3.14.86 \(\int \frac {\sqrt {c+d x}}{(a+b x)^6} \, dx\) [1386]

Optimal. Leaf size=218 \[ -\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac {7 d^2 \sqrt {c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac {7 d^3 \sqrt {c+d x}}{192 b (b c-a d)^3 (a+b x)^2}+\frac {7 d^4 \sqrt {c+d x}}{128 b (b c-a d)^4 (a+b x)}-\frac {7 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}} \]

[Out]

-7/128*d^5*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(3/2)/(-a*d+b*c)^(9/2)-1/5*(d*x+c)^(1/2)/b/(b*x+a
)^5-1/40*d*(d*x+c)^(1/2)/b/(-a*d+b*c)/(b*x+a)^4+7/240*d^2*(d*x+c)^(1/2)/b/(-a*d+b*c)^2/(b*x+a)^3-7/192*d^3*(d*
x+c)^(1/2)/b/(-a*d+b*c)^3/(b*x+a)^2+7/128*d^4*(d*x+c)^(1/2)/b/(-a*d+b*c)^4/(b*x+a)

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Rubi [A]
time = 0.10, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {43, 44, 65, 214} \begin {gather*} -\frac {7 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}}+\frac {7 d^4 \sqrt {c+d x}}{128 b (a+b x) (b c-a d)^4}-\frac {7 d^3 \sqrt {c+d x}}{192 b (a+b x)^2 (b c-a d)^3}+\frac {7 d^2 \sqrt {c+d x}}{240 b (a+b x)^3 (b c-a d)^2}-\frac {d \sqrt {c+d x}}{40 b (a+b x)^4 (b c-a d)}-\frac {\sqrt {c+d x}}{5 b (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(a + b*x)^6,x]

[Out]

-1/5*Sqrt[c + d*x]/(b*(a + b*x)^5) - (d*Sqrt[c + d*x])/(40*b*(b*c - a*d)*(a + b*x)^4) + (7*d^2*Sqrt[c + d*x])/
(240*b*(b*c - a*d)^2*(a + b*x)^3) - (7*d^3*Sqrt[c + d*x])/(192*b*(b*c - a*d)^3*(a + b*x)^2) + (7*d^4*Sqrt[c +
d*x])/(128*b*(b*c - a*d)^4*(a + b*x)) - (7*d^5*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(128*b^(3/2)*
(b*c - a*d)^(9/2))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d x}}{(a+b x)^6} \, dx &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}+\frac {d \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx}{10 b}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}-\frac {\left (7 d^2\right ) \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx}{80 b (b c-a d)}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac {7 d^2 \sqrt {c+d x}}{240 b (b c-a d)^2 (a+b x)^3}+\frac {\left (7 d^3\right ) \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{96 b (b c-a d)^2}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac {7 d^2 \sqrt {c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac {7 d^3 \sqrt {c+d x}}{192 b (b c-a d)^3 (a+b x)^2}-\frac {\left (7 d^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{128 b (b c-a d)^3}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac {7 d^2 \sqrt {c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac {7 d^3 \sqrt {c+d x}}{192 b (b c-a d)^3 (a+b x)^2}+\frac {7 d^4 \sqrt {c+d x}}{128 b (b c-a d)^4 (a+b x)}+\frac {\left (7 d^5\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{256 b (b c-a d)^4}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac {7 d^2 \sqrt {c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac {7 d^3 \sqrt {c+d x}}{192 b (b c-a d)^3 (a+b x)^2}+\frac {7 d^4 \sqrt {c+d x}}{128 b (b c-a d)^4 (a+b x)}+\frac {\left (7 d^4\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{128 b (b c-a d)^4}\\ &=-\frac {\sqrt {c+d x}}{5 b (a+b x)^5}-\frac {d \sqrt {c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac {7 d^2 \sqrt {c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac {7 d^3 \sqrt {c+d x}}{192 b (b c-a d)^3 (a+b x)^2}+\frac {7 d^4 \sqrt {c+d x}}{128 b (b c-a d)^4 (a+b x)}-\frac {7 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 1.41, size = 224, normalized size = 1.03 \begin {gather*} \frac {\sqrt {c+d x} \left (-105 a^4 d^4+10 a^3 b d^3 (121 c+79 d x)+2 a^2 b^2 d^2 \left (-1052 c^2-289 c d x+448 d^2 x^2\right )+2 a b^3 d \left (744 c^3+128 c^2 d x-161 c d^2 x^2+245 d^3 x^3\right )+b^4 \left (-384 c^4-48 c^3 d x+56 c^2 d^2 x^2-70 c d^3 x^3+105 d^4 x^4\right )\right )}{1920 b (b c-a d)^4 (a+b x)^5}+\frac {7 d^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{128 b^{3/2} (-b c+a d)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]/(a + b*x)^6,x]

[Out]

(Sqrt[c + d*x]*(-105*a^4*d^4 + 10*a^3*b*d^3*(121*c + 79*d*x) + 2*a^2*b^2*d^2*(-1052*c^2 - 289*c*d*x + 448*d^2*
x^2) + 2*a*b^3*d*(744*c^3 + 128*c^2*d*x - 161*c*d^2*x^2 + 245*d^3*x^3) + b^4*(-384*c^4 - 48*c^3*d*x + 56*c^2*d
^2*x^2 - 70*c*d^3*x^3 + 105*d^4*x^4)))/(1920*b*(b*c - a*d)^4*(a + b*x)^5) + (7*d^5*ArcTan[(Sqrt[b]*Sqrt[c + d*
x])/Sqrt[-(b*c) + a*d]])/(128*b^(3/2)*(-(b*c) + a*d)^(9/2))

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Maple [A]
time = 0.16, size = 293, normalized size = 1.34

method	result	size
derivativedivides	\(2 d^{5} \left (\frac {\frac {7 b^{3} \left (d x +c \right )^{\frac {9}{2}}}{256 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {49 b^{2} \left (d x +c \right )^{\frac {7}{2}}}{384 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {7 b \left (d x +c \right )^{\frac {5}{2}}}{30 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {79 \left (d x +c \right )^{\frac {3}{2}}}{384 \left (a d -b c \right )}-\frac {7 \sqrt {d x +c}}{256 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{5}}+\frac {7 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{256 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\)	\(293\)
default	\(2 d^{5} \left (\frac {\frac {7 b^{3} \left (d x +c \right )^{\frac {9}{2}}}{256 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {49 b^{2} \left (d x +c \right )^{\frac {7}{2}}}{384 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {7 b \left (d x +c \right )^{\frac {5}{2}}}{30 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {79 \left (d x +c \right )^{\frac {3}{2}}}{384 \left (a d -b c \right )}-\frac {7 \sqrt {d x +c}}{256 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{5}}+\frac {7 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{256 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\)	\(293\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(b*x+a)^6,x,method=_RETURNVERBOSE)

[Out]

2*d^5*((7/256*b^3/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(9/2)+49/384*b^2/(a^
3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(7/2)+7/30*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(5/2)+79/3
84/(a*d-b*c)*(d*x+c)^(3/2)-7/256*(d*x+c)^(1/2)/b)/((d*x+c)*b+a*d-b*c)^5+7/256/b/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b
^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)))

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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((d*x+c)^(1/2)/(b*x+a)^6,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(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 830 vs. \(2 (186) = 372\).
time = 0.45, size = 1673, normalized size = 7.67 \begin {gather*} \left [\frac {105 \, {\left (b^{5} d^{5} x^{5} + 5 \, a b^{4} d^{5} x^{4} + 10 \, a^{2} b^{3} d^{5} x^{3} + 10 \, a^{3} b^{2} d^{5} x^{2} + 5 \, a^{4} b d^{5} x + a^{5} d^{5}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 2 \, {\left (384 \, b^{6} c^{5} - 1872 \, a b^{5} c^{4} d + 3592 \, a^{2} b^{4} c^{3} d^{2} - 3314 \, a^{3} b^{3} c^{2} d^{3} + 1315 \, a^{4} b^{2} c d^{4} - 105 \, a^{5} b d^{5} - 105 \, {\left (b^{6} c d^{4} - a b^{5} d^{5}\right )} x^{4} + 70 \, {\left (b^{6} c^{2} d^{3} - 8 \, a b^{5} c d^{4} + 7 \, a^{2} b^{4} d^{5}\right )} x^{3} - 14 \, {\left (4 \, b^{6} c^{3} d^{2} - 27 \, a b^{5} c^{2} d^{3} + 87 \, a^{2} b^{4} c d^{4} - 64 \, a^{3} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (24 \, b^{6} c^{4} d - 152 \, a b^{5} c^{3} d^{2} + 417 \, a^{2} b^{4} c^{2} d^{3} - 684 \, a^{3} b^{3} c d^{4} + 395 \, a^{4} b^{2} d^{5}\right )} x\right )} \sqrt {d x + c}}{3840 \, {\left (a^{5} b^{7} c^{5} - 5 \, a^{6} b^{6} c^{4} d + 10 \, a^{7} b^{5} c^{3} d^{2} - 10 \, a^{8} b^{4} c^{2} d^{3} + 5 \, a^{9} b^{3} c d^{4} - a^{10} b^{2} d^{5} + {\left (b^{12} c^{5} - 5 \, a b^{11} c^{4} d + 10 \, a^{2} b^{10} c^{3} d^{2} - 10 \, a^{3} b^{9} c^{2} d^{3} + 5 \, a^{4} b^{8} c d^{4} - a^{5} b^{7} d^{5}\right )} x^{5} + 5 \, {\left (a b^{11} c^{5} - 5 \, a^{2} b^{10} c^{4} d + 10 \, a^{3} b^{9} c^{3} d^{2} - 10 \, a^{4} b^{8} c^{2} d^{3} + 5 \, a^{5} b^{7} c d^{4} - a^{6} b^{6} d^{5}\right )} x^{4} + 10 \, {\left (a^{2} b^{10} c^{5} - 5 \, a^{3} b^{9} c^{4} d + 10 \, a^{4} b^{8} c^{3} d^{2} - 10 \, a^{5} b^{7} c^{2} d^{3} + 5 \, a^{6} b^{6} c d^{4} - a^{7} b^{5} d^{5}\right )} x^{3} + 10 \, {\left (a^{3} b^{9} c^{5} - 5 \, a^{4} b^{8} c^{4} d + 10 \, a^{5} b^{7} c^{3} d^{2} - 10 \, a^{6} b^{6} c^{2} d^{3} + 5 \, a^{7} b^{5} c d^{4} - a^{8} b^{4} d^{5}\right )} x^{2} + 5 \, {\left (a^{4} b^{8} c^{5} - 5 \, a^{5} b^{7} c^{4} d + 10 \, a^{6} b^{6} c^{3} d^{2} - 10 \, a^{7} b^{5} c^{2} d^{3} + 5 \, a^{8} b^{4} c d^{4} - a^{9} b^{3} d^{5}\right )} x\right )}}, \frac {105 \, {\left (b^{5} d^{5} x^{5} + 5 \, a b^{4} d^{5} x^{4} + 10 \, a^{2} b^{3} d^{5} x^{3} + 10 \, a^{3} b^{2} d^{5} x^{2} + 5 \, a^{4} b d^{5} x + a^{5} d^{5}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - {\left (384 \, b^{6} c^{5} - 1872 \, a b^{5} c^{4} d + 3592 \, a^{2} b^{4} c^{3} d^{2} - 3314 \, a^{3} b^{3} c^{2} d^{3} + 1315 \, a^{4} b^{2} c d^{4} - 105 \, a^{5} b d^{5} - 105 \, {\left (b^{6} c d^{4} - a b^{5} d^{5}\right )} x^{4} + 70 \, {\left (b^{6} c^{2} d^{3} - 8 \, a b^{5} c d^{4} + 7 \, a^{2} b^{4} d^{5}\right )} x^{3} - 14 \, {\left (4 \, b^{6} c^{3} d^{2} - 27 \, a b^{5} c^{2} d^{3} + 87 \, a^{2} b^{4} c d^{4} - 64 \, a^{3} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (24 \, b^{6} c^{4} d - 152 \, a b^{5} c^{3} d^{2} + 417 \, a^{2} b^{4} c^{2} d^{3} - 684 \, a^{3} b^{3} c d^{4} + 395 \, a^{4} b^{2} d^{5}\right )} x\right )} \sqrt {d x + c}}{1920 \, {\left (a^{5} b^{7} c^{5} - 5 \, a^{6} b^{6} c^{4} d + 10 \, a^{7} b^{5} c^{3} d^{2} - 10 \, a^{8} b^{4} c^{2} d^{3} + 5 \, a^{9} b^{3} c d^{4} - a^{10} b^{2} d^{5} + {\left (b^{12} c^{5} - 5 \, a b^{11} c^{4} d + 10 \, a^{2} b^{10} c^{3} d^{2} - 10 \, a^{3} b^{9} c^{2} d^{3} + 5 \, a^{4} b^{8} c d^{4} - a^{5} b^{7} d^{5}\right )} x^{5} + 5 \, {\left (a b^{11} c^{5} - 5 \, a^{2} b^{10} c^{4} d + 10 \, a^{3} b^{9} c^{3} d^{2} - 10 \, a^{4} b^{8} c^{2} d^{3} + 5 \, a^{5} b^{7} c d^{4} - a^{6} b^{6} d^{5}\right )} x^{4} + 10 \, {\left (a^{2} b^{10} c^{5} - 5 \, a^{3} b^{9} c^{4} d + 10 \, a^{4} b^{8} c^{3} d^{2} - 10 \, a^{5} b^{7} c^{2} d^{3} + 5 \, a^{6} b^{6} c d^{4} - a^{7} b^{5} d^{5}\right )} x^{3} + 10 \, {\left (a^{3} b^{9} c^{5} - 5 \, a^{4} b^{8} c^{4} d + 10 \, a^{5} b^{7} c^{3} d^{2} - 10 \, a^{6} b^{6} c^{2} d^{3} + 5 \, a^{7} b^{5} c d^{4} - a^{8} b^{4} d^{5}\right )} x^{2} + 5 \, {\left (a^{4} b^{8} c^{5} - 5 \, a^{5} b^{7} c^{4} d + 10 \, a^{6} b^{6} c^{3} d^{2} - 10 \, a^{7} b^{5} c^{2} d^{3} + 5 \, a^{8} b^{4} c d^{4} - a^{9} b^{3} d^{5}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^6,x, algorithm="fricas")

[Out]

[1/3840*(105*(b^5*d^5*x^5 + 5*a*b^4*d^5*x^4 + 10*a^2*b^3*d^5*x^3 + 10*a^3*b^2*d^5*x^2 + 5*a^4*b*d^5*x + a^5*d^
5)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) - 2*(384*b^6
*c^5 - 1872*a*b^5*c^4*d + 3592*a^2*b^4*c^3*d^2 - 3314*a^3*b^3*c^2*d^3 + 1315*a^4*b^2*c*d^4 - 105*a^5*b*d^5 - 1
05*(b^6*c*d^4 - a*b^5*d^5)*x^4 + 70*(b^6*c^2*d^3 - 8*a*b^5*c*d^4 + 7*a^2*b^4*d^5)*x^3 - 14*(4*b^6*c^3*d^2 - 27
*a*b^5*c^2*d^3 + 87*a^2*b^4*c*d^4 - 64*a^3*b^3*d^5)*x^2 + 2*(24*b^6*c^4*d - 152*a*b^5*c^3*d^2 + 417*a^2*b^4*c^
2*d^3 - 684*a^3*b^3*c*d^4 + 395*a^4*b^2*d^5)*x)*sqrt(d*x + c))/(a^5*b^7*c^5 - 5*a^6*b^6*c^4*d + 10*a^7*b^5*c^3
*d^2 - 10*a^8*b^4*c^2*d^3 + 5*a^9*b^3*c*d^4 - a^10*b^2*d^5 + (b^12*c^5 - 5*a*b^11*c^4*d + 10*a^2*b^10*c^3*d^2
- 10*a^3*b^9*c^2*d^3 + 5*a^4*b^8*c*d^4 - a^5*b^7*d^5)*x^5 + 5*(a*b^11*c^5 - 5*a^2*b^10*c^4*d + 10*a^3*b^9*c^3*
d^2 - 10*a^4*b^8*c^2*d^3 + 5*a^5*b^7*c*d^4 - a^6*b^6*d^5)*x^4 + 10*(a^2*b^10*c^5 - 5*a^3*b^9*c^4*d + 10*a^4*b^
8*c^3*d^2 - 10*a^5*b^7*c^2*d^3 + 5*a^6*b^6*c*d^4 - a^7*b^5*d^5)*x^3 + 10*(a^3*b^9*c^5 - 5*a^4*b^8*c^4*d + 10*a
^5*b^7*c^3*d^2 - 10*a^6*b^6*c^2*d^3 + 5*a^7*b^5*c*d^4 - a^8*b^4*d^5)*x^2 + 5*(a^4*b^8*c^5 - 5*a^5*b^7*c^4*d +
10*a^6*b^6*c^3*d^2 - 10*a^7*b^5*c^2*d^3 + 5*a^8*b^4*c*d^4 - a^9*b^3*d^5)*x), 1/1920*(105*(b^5*d^5*x^5 + 5*a*b^
4*d^5*x^4 + 10*a^2*b^3*d^5*x^3 + 10*a^3*b^2*d^5*x^2 + 5*a^4*b*d^5*x + a^5*d^5)*sqrt(-b^2*c + a*b*d)*arctan(sqr
t(-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + b*c)) - (384*b^6*c^5 - 1872*a*b^5*c^4*d + 3592*a^2*b^4*c^3*d^2 - 3314
*a^3*b^3*c^2*d^3 + 1315*a^4*b^2*c*d^4 - 105*a^5*b*d^5 - 105*(b^6*c*d^4 - a*b^5*d^5)*x^4 + 70*(b^6*c^2*d^3 - 8*
a*b^5*c*d^4 + 7*a^2*b^4*d^5)*x^3 - 14*(4*b^6*c^3*d^2 - 27*a*b^5*c^2*d^3 + 87*a^2*b^4*c*d^4 - 64*a^3*b^3*d^5)*x
^2 + 2*(24*b^6*c^4*d - 152*a*b^5*c^3*d^2 + 417*a^2*b^4*c^2*d^3 - 684*a^3*b^3*c*d^4 + 395*a^4*b^2*d^5)*x)*sqrt(
d*x + c))/(a^5*b^7*c^5 - 5*a^6*b^6*c^4*d + 10*a^7*b^5*c^3*d^2 - 10*a^8*b^4*c^2*d^3 + 5*a^9*b^3*c*d^4 - a^10*b^
2*d^5 + (b^12*c^5 - 5*a*b^11*c^4*d + 10*a^2*b^10*c^3*d^2 - 10*a^3*b^9*c^2*d^3 + 5*a^4*b^8*c*d^4 - a^5*b^7*d^5)
*x^5 + 5*(a*b^11*c^5 - 5*a^2*b^10*c^4*d + 10*a^3*b^9*c^3*d^2 - 10*a^4*b^8*c^2*d^3 + 5*a^5*b^7*c*d^4 - a^6*b^6*
d^5)*x^4 + 10*(a^2*b^10*c^5 - 5*a^3*b^9*c^4*d + 10*a^4*b^8*c^3*d^2 - 10*a^5*b^7*c^2*d^3 + 5*a^6*b^6*c*d^4 - a^
7*b^5*d^5)*x^3 + 10*(a^3*b^9*c^5 - 5*a^4*b^8*c^4*d + 10*a^5*b^7*c^3*d^2 - 10*a^6*b^6*c^2*d^3 + 5*a^7*b^5*c*d^4
- a^8*b^4*d^5)*x^2 + 5*(a^4*b^8*c^5 - 5*a^5*b^7*c^4*d + 10*a^6*b^6*c^3*d^2 - 10*a^7*b^5*c^2*d^3 + 5*a^8*b^4*c
*d^4 - a^9*b^3*d^5)*x)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2)/(b*x+a)**6,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (186) = 372\).
time = 2.11, size = 432, normalized size = 1.98 \begin {gather*} \frac {7 \, d^{5} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{128 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \sqrt {-b^{2} c + a b d}} + \frac {105 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{4} d^{5} - 490 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{4} c d^{5} + 896 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4} c^{2} d^{5} - 790 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{4} c^{3} d^{5} - 105 \, \sqrt {d x + c} b^{4} c^{4} d^{5} + 490 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{3} d^{6} - 1792 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{3} c d^{6} + 2370 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} c^{2} d^{6} + 420 \, \sqrt {d x + c} a b^{3} c^{3} d^{6} + 896 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{2} d^{7} - 2370 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{2} c d^{7} - 630 \, \sqrt {d x + c} a^{2} b^{2} c^{2} d^{7} + 790 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b d^{8} + 420 \, \sqrt {d x + c} a^{3} b c d^{8} - 105 \, \sqrt {d x + c} a^{4} d^{9}}{1920 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^6,x, algorithm="giac")

[Out]

7/128*d^5*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b
^2*c*d^3 + a^4*b*d^4)*sqrt(-b^2*c + a*b*d)) + 1/1920*(105*(d*x + c)^(9/2)*b^4*d^5 - 490*(d*x + c)^(7/2)*b^4*c*
d^5 + 896*(d*x + c)^(5/2)*b^4*c^2*d^5 - 790*(d*x + c)^(3/2)*b^4*c^3*d^5 - 105*sqrt(d*x + c)*b^4*c^4*d^5 + 490*
(d*x + c)^(7/2)*a*b^3*d^6 - 1792*(d*x + c)^(5/2)*a*b^3*c*d^6 + 2370*(d*x + c)^(3/2)*a*b^3*c^2*d^6 + 420*sqrt(d
*x + c)*a*b^3*c^3*d^6 + 896*(d*x + c)^(5/2)*a^2*b^2*d^7 - 2370*(d*x + c)^(3/2)*a^2*b^2*c*d^7 - 630*sqrt(d*x +
c)*a^2*b^2*c^2*d^7 + 790*(d*x + c)^(3/2)*a^3*b*d^8 + 420*sqrt(d*x + c)*a^3*b*c*d^8 - 105*sqrt(d*x + c)*a^4*d^9
)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*((d*x + c)*b - b*c + a*d)^5)

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Mupad [B]
time = 0.49, size = 401, normalized size = 1.84 \begin {gather*} \frac {\frac {79\,d^5\,{\left (c+d\,x\right )}^{3/2}}{192\,\left (a\,d-b\,c\right )}-\frac {7\,d^5\,\sqrt {c+d\,x}}{128\,b}+\frac {49\,b^2\,d^5\,{\left (c+d\,x\right )}^{7/2}}{192\,{\left (a\,d-b\,c\right )}^3}+\frac {7\,b^3\,d^5\,{\left (c+d\,x\right )}^{9/2}}{128\,{\left (a\,d-b\,c\right )}^4}+\frac {7\,b\,d^5\,{\left (c+d\,x\right )}^{5/2}}{15\,{\left (a\,d-b\,c\right )}^2}}{b^5\,{\left (c+d\,x\right )}^5-{\left (c+d\,x\right )}^2\,\left (-10\,a^3\,b^2\,d^3+30\,a^2\,b^3\,c\,d^2-30\,a\,b^4\,c^2\,d+10\,b^5\,c^3\right )-\left (5\,b^5\,c-5\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^4+a^5\,d^5-b^5\,c^5+{\left (c+d\,x\right )}^3\,\left (10\,a^2\,b^3\,d^2-20\,a\,b^4\,c\,d+10\,b^5\,c^2\right )+\left (c+d\,x\right )\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )-10\,a^2\,b^3\,c^3\,d^2+10\,a^3\,b^2\,c^2\,d^3+5\,a\,b^4\,c^4\,d-5\,a^4\,b\,c\,d^4}+\frac {7\,d^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{128\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(1/2)/(a + b*x)^6,x)

[Out]

((79*d^5*(c + d*x)^(3/2))/(192*(a*d - b*c)) - (7*d^5*(c + d*x)^(1/2))/(128*b) + (49*b^2*d^5*(c + d*x)^(7/2))/(
192*(a*d - b*c)^3) + (7*b^3*d^5*(c + d*x)^(9/2))/(128*(a*d - b*c)^4) + (7*b*d^5*(c + d*x)^(5/2))/(15*(a*d - b*
c)^2))/(b^5*(c + d*x)^5 - (c + d*x)^2*(10*b^5*c^3 - 10*a^3*b^2*d^3 + 30*a^2*b^3*c*d^2 - 30*a*b^4*c^2*d) - (5*b
^5*c - 5*a*b^4*d)*(c + d*x)^4 + a^5*d^5 - b^5*c^5 + (c + d*x)^3*(10*b^5*c^2 + 10*a^2*b^3*d^2 - 20*a*b^4*c*d) +
(c + d*x)*(5*b^5*c^4 + 5*a^4*b*d^4 - 20*a^3*b^2*c*d^3 + 30*a^2*b^3*c^2*d^2 - 20*a*b^4*c^3*d) - 10*a^2*b^3*c^3
*d^2 + 10*a^3*b^2*c^2*d^3 + 5*a*b^4*c^4*d - 5*a^4*b*c*d^4) + (7*d^5*atan((b^(1/2)*(c + d*x)^(1/2))/(a*d - b*c)
^(1/2)))/(128*b^(3/2)*(a*d - b*c)^(9/2))

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