*x)^
4) + Sqrt[a + b*x + c*x^2]/(32*c*(b^2 - 4*a*c)^2*d^7*(b + 2*c*x)^2) + ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])
/Sqrt[b^2 - 4*a*c]]/(64*c^(3/2)*(b^2 - 4*a*c)^(5/2)*d^7)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 698

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

Rule 702

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

Rule 707

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

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx &=-\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\int \frac {1}{(b d+2 c d x)^5 \sqrt {a+b x+c x^2}} \, dx}{24 c d^2}\\ &=-\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{48 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^4}+\frac {\int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx}{32 c \left (b^2-4 a c\right ) d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{48 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{32 c \left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2}+\frac {\int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{64 c \left (b^2-4 a c\right )^2 d^6}\\ &=-\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{48 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{32 c \left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2}+\frac {\text {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{16 \left (b^2-4 a c\right )^2 d^6}\\ &=-\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{48 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{32 c \left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2}+\frac {\tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{64 c^{3/2} \left (b^2-4 a c\right )^{5/2} d^7}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.03, size = 62, normalized size = 0.35 \begin {gather*} \frac {2 (a+x (b+c x))^{3/2} \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{3 \left (b^2-4 a c\right )^4 d^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + x*(b + c*x))^(3/2)*Hypergeometric2F1[3/2, 4, 5/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(3*(b^2 - 4
*a*c)^4*d^7)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(367\) vs. \(2(151)=302\).
time = 0.69, size = 368, normalized size = 2.10

method	result	size
default	\(\frac {-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6}}-\frac {2 c^{2} \left (-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}-\frac {c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {2 c^{2} \left (\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}}{128 d^{7} c^{7}}\)	\(368\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128/d^7/c^7*(-2/3/(4*a*c-b^2)*c/(x+1/2*b/c)^6*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)-2*c^2/(4*a*c-b^2)*(-
1/(4*a*c-b^2)*c/(x+1/2*b/c)^4*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)-c^2/(4*a*c-b^2)*(-2/(4*a*c-b^2)*c/(x+1
/2*b/c)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)+2*c^2/(4*a*c-b^2)*(1/2*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(
1/2)-1/2*(4*a*c-b^2)/c/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*
c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))))))

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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((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^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 [B] Leaf count of result is larger than twice the leaf count of optimal. 595 vs. \(2 (151) = 302\).
time = 5.91, size = 1220, normalized size = 6.97 \begin {gather*} \left [-\frac {3 \, {\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \sqrt {-b^{2} c + 4 \, a c^{2}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {-b^{2} c + 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, {\left (3 \, b^{6} c - 68 \, a b^{4} c^{2} + 352 \, a^{2} b^{2} c^{3} - 512 \, a^{3} c^{4} - 48 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} - 96 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3} - 16 \, {\left (5 \, b^{4} c^{3} - 22 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} x^{2} - 32 \, {\left (b^{5} c^{2} - 5 \, a b^{3} c^{3} + 4 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, {\left (64 \, {\left (b^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{7} x^{6} + 192 \, {\left (b^{7} c^{7} - 12 \, a b^{5} c^{8} + 48 \, a^{2} b^{3} c^{9} - 64 \, a^{3} b c^{10}\right )} d^{7} x^{5} + 240 \, {\left (b^{8} c^{6} - 12 \, a b^{6} c^{7} + 48 \, a^{2} b^{4} c^{8} - 64 \, a^{3} b^{2} c^{9}\right )} d^{7} x^{4} + 160 \, {\left (b^{9} c^{5} - 12 \, a b^{7} c^{6} + 48 \, a^{2} b^{5} c^{7} - 64 \, a^{3} b^{3} c^{8}\right )} d^{7} x^{3} + 60 \, {\left (b^{10} c^{4} - 12 \, a b^{8} c^{5} + 48 \, a^{2} b^{6} c^{6} - 64 \, a^{3} b^{4} c^{7}\right )} d^{7} x^{2} + 12 \, {\left (b^{11} c^{3} - 12 \, a b^{9} c^{4} + 48 \, a^{2} b^{7} c^{5} - 64 \, a^{3} b^{5} c^{6}\right )} d^{7} x + {\left (b^{12} c^{2} - 12 \, a b^{10} c^{3} + 48 \, a^{2} b^{8} c^{4} - 64 \, a^{3} b^{6} c^{5}\right )} d^{7}\right )}}, -\frac {3 \, {\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \sqrt {b^{2} c - 4 \, a c^{2}} \arctan \left (\frac {\sqrt {b^{2} c - 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (3 \, b^{6} c - 68 \, a b^{4} c^{2} + 352 \, a^{2} b^{2} c^{3} - 512 \, a^{3} c^{4} - 48 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} - 96 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3} - 16 \, {\left (5 \, b^{4} c^{3} - 22 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} x^{2} - 32 \, {\left (b^{5} c^{2} - 5 \, a b^{3} c^{3} + 4 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{192 \, {\left (64 \, {\left (b^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{7} x^{6} + 192 \, {\left (b^{7} c^{7} - 12 \, a b^{5} c^{8} + 48 \, a^{2} b^{3} c^{9} - 64 \, a^{3} b c^{10}\right )} d^{7} x^{5} + 240 \, {\left (b^{8} c^{6} - 12 \, a b^{6} c^{7} + 48 \, a^{2} b^{4} c^{8} - 64 \, a^{3} b^{2} c^{9}\right )} d^{7} x^{4} + 160 \, {\left (b^{9} c^{5} - 12 \, a b^{7} c^{6} + 48 \, a^{2} b^{5} c^{7} - 64 \, a^{3} b^{3} c^{8}\right )} d^{7} x^{3} + 60 \, {\left (b^{10} c^{4} - 12 \, a b^{8} c^{5} + 48 \, a^{2} b^{6} c^{6} - 64 \, a^{3} b^{4} c^{7}\right )} d^{7} x^{2} + 12 \, {\left (b^{11} c^{3} - 12 \, a b^{9} c^{4} + 48 \, a^{2} b^{7} c^{5} - 64 \, a^{3} b^{5} c^{6}\right )} d^{7} x + {\left (b^{12} c^{2} - 12 \, a b^{10} c^{3} + 48 \, a^{2} b^{8} c^{4} - 64 \, a^{3} b^{6} c^{5}\right )} d^{7}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/384*(3*(64*c^6*x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^6
)*sqrt(-b^2*c + 4*a*c^2)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x +
a))/(4*c^2*x^2 + 4*b*c*x + b^2)) + 4*(3*b^6*c - 68*a*b^4*c^2 + 352*a^2*b^2*c^3 - 512*a^3*c^4 - 48*(b^2*c^5 -
4*a*c^6)*x^4 - 96*(b^3*c^4 - 4*a*b*c^5)*x^3 - 16*(5*b^4*c^3 - 22*a*b^2*c^4 + 8*a^2*c^5)*x^2 - 32*(b^5*c^2 - 5*
a*b^3*c^3 + 4*a^2*b*c^4)*x)*sqrt(c*x^2 + b*x + a))/(64*(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11
)*d^7*x^6 + 192*(b^7*c^7 - 12*a*b^5*c^8 + 48*a^2*b^3*c^9 - 64*a^3*b*c^10)*d^7*x^5 + 240*(b^8*c^6 - 12*a*b^6*c^
7 + 48*a^2*b^4*c^8 - 64*a^3*b^2*c^9)*d^7*x^4 + 160*(b^9*c^5 - 12*a*b^7*c^6 + 48*a^2*b^5*c^7 - 64*a^3*b^3*c^8)*
d^7*x^3 + 60*(b^10*c^4 - 12*a*b^8*c^5 + 48*a^2*b^6*c^6 - 64*a^3*b^4*c^7)*d^7*x^2 + 12*(b^11*c^3 - 12*a*b^9*c^4
+ 48*a^2*b^7*c^5 - 64*a^3*b^5*c^6)*d^7*x + (b^12*c^2 - 12*a*b^10*c^3 + 48*a^2*b^8*c^4 - 64*a^3*b^6*c^5)*d^7),
-1/192*(3*(64*c^6*x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^6
)*sqrt(b^2*c - 4*a*c^2)*arctan(1/2*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a)/(c^2*x^2 + b*c*x + a*c)) + 2*(3
*b^6*c - 68*a*b^4*c^2 + 352*a^2*b^2*c^3 - 512*a^3*c^4 - 48*(b^2*c^5 - 4*a*c^6)*x^4 - 96*(b^3*c^4 - 4*a*b*c^5)*
x^3 - 16*(5*b^4*c^3 - 22*a*b^2*c^4 + 8*a^2*c^5)*x^2 - 32*(b^5*c^2 - 5*a*b^3*c^3 + 4*a^2*b*c^4)*x)*sqrt(c*x^2 +
b*x + a))/(64*(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^7*x^6 + 192*(b^7*c^7 - 12*a*b^5*c^8
+ 48*a^2*b^3*c^9 - 64*a^3*b*c^10)*d^7*x^5 + 240*(b^8*c^6 - 12*a*b^6*c^7 + 48*a^2*b^4*c^8 - 64*a^3*b^2*c^9)*d^7
*x^4 + 160*(b^9*c^5 - 12*a*b^7*c^6 + 48*a^2*b^5*c^7 - 64*a^3*b^3*c^8)*d^7*x^3 + 60*(b^10*c^4 - 12*a*b^8*c^5 +
48*a^2*b^6*c^6 - 64*a^3*b^4*c^7)*d^7*x^2 + 12*(b^11*c^3 - 12*a*b^9*c^4 + 48*a^2*b^7*c^5 - 64*a^3*b^5*c^6)*d^7*
x + (b^12*c^2 - 12*a*b^10*c^3 + 48*a^2*b^8*c^4 - 64*a^3*b^6*c^5)*d^7)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx}{d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*c**4*x
**4 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 128*c**7*x**7), x)/d**7

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{128,[7]%%%},[14,7,0,0]%%%}+%%%{%%{[%%%{-896,[6]%%%},
0]:[1,0,%%%

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (b\,d+2\,c\,d\,x\right )}^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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