3.15.10 \(\int \frac {(c+d x)^{5/2}}{(a+b x)^6} \, dx\) [1410]

Optimal. Leaf size=198 \[ -\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (b c-a d) (a+b x)^2}+\frac {3 d^4 \sqrt {c+d x}}{128 b^3 (b c-a d)^2 (a+b x)}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}-\frac {3 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{7/2} (b c-a d)^{5/2}} \]

[Out]

-1/8*d*(d*x+c)^(3/2)/b^2/(b*x+a)^4-1/5*(d*x+c)^(5/2)/b/(b*x+a)^5-3/128*d^5*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d
+b*c)^(1/2))/b^(7/2)/(-a*d+b*c)^(5/2)-1/16*d^2*(d*x+c)^(1/2)/b^3/(b*x+a)^3-1/64*d^3*(d*x+c)^(1/2)/b^3/(-a*d+b*
c)/(b*x+a)^2+3/128*d^4*(d*x+c)^(1/2)/b^3/(-a*d+b*c)^2/(b*x+a)

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Rubi [A]
time = 0.06, antiderivative size = 198, 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 {3 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{7/2} (b c-a d)^{5/2}}+\frac {3 d^4 \sqrt {c+d x}}{128 b^3 (a+b x) (b c-a d)^2}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (a+b x)^2 (b c-a d)}-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2)/(a + b*x)^6,x]

[Out]

-1/16*(d^2*Sqrt[c + d*x])/(b^3*(a + b*x)^3) - (d^3*Sqrt[c + d*x])/(64*b^3*(b*c - a*d)*(a + b*x)^2) + (3*d^4*Sq
rt[c + d*x])/(128*b^3*(b*c - a*d)^2*(a + b*x)) - (d*(c + d*x)^(3/2))/(8*b^2*(a + b*x)^4) - (c + d*x)^(5/2)/(5*
b*(a + b*x)^5) - (3*d^5*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(128*b^(7/2)*(b*c - a*d)^(5/2))

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 {(c+d x)^{5/2}}{(a+b x)^6} \, dx &=-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}+\frac {d \int \frac {(c+d x)^{3/2}}{(a+b x)^5} \, dx}{2 b}\\ &=-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}+\frac {\left (3 d^2\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx}{16 b^2}\\ &=-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}+\frac {d^3 \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{32 b^3}\\ &=-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (b c-a d) (a+b x)^2}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}-\frac {\left (3 d^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{128 b^3 (b c-a d)}\\ &=-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (b c-a d) (a+b x)^2}+\frac {3 d^4 \sqrt {c+d x}}{128 b^3 (b c-a d)^2 (a+b x)}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}+\frac {\left (3 d^5\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{256 b^3 (b c-a d)^2}\\ &=-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (b c-a d) (a+b x)^2}+\frac {3 d^4 \sqrt {c+d x}}{128 b^3 (b c-a d)^2 (a+b x)}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}+\frac {\left (3 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^3 (b c-a d)^2}\\ &=-\frac {d^2 \sqrt {c+d x}}{16 b^3 (a+b x)^3}-\frac {d^3 \sqrt {c+d x}}{64 b^3 (b c-a d) (a+b x)^2}+\frac {3 d^4 \sqrt {c+d x}}{128 b^3 (b c-a d)^2 (a+b x)}-\frac {d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(c+d x)^{5/2}}{5 b (a+b x)^5}-\frac {3 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{7/2} (b c-a d)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 1.32, size = 222, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {c+d x} \left (15 a^4 d^4+10 a^3 b d^3 (c+7 d x)+2 a^2 b^2 d^2 \left (4 c^2+23 c d x+64 d^2 x^2\right )-2 a b^3 d \left (88 c^3+256 c^2 d x+233 c d^2 x^2+35 d^3 x^3\right )+b^4 \left (128 c^4+336 c^3 d x+248 c^2 d^2 x^2+10 c d^3 x^3-15 d^4 x^4\right )\right )}{640 b^3 (b c-a d)^2 (a+b x)^5}+\frac {3 d^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{128 b^{7/2} (-b c+a d)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/2)/(a + b*x)^6,x]

[Out]

-1/640*(Sqrt[c + d*x]*(15*a^4*d^4 + 10*a^3*b*d^3*(c + 7*d*x) + 2*a^2*b^2*d^2*(4*c^2 + 23*c*d*x + 64*d^2*x^2) -
2*a*b^3*d*(88*c^3 + 256*c^2*d*x + 233*c*d^2*x^2 + 35*d^3*x^3) + b^4*(128*c^4 + 336*c^3*d*x + 248*c^2*d^2*x^2
+ 10*c*d^3*x^3 - 15*d^4*x^4)))/(b^3*(b*c - a*d)^2*(a + b*x)^5) + (3*d^5*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(
b*c) + a*d]])/(128*b^(7/2)*(-(b*c) + a*d)^(5/2))

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Maple [A]
time = 0.16, size = 205, normalized size = 1.04

method	result	size
derivativedivides	\(2 d^{5} \left (\frac {\frac {3 b \left (d x +c \right )^{\frac {9}{2}}}{256 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {7 \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a d -b c \right )}-\frac {\left (d x +c \right )^{\frac {5}{2}}}{10 b}-\frac {7 \left (a d -b c \right ) \left (d x +c \right )^{\frac {3}{2}}}{128 b^{2}}-\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d x +c}}{256 b^{3}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{5}}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{256 b^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\)	\(205\)
default	\(2 d^{5} \left (\frac {\frac {3 b \left (d x +c \right )^{\frac {9}{2}}}{256 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {7 \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a d -b c \right )}-\frac {\left (d x +c \right )^{\frac {5}{2}}}{10 b}-\frac {7 \left (a d -b c \right ) \left (d x +c \right )^{\frac {3}{2}}}{128 b^{2}}-\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d x +c}}{256 b^{3}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{5}}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{256 b^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)/(b*x+a)^6,x,method=_RETURNVERBOSE)

[Out]

2*d^5*((3/256*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(9/2)+7/128/(a*d-b*c)*(d*x+c)^(7/2)-1/10*(d*x+c)^(5/2)/b-7
/128*(a*d-b*c)/b^2*(d*x+c)^(3/2)-3/256/b^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(1/2))/((d*x+c)*b+a*d-b*c)^5+3/
256/b^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)/((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)^(5/2)/(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. 662 vs. \(2 (166) = 332\).
time = 0.45, size = 1337, normalized size = 6.75 \begin {gather*} \left [\frac {15 \, {\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 (128 \, b^{6} c^{5} - 304 \, a b^{5} c^{4} d + 184 \, a^{2} b^{4} c^{3} d^{2} + 2 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - 15 \, a^{5} b d^{5} - 15 \, {\left (b^{6} c d^{4} - a b^{5} d^{5}\right )} x^{4} + 10 \, {\left (b^{6} c^{2} d^{3} - 8 \, a b^{5} c d^{4} + 7 \, a^{2} b^{4} d^{5}\right )} x^{3} + 2 \, {\left (124 \, b^{6} c^{3} d^{2} - 357 \, a b^{5} c^{2} d^{3} + 297 \, a^{2} b^{4} c d^{4} - 64 \, a^{3} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (168 \, b^{6} c^{4} d - 424 \, a b^{5} c^{3} d^{2} + 279 \, a^{2} b^{4} c^{2} d^{3} + 12 \, a^{3} b^{3} c d^{4} - 35 \, a^{4} b^{2} d^{5}\right )} x\right )} \sqrt {d x + c}}{1280 \, {\left (a^{5} b^{7} c^{3} - 3 \, a^{6} b^{6} c^{2} d + 3 \, a^{7} b^{5} c d^{2} - a^{8} b^{4} d^{3} + {\left (b^{12} c^{3} - 3 \, a b^{11} c^{2} d + 3 \, a^{2} b^{10} c d^{2} - a^{3} b^{9} d^{3}\right )} x^{5} + 5 \, {\left (a b^{11} c^{3} - 3 \, a^{2} b^{10} c^{2} d + 3 \, a^{3} b^{9} c d^{2} - a^{4} b^{8} d^{3}\right )} x^{4} + 10 \, {\left (a^{2} b^{10} c^{3} - 3 \, a^{3} b^{9} c^{2} d + 3 \, a^{4} b^{8} c d^{2} - a^{5} b^{7} d^{3}\right )} x^{3} + 10 \, {\left (a^{3} b^{9} c^{3} - 3 \, a^{4} b^{8} c^{2} d + 3 \, a^{5} b^{7} c d^{2} - a^{6} b^{6} d^{3}\right )} x^{2} + 5 \, {\left (a^{4} b^{8} c^{3} - 3 \, a^{5} b^{7} c^{2} d + 3 \, a^{6} b^{6} c d^{2} - a^{7} b^{5} d^{3}\right )} x\right )}}, \frac {15 \, {\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 (128 \, b^{6} c^{5} - 304 \, a b^{5} c^{4} d + 184 \, a^{2} b^{4} c^{3} d^{2} + 2 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - 15 \, a^{5} b d^{5} - 15 \, {\left (b^{6} c d^{4} - a b^{5} d^{5}\right )} x^{4} + 10 \, {\left (b^{6} c^{2} d^{3} - 8 \, a b^{5} c d^{4} + 7 \, a^{2} b^{4} d^{5}\right )} x^{3} + 2 \, {\left (124 \, b^{6} c^{3} d^{2} - 357 \, a b^{5} c^{2} d^{3} + 297 \, a^{2} b^{4} c d^{4} - 64 \, a^{3} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (168 \, b^{6} c^{4} d - 424 \, a b^{5} c^{3} d^{2} + 279 \, a^{2} b^{4} c^{2} d^{3} + 12 \, a^{3} b^{3} c d^{4} - 35 \, a^{4} b^{2} d^{5}\right )} x\right )} \sqrt {d x + c}}{640 \, {\left (a^{5} b^{7} c^{3} - 3 \, a^{6} b^{6} c^{2} d + 3 \, a^{7} b^{5} c d^{2} - a^{8} b^{4} d^{3} + {\left (b^{12} c^{3} - 3 \, a b^{11} c^{2} d + 3 \, a^{2} b^{10} c d^{2} - a^{3} b^{9} d^{3}\right )} x^{5} + 5 \, {\left (a b^{11} c^{3} - 3 \, a^{2} b^{10} c^{2} d + 3 \, a^{3} b^{9} c d^{2} - a^{4} b^{8} d^{3}\right )} x^{4} + 10 \, {\left (a^{2} b^{10} c^{3} - 3 \, a^{3} b^{9} c^{2} d + 3 \, a^{4} b^{8} c d^{2} - a^{5} b^{7} d^{3}\right )} x^{3} + 10 \, {\left (a^{3} b^{9} c^{3} - 3 \, a^{4} b^{8} c^{2} d + 3 \, a^{5} b^{7} c d^{2} - a^{6} b^{6} d^{3}\right )} x^{2} + 5 \, {\left (a^{4} b^{8} c^{3} - 3 \, a^{5} b^{7} c^{2} d + 3 \, a^{6} b^{6} c d^{2} - a^{7} b^{5} d^{3}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/(b*x+a)^6,x, algorithm="fricas")

[Out]

[1/1280*(15*(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*(128*b^6*
c^5 - 304*a*b^5*c^4*d + 184*a^2*b^4*c^3*d^2 + 2*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - 15*a^5*b*d^5 - 15*(b^6*c*d
^4 - a*b^5*d^5)*x^4 + 10*(b^6*c^2*d^3 - 8*a*b^5*c*d^4 + 7*a^2*b^4*d^5)*x^3 + 2*(124*b^6*c^3*d^2 - 357*a*b^5*c^
2*d^3 + 297*a^2*b^4*c*d^4 - 64*a^3*b^3*d^5)*x^2 + 2*(168*b^6*c^4*d - 424*a*b^5*c^3*d^2 + 279*a^2*b^4*c^2*d^3 +
12*a^3*b^3*c*d^4 - 35*a^4*b^2*d^5)*x)*sqrt(d*x + c))/(a^5*b^7*c^3 - 3*a^6*b^6*c^2*d + 3*a^7*b^5*c*d^2 - a^8*b
^4*d^3 + (b^12*c^3 - 3*a*b^11*c^2*d + 3*a^2*b^10*c*d^2 - a^3*b^9*d^3)*x^5 + 5*(a*b^11*c^3 - 3*a^2*b^10*c^2*d +
3*a^3*b^9*c*d^2 - a^4*b^8*d^3)*x^4 + 10*(a^2*b^10*c^3 - 3*a^3*b^9*c^2*d + 3*a^4*b^8*c*d^2 - a^5*b^7*d^3)*x^3
+ 10*(a^3*b^9*c^3 - 3*a^4*b^8*c^2*d + 3*a^5*b^7*c*d^2 - a^6*b^6*d^3)*x^2 + 5*(a^4*b^8*c^3 - 3*a^5*b^7*c^2*d +
3*a^6*b^6*c*d^2 - a^7*b^5*d^3)*x), 1/640*(15*(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(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + b*c
)) - (128*b^6*c^5 - 304*a*b^5*c^4*d + 184*a^2*b^4*c^3*d^2 + 2*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - 15*a^5*b*d^5
- 15*(b^6*c*d^4 - a*b^5*d^5)*x^4 + 10*(b^6*c^2*d^3 - 8*a*b^5*c*d^4 + 7*a^2*b^4*d^5)*x^3 + 2*(124*b^6*c^3*d^2
- 357*a*b^5*c^2*d^3 + 297*a^2*b^4*c*d^4 - 64*a^3*b^3*d^5)*x^2 + 2*(168*b^6*c^4*d - 424*a*b^5*c^3*d^2 + 279*a^2
*b^4*c^2*d^3 + 12*a^3*b^3*c*d^4 - 35*a^4*b^2*d^5)*x)*sqrt(d*x + c))/(a^5*b^7*c^3 - 3*a^6*b^6*c^2*d + 3*a^7*b^5
*c*d^2 - a^8*b^4*d^3 + (b^12*c^3 - 3*a*b^11*c^2*d + 3*a^2*b^10*c*d^2 - a^3*b^9*d^3)*x^5 + 5*(a*b^11*c^3 - 3*a^
2*b^10*c^2*d + 3*a^3*b^9*c*d^2 - a^4*b^8*d^3)*x^4 + 10*(a^2*b^10*c^3 - 3*a^3*b^9*c^2*d + 3*a^4*b^8*c*d^2 - a^5
*b^7*d^3)*x^3 + 10*(a^3*b^9*c^3 - 3*a^4*b^8*c^2*d + 3*a^5*b^7*c*d^2 - a^6*b^6*d^3)*x^2 + 5*(a^4*b^8*c^3 - 3*a^
5*b^7*c^2*d + 3*a^6*b^6*c*d^2 - a^7*b^5*d^3)*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)**(5/2)/(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. 380 vs. \(2 (166) = 332\).
time = 1.51, size = 380, normalized size = 1.92 \begin {gather*} \frac {3 \, d^{5} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{128 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {15 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{4} d^{5} - 70 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{4} c d^{5} - 128 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4} c^{2} d^{5} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{4} c^{3} d^{5} - 15 \, \sqrt {d x + c} b^{4} c^{4} d^{5} + 70 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{3} d^{6} + 256 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{3} c d^{6} - 210 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} c^{2} d^{6} + 60 \, \sqrt {d x + c} a b^{3} c^{3} d^{6} - 128 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{2} d^{7} + 210 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{2} c d^{7} - 90 \, \sqrt {d x + c} a^{2} b^{2} c^{2} d^{7} - 70 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b d^{8} + 60 \, \sqrt {d x + c} a^{3} b c d^{8} - 15 \, \sqrt {d x + c} a^{4} d^{9}}{640 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\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)^(5/2)/(b*x+a)^6,x, algorithm="giac")

[Out]

3/128*d^5*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*sqrt(-b^2*c + a*
b*d)) + 1/640*(15*(d*x + c)^(9/2)*b^4*d^5 - 70*(d*x + c)^(7/2)*b^4*c*d^5 - 128*(d*x + c)^(5/2)*b^4*c^2*d^5 + 7
0*(d*x + c)^(3/2)*b^4*c^3*d^5 - 15*sqrt(d*x + c)*b^4*c^4*d^5 + 70*(d*x + c)^(7/2)*a*b^3*d^6 + 256*(d*x + c)^(5
/2)*a*b^3*c*d^6 - 210*(d*x + c)^(3/2)*a*b^3*c^2*d^6 + 60*sqrt(d*x + c)*a*b^3*c^3*d^6 - 128*(d*x + c)^(5/2)*a^2
*b^2*d^7 + 210*(d*x + c)^(3/2)*a^2*b^2*c*d^7 - 90*sqrt(d*x + c)*a^2*b^2*c^2*d^7 - 70*(d*x + c)^(3/2)*a^3*b*d^8
+ 60*sqrt(d*x + c)*a^3*b*c*d^8 - 15*sqrt(d*x + c)*a^4*d^9)/((b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*((d*x + c)*
b - b*c + a*d)^5)

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Mupad [B]
time = 0.50, size = 411, normalized size = 2.08 \begin {gather*} \frac {3\,d^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{128\,b^{7/2}\,{\left (a\,d-b\,c\right )}^{5/2}}-\frac {\frac {d^5\,{\left (c+d\,x\right )}^{5/2}}{5\,b}-\frac {7\,d^5\,{\left (c+d\,x\right )}^{7/2}}{64\,\left (a\,d-b\,c\right )}+\frac {3\,d^5\,\sqrt {c+d\,x}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{128\,b^3}+\frac {7\,d^5\,\left (a\,d-b\,c\right )\,{\left (c+d\,x\right )}^{3/2}}{64\,b^2}-\frac {3\,b\,d^5\,{\left (c+d\,x\right )}^{9/2}}{128\,{\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(5/2)/(a + b*x)^6,x)

[Out]

(3*d^5*atan((b^(1/2)*(c + d*x)^(1/2))/(a*d - b*c)^(1/2)))/(128*b^(7/2)*(a*d - b*c)^(5/2)) - ((d^5*(c + d*x)^(5
/2))/(5*b) - (7*d^5*(c + d*x)^(7/2))/(64*(a*d - b*c)) + (3*d^5*(c + d*x)^(1/2)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)
)/(128*b^3) + (7*d^5*(a*d - b*c)*(c + d*x)^(3/2))/(64*b^2) - (3*b*d^5*(c + d*x)^(9/2))/(128*(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)

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