3.14.83 \(\int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx\) [1383]

Optimal. Leaf size=110 \[ -\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}} \]

[Out]

1/4*d^2*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(3/2)/(-a*d+b*c)^(3/2)-1/2*(d*x+c)^(1/2)/b/(b*x+a)^2
-1/4*d*(d*x+c)^(1/2)/b/(-a*d+b*c)/(b*x+a)

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Rubi [A]
time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}}-\frac {d \sqrt {c+d x}}{4 b (a+b x) (b c-a d)}-\frac {\sqrt {c+d x}}{2 b (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(a + b*x)^3,x]

[Out]

-1/2*Sqrt[c + d*x]/(b*(a + b*x)^2) - (d*Sqrt[c + d*x])/(4*b*(b*c - a*d)*(a + b*x)) + (d^2*ArcTanh[(Sqrt[b]*Sqr
t[c + d*x])/Sqrt[b*c - a*d]])/(4*b^(3/2)*(b*c - a*d)^(3/2))

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)^3} \, dx &=-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}+\frac {d \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{4 b}\\ &=-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}-\frac {d^2 \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{8 b (b c-a d)}\\ &=-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}-\frac {d \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 b (b c-a d)}\\ &=-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 99, normalized size = 0.90 \begin {gather*} \frac {\frac {\sqrt {b} \sqrt {c+d x} (2 b c-a d+b d x)}{(-b c+a d) (a+b x)^2}+\frac {d^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}}{4 b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]/(a + b*x)^3,x]

[Out]

((Sqrt[b]*Sqrt[c + d*x]*(2*b*c - a*d + b*d*x))/((-(b*c) + a*d)*(a + b*x)^2) + (d^2*ArcTan[(Sqrt[b]*Sqrt[c + d*
x])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(3/2))/(4*b^(3/2))

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(1383\) vs. \(2(110)=220\).
time = 57.12, size = 1307, normalized size = 11.88 \begin {gather*} \frac {8 d \left (a^4 d^2-2 a^3 b c d+2 a^3 b d^2 x+a^2 b^2 c^2-4 a^2 b^2 c d x+a^2 b^2 d^2 x^2+2 a b^3 c^2 x-2 a b^3 c d x^2+b^4 c^2 x^2\right ) \sqrt {c+d x}+d^2 \left (-3 a d \text {Log}\left [a^3 d^3 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}-3 a^2 b c d^2 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}+3 a b^2 c^2 d \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}-b^3 c^3 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}+\sqrt {c+d x}\right ] \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}+3 a d \text {Log}\left [-a^3 d^3 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}+3 a^2 b c d^2 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}-3 a b^2 c^2 d \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}+b^3 c^3 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}+\sqrt {c+d x}\right ] \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}-4 \text {Log}\left [-a^2 d^2 \sqrt {-\frac {1}{b \left (a d-b c\right )^3}}+2 a b c d \sqrt {-\frac {1}{b \left (a d-b c\right )^3}}-b^2 c^2 \sqrt {-\frac {1}{b \left (a d-b c\right )^3}}+\sqrt {c+d x}\right ] \sqrt {-\frac {1}{b \left (a d-b c\right )^3}}+4 \text {Log}\left [a^2 d^2 \sqrt {-\frac {1}{b \left (a d-b c\right )^3}}-2 a b c d \sqrt {-\frac {1}{b \left (a d-b c\right )^3}}+b^2 c^2 \sqrt {-\frac {1}{b \left (a d-b c\right )^3}}+\sqrt {c+d x}\right ] \sqrt {-\frac {1}{b \left (a d-b c\right )^3}}\right ) \left (a^2 d-a b c+a b d x-b^2 c x\right ) \left (a^4 d^2-2 a^3 b c d+2 a^3 b d^2 x+a^2 b^2 c^2-4 a^2 b^2 c d x+a^2 b^2 d^2 x^2+2 a b^3 c^2 x-2 a b^3 c d x^2+b^4 c^2 x^2\right )-10 a^2 d^2 \left (a^2 d-a b c+a b d x-b^2 c x\right ) \sqrt {c+d x}+2 b \left (-3 a d \left (c+d x\right )+3 b c \left (c+d x\right )-5 b c^2\right ) \left (a^2 d-a b c+a b d x-b^2 c x\right ) \sqrt {c+d x}+20 a b c d \left (a^2 d-a b c+a b d x-b^2 c x\right ) \sqrt {c+d x}+3 b c d^2 \left (a^2 d-a b c+a b d x-b^2 c x\right ) \left (a^4 d^2-2 a^3 b c d+2 a^3 b d^2 x+a^2 b^2 c^2-4 a^2 b^2 c d x+a^2 b^2 d^2 x^2+2 a b^3 c^2 x-2 a b^3 c d x^2+b^4 c^2 x^2\right ) \left (\text {Log}\left [a^3 d^3 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}-3 a^2 b c d^2 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}+3 a b^2 c^2 d \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}-b^3 c^3 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}+\sqrt {c+d x}\right ]-\text {Log}\left [-a^3 d^3 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}+3 a^2 b c d^2 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}-3 a b^2 c^2 d \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}+b^3 c^3 \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}+\sqrt {c+d x}\right ]\right ) \sqrt {-\frac {1}{b \left (a d-b c\right )^5}}}{8 b \left (a^2 d-a b c+a b d x-b^2 c x\right ) \left (a^4 d^2-2 a^3 b c d+2 a^3 b d^2 x+a^2 b^2 c^2-4 a^2 b^2 c d x+a^2 b^2 d^2 x^2+2 a b^3 c^2 x-2 a b^3 c d x^2+b^4 c^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(c + d*x)^(1/2)/(a + b*x)^3,x]')

[Out]

(8 d (a ^ 4 d ^ 2 - 2 a ^ 3 b c d + 2 a ^ 3 b d ^ 2 x + a ^ 2 b ^ 2 c ^ 2 - 4 a ^ 2 b ^ 2 c d x + a ^ 2 b ^ 2
d ^ 2 x ^ 2 + 2 a b ^ 3 c ^ 2 x - 2 a b ^ 3 c d x ^ 2 + b ^ 4 c ^ 2 x ^ 2) Sqrt[c + d x] + d ^ 2 (-3 a d Log[a
^ 3 d ^ 3 Sqrt[-1 / (b (a d - b c) ^ 5)] - 3 a ^ 2 b c d ^ 2 Sqrt[-1 / (b (a d - b c) ^ 5)] + 3 a b ^ 2 c ^ 2
d Sqrt[-1 / (b (a d - b c) ^ 5)] - b ^ 3 c ^ 3 Sqrt[-1 / (b (a d - b c) ^ 5)] + Sqrt[c + d x]] Sqrt[-1 / (b (
a d - b c) ^ 5)] + 3 a d Log[-a ^ 3 d ^ 3 Sqrt[-1 / (b (a d - b c) ^ 5)] + 3 a ^ 2 b c d ^ 2 Sqrt[-1 / (b (a d
- b c) ^ 5)] - 3 a b ^ 2 c ^ 2 d Sqrt[-1 / (b (a d - b c) ^ 5)] + b ^ 3 c ^ 3 Sqrt[-1 / (b (a d - b c) ^ 5)]
+ Sqrt[c + d x]] Sqrt[-1 / (b (a d - b c) ^ 5)] - 4 Log[-a ^ 2 d ^ 2 Sqrt[-1 / (b (a d - b c) ^ 3)] + 2 a b c
d Sqrt[-1 / (b (a d - b c) ^ 3)] - b ^ 2 c ^ 2 Sqrt[-1 / (b (a d - b c) ^ 3)] + Sqrt[c + d x]] Sqrt[-1 / (b (a
d - b c) ^ 3)] + 4 Log[a ^ 2 d ^ 2 Sqrt[-1 / (b (a d - b c) ^ 3)] - 2 a b c d Sqrt[-1 / (b (a d - b c) ^ 3)]
+ b ^ 2 c ^ 2 Sqrt[-1 / (b (a d - b c) ^ 3)] + Sqrt[c + d x]] Sqrt[-1 / (b (a d - b c) ^ 3)]) (a ^ 2 d - a b c
+ a b d x - b ^ 2 c x) (a ^ 4 d ^ 2 - 2 a ^ 3 b c d + 2 a ^ 3 b d ^ 2 x + a ^ 2 b ^ 2 c ^ 2 - 4 a ^ 2 b ^ 2 c
d x + a ^ 2 b ^ 2 d ^ 2 x ^ 2 + 2 a b ^ 3 c ^ 2 x - 2 a b ^ 3 c d x ^ 2 + b ^ 4 c ^ 2 x ^ 2) - 10 a ^ 2 d ^ 2
(a ^ 2 d - a b c + a b d x - b ^ 2 c x) Sqrt[c + d x] + 2 b (-3 a d (c + d x) + 3 b c (c + d x) - 5 b c ^ 2)
(a ^ 2 d - a b c + a b d x - b ^ 2 c x) Sqrt[c + d x] + 20 a b c d (a ^ 2 d - a b c + a b d x - b ^ 2 c x) Sqr
t[c + d x] + 3 b c d ^ 2 (a ^ 2 d - a b c + a b d x - b ^ 2 c x) (a ^ 4 d ^ 2 - 2 a ^ 3 b c d + 2 a ^ 3 b d ^
2 x + a ^ 2 b ^ 2 c ^ 2 - 4 a ^ 2 b ^ 2 c d x + a ^ 2 b ^ 2 d ^ 2 x ^ 2 + 2 a b ^ 3 c ^ 2 x - 2 a b ^ 3 c d x
^ 2 + b ^ 4 c ^ 2 x ^ 2) (Log[a ^ 3 d ^ 3 Sqrt[-1 / (b (a d - b c) ^ 5)] - 3 a ^ 2 b c d ^ 2 Sqrt[-1 / (b (a d
- b c) ^ 5)] + 3 a b ^ 2 c ^ 2 d Sqrt[-1 / (b (a d - b c) ^ 5)] - b ^ 3 c ^ 3 Sqrt[-1 / (b (a d - b c) ^ 5)]
+ Sqrt[c + d x]] - Log[-a ^ 3 d ^ 3 Sqrt[-1 / (b (a d - b c) ^ 5)] + 3 a ^ 2 b c d ^ 2 Sqrt[-1 / (b (a d - b c
) ^ 5)] - 3 a b ^ 2 c ^ 2 d Sqrt[-1 / (b (a d - b c) ^ 5)] + b ^ 3 c ^ 3 Sqrt[-1 / (b (a d - b c) ^ 5)] + Sqrt
[c + d x]]) Sqrt[-1 / (b (a d - b c) ^ 5)]) / (8 b (a ^ 2 d - a b c + a b d x - b ^ 2 c x) (a ^ 4 d ^ 2 - 2 a
^ 3 b c d + 2 a ^ 3 b d ^ 2 x + a ^ 2 b ^ 2 c ^ 2 - 4 a ^ 2 b ^ 2 c d x + a ^ 2 b ^ 2 d ^ 2 x ^ 2 + 2 a b ^ 3
c ^ 2 x - 2 a b ^ 3 c d x ^ 2 + b ^ 4 c ^ 2 x ^ 2))

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Maple [A]
time = 0.14, size = 106, normalized size = 0.96

method	result	size
derivativedivides	\(2 d^{2} \left (\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\sqrt {d x +c}}{8 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) b \sqrt {\left (a d -b c \right ) b}}\right )\)	\(106\)
default	\(2 d^{2} \left (\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\sqrt {d x +c}}{8 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) b \sqrt {\left (a d -b c \right ) b}}\right )\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

2*d^2*((1/8/(a*d-b*c)*(d*x+c)^(3/2)-1/8*(d*x+c)^(1/2)/b)/((d*x+c)*b+a*d-b*c)^2+1/8/(a*d-b*c)/b/((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)^3,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. 221 vs. \(2 (90) = 180\).
time = 0.31, size = 456, normalized size = 4.15 \begin {gather*} \left [-\frac {{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\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 (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{8 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )}}, -\frac {{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\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 (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{4 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\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)^3,x, algorithm="fricas")

[Out]

[-1/8*((b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*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*(2*b^3*c^2 - 3*a*b^2*c*d + a^2*b*d^2 + (b^3*c*d - a*b^2*d^2)*x)*sqrt(d*x + c
))/(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^2 + 2*(a*b^5*c^2 - 2*a
^2*b^4*c*d + a^3*b^3*d^2)*x), -1/4*((b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^
2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + b*c)) + (2*b^3*c^2 - 3*a*b^2*c*d + a^2*b*d^2 + (b^3*c*d - a*b^2*d^2)*x)*sq
rt(d*x + c))/(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^2 + 2*(a*b^5
*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1658 vs. \(2 (88) = 176\).
time = 95.19, size = 1658, normalized size = 15.07

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2)/(b*x+a)**3,x)

[Out]

-10*a**2*d**4*sqrt(c + d*x)/(8*a**4*b*d**4 - 16*a**3*b**2*c*d**3 + 16*a**3*b**2*d**4*x - 48*a**2*b**3*c*d**3*x
+ 8*a**2*b**3*d**2*(c + d*x)**2 + 16*a*b**4*c**3*d + 48*a*b**4*c**2*d**2*x - 16*a*b**4*c*d*(c + d*x)**2 - 8*b
**5*c**4 - 16*b**5*c**3*d*x + 8*b**5*c**2*(c + d*x)**2) + 20*a*c*d**3*sqrt(c + d*x)/(8*a**4*d**4 - 16*a**3*b*c
*d**3 + 16*a**3*b*d**4*x - 48*a**2*b**2*c*d**3*x + 8*a**2*b**2*d**2*(c + d*x)**2 + 16*a*b**3*c**3*d + 48*a*b**
3*c**2*d**2*x - 16*a*b**3*c*d*(c + d*x)**2 - 8*b**4*c**4 - 16*b**4*c**3*d*x + 8*b**4*c**2*(c + d*x)**2) - 6*a*
d**3*(c + d*x)**(3/2)/(8*a**4*d**4 - 16*a**3*b*c*d**3 + 16*a**3*b*d**4*x - 48*a**2*b**2*c*d**3*x + 8*a**2*b**2
*d**2*(c + d*x)**2 + 16*a*b**3*c**3*d + 48*a*b**3*c**2*d**2*x - 16*a*b**3*c*d*(c + d*x)**2 - 8*b**4*c**4 - 16*
b**4*c**3*d*x + 8*b**4*c**2*(c + d*x)**2) + 3*a*d**3*sqrt(-1/(b*(a*d - b*c)**5))*log(-a**3*d**3*sqrt(-1/(b*(a*
d - b*c)**5)) + 3*a**2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a*b**2*c**2*d*sqrt(-1/(b*(a*d - b*c)**5)) + b*
*3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/(8*b) - 3*a*d**3*sqrt(-1/(b*(a*d - b*c)**5))*log(a**3*d**
3*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a**2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a*b**2*c**2*d*sqrt(-1/(b*(a*d
- b*c)**5)) - b**3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/(8*b) - 10*b*c**2*d**2*sqrt(c + d*x)/(8*a
**4*d**4 - 16*a**3*b*c*d**3 + 16*a**3*b*d**4*x - 48*a**2*b**2*c*d**3*x + 8*a**2*b**2*d**2*(c + d*x)**2 + 16*a*
b**3*c**3*d + 48*a*b**3*c**2*d**2*x - 16*a*b**3*c*d*(c + d*x)**2 - 8*b**4*c**4 - 16*b**4*c**3*d*x + 8*b**4*c**
2*(c + d*x)**2) + 6*b*c*d**2*(c + d*x)**(3/2)/(8*a**4*d**4 - 16*a**3*b*c*d**3 + 16*a**3*b*d**4*x - 48*a**2*b**
2*c*d**3*x + 8*a**2*b**2*d**2*(c + d*x)**2 + 16*a*b**3*c**3*d + 48*a*b**3*c**2*d**2*x - 16*a*b**3*c*d*(c + d*x
)**2 - 8*b**4*c**4 - 16*b**4*c**3*d*x + 8*b**4*c**2*(c + d*x)**2) - 3*c*d**2*sqrt(-1/(b*(a*d - b*c)**5))*log(-
a**3*d**3*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a**2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a*b**2*c**2*d*sqrt(-1/
(b*(a*d - b*c)**5)) + b**3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/8 + 3*c*d**2*sqrt(-1/(b*(a*d - b*
c)**5))*log(a**3*d**3*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a**2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a*b**2*c**
2*d*sqrt(-1/(b*(a*d - b*c)**5)) - b**3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/8 + 2*d**2*sqrt(c + d
*x)/(2*a**2*b*d**2 - 2*a*b**2*c*d + 2*a*b**2*d**2*x - 2*b**3*c*d*x) - d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(-a*
*2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)
**3)) + sqrt(c + d*x))/(2*b) + d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*
a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b)

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Giac [A]
time = 0.00, size = 149, normalized size = 1.35 \begin {gather*} \frac {\sqrt {c+d x} \left (c+d x\right ) b d^{2}+\sqrt {c+d x} c b d^{2}-\sqrt {c+d x} d^{3} a}{\left (-4 c b^{2}+4 b d a\right ) \left (\left (c+d x\right ) b-c b+d a\right )^{2}}-\frac {d^{2} \arctan \left (\frac {b \sqrt {c+d x}}{\sqrt {-b^{2} c+a b d}}\right )}{2 \left (2 c b^{2}-2 b d a\right ) \sqrt {-b^{2} c+a b d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^3,x)

[Out]

-1/4*d^2*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^2*c - a*b*d)*sqrt(-b^2*c + a*b*d)) - 1/4*((d*x + c)^
(3/2)*b*d^2 + sqrt(d*x + c)*b*c*d^2 - sqrt(d*x + c)*a*d^3)/((b^2*c - a*b*d)*((d*x + c)*b - b*c + a*d)^2)

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Mupad [B]
time = 0.30, size = 135, normalized size = 1.23 \begin {gather*} \frac {d^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{4\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {\frac {d^2\,\sqrt {c+d\,x}}{4\,b}-\frac {d^2\,{\left (c+d\,x\right )}^{3/2}}{4\,\left (a\,d-b\,c\right )}}{b^2\,{\left (c+d\,x\right )}^2-\left (2\,b^2\,c-2\,a\,b\,d\right )\,\left (c+d\,x\right )+a^2\,d^2+b^2\,c^2-2\,a\,b\,c\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(1/2)/(a + b*x)^3,x)

[Out]

(d^2*atan((b^(1/2)*(c + d*x)^(1/2))/(a*d - b*c)^(1/2)))/(4*b^(3/2)*(a*d - b*c)^(3/2)) - ((d^2*(c + d*x)^(1/2))
/(4*b) - (d^2*(c + d*x)^(3/2))/(4*(a*d - b*c)))/(b^2*(c + d*x)^2 - (2*b^2*c - 2*a*b*d)*(c + d*x) + a^2*d^2 + b
^2*c^2 - 2*a*b*c*d)

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