3.18.17 \(\int \frac {(c+d x)^{3/2}}{(a+b x)^2 (e+f x)} \, dx\) [1717]

Optimal. Leaf size=170 \[ -\frac {(b c-a d) \sqrt {c+d x}}{b (b e-a f) (a+b x)}+\frac {2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} (b e-a f)^2}-\frac {\sqrt {b c-a d} (3 b d e-2 b c f-a d f) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b e-a f)^2} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 162, 65, 214, 211} \begin {gather*} \frac {2 (d e-c f)^{3/2} \text {ArcTan}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} (b e-a f)^2}-\frac {\sqrt {b c-a d} (-a d f-2 b c f+3 b d e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b e-a f)^2}-\frac {\sqrt {c+d x} (b c-a d)}{b (a+b x) (b e-a f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(3/2)/((a + b*x)^2*(e + f*x)),x]

[Out]

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

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 100

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

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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

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Mathematica [A]
time = 0.66, size = 156, normalized size = 0.92 \begin {gather*} \frac {\frac {(-b c+a d) (b e-a f) \sqrt {c+d x}}{b (a+b x)}+\frac {\sqrt {-b c+a d} (-3 b d e+2 b c f+a d f) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{3/2}}+\frac {2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f}}}{(b e-a f)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(3/2)/((a + b*x)^2*(e + f*x)),x]

[Out]

(((-(b*c) + a*d)*(b*e - a*f)*Sqrt[c + d*x])/(b*(a + b*x)) + (Sqrt[-(b*c) + a*d]*(-3*b*d*e + 2*b*c*f + a*d*f)*A
rcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/b^(3/2) + (2*(d*e - c*f)^(3/2)*ArcTan[(Sqrt[f]*Sqrt[c + d*x
])/Sqrt[d*e - c*f]])/Sqrt[f])/(b*e - a*f)^2

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Maple [A]
time = 0.09, size = 181, normalized size = 1.06

method	result	size
derivativedivides	\(2 d^{2} \left (-\frac {\left (c f -d e \right )^{2} \arctanh \left (\frac {f \sqrt {d x +c}}{\sqrt {\left (c f -d e \right ) f}}\right )}{d^{2} \left (a f -b e \right )^{2} \sqrt {\left (c f -d e \right ) f}}+\frac {\left (a d -b c \right ) \left (-\frac {d \left (a f -b e \right ) \sqrt {d x +c}}{2 b \left (b \left (d x +c \right )+a d -b c \right )}+\frac {\left (a d f +2 b c f -3 b d e \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{d^{2} \left (a f -b e \right )^{2}}\right )\)	\(181\)
default	\(2 d^{2} \left (-\frac {\left (c f -d e \right )^{2} \arctanh \left (\frac {f \sqrt {d x +c}}{\sqrt {\left (c f -d e \right ) f}}\right )}{d^{2} \left (a f -b e \right )^{2} \sqrt {\left (c f -d e \right ) f}}+\frac {\left (a d -b c \right ) \left (-\frac {d \left (a f -b e \right ) \sqrt {d x +c}}{2 b \left (b \left (d x +c \right )+a d -b c \right )}+\frac {\left (a d f +2 b c f -3 b d e \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{d^{2} \left (a f -b e \right )^{2}}\right )\)	\(181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d^2*(-(c*f-d*e)^2/d^2/(a*f-b*e)^2/((c*f-d*e)*f)^(1/2)*arctanh(f*(d*x+c)^(1/2)/((c*f-d*e)*f)^(1/2))+(a*d-b*c)
/d^2/(a*f-b*e)^2*(-1/2*d*(a*f-b*e)/b*(d*x+c)^(1/2)/(b*(d*x+c)+a*d-b*c)+1/2*(a*d*f+2*b*c*f-3*b*d*e)/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)^(3/2)/(b*x+a)^2/(f*x+e),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(c*f-%e*d>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 2.63, size = 1139, normalized size = 6.70 \begin {gather*} \left [-\frac {{\left ({\left (2 \, b^{2} c + a b d\right )} f x + {\left (2 \, a b c + a^{2} d\right )} f - 3 \, {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (b^{2} c f x + a b c f - {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {\frac {c f - d e}{f}} \log \left (\frac {d f x + 2 \, c f + 2 \, \sqrt {d x + c} f \sqrt {\frac {c f - d e}{f}} - d e}{f x + e}\right ) - 2 \, {\left ({\left (a b c - a^{2} d\right )} f - {\left (b^{2} c - a b d\right )} e\right )} \sqrt {d x + c}}{2 \, {\left (a^{2} b^{2} f^{2} x + a^{3} b f^{2} + {\left (b^{4} x + a b^{3}\right )} e^{2} - 2 \, {\left (a b^{3} f x + a^{2} b^{2} f\right )} e\right )}}, -\frac {4 \, {\left (b^{2} c f x + a b c f - {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {-\frac {c f - d e}{f}} \arctan \left (-\frac {\sqrt {d x + c} f \sqrt {-\frac {c f - d e}{f}}}{c f - d e}\right ) + {\left ({\left (2 \, b^{2} c + a b d\right )} f x + {\left (2 \, a b c + a^{2} d\right )} f - 3 \, {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) - 2 \, {\left ({\left (a b c - a^{2} d\right )} f - {\left (b^{2} c - a b d\right )} e\right )} \sqrt {d x + c}}{2 \, {\left (a^{2} b^{2} f^{2} x + a^{3} b f^{2} + {\left (b^{4} x + a b^{3}\right )} e^{2} - 2 \, {\left (a b^{3} f x + a^{2} b^{2} f\right )} e\right )}}, \frac {{\left ({\left (2 \, b^{2} c + a b d\right )} f x + {\left (2 \, a b c + a^{2} d\right )} f - 3 \, {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (b^{2} c f x + a b c f - {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {\frac {c f - d e}{f}} \log \left (\frac {d f x + 2 \, c f + 2 \, \sqrt {d x + c} f \sqrt {\frac {c f - d e}{f}} - d e}{f x + e}\right ) + {\left ({\left (a b c - a^{2} d\right )} f - {\left (b^{2} c - a b d\right )} e\right )} \sqrt {d x + c}}{a^{2} b^{2} f^{2} x + a^{3} b f^{2} + {\left (b^{4} x + a b^{3}\right )} e^{2} - 2 \, {\left (a b^{3} f x + a^{2} b^{2} f\right )} e}, \frac {{\left ({\left (2 \, b^{2} c + a b d\right )} f x + {\left (2 \, a b c + a^{2} d\right )} f - 3 \, {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - 2 \, {\left (b^{2} c f x + a b c f - {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {-\frac {c f - d e}{f}} \arctan \left (-\frac {\sqrt {d x + c} f \sqrt {-\frac {c f - d e}{f}}}{c f - d e}\right ) + {\left ({\left (a b c - a^{2} d\right )} f - {\left (b^{2} c - a b d\right )} e\right )} \sqrt {d x + c}}{a^{2} b^{2} f^{2} x + a^{3} b f^{2} + {\left (b^{4} x + a b^{3}\right )} e^{2} - 2 \, {\left (a b^{3} f x + a^{2} b^{2} f\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^2/(f*x+e),x, algorithm="fricas")

[Out]

[-1/2*(((2*b^2*c + a*b*d)*f*x + (2*a*b*c + a^2*d)*f - 3*(b^2*d*x + a*b*d)*e)*sqrt((b*c - a*d)/b)*log((b*d*x +
2*b*c - a*d - 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*(b^2*c*f*x + a*b*c*f - (b^2*d*x + a*b*d)*e
)*sqrt((c*f - d*e)/f)*log((d*f*x + 2*c*f + 2*sqrt(d*x + c)*f*sqrt((c*f - d*e)/f) - d*e)/(f*x + e)) - 2*((a*b*c
- a^2*d)*f - (b^2*c - a*b*d)*e)*sqrt(d*x + c))/(a^2*b^2*f^2*x + a^3*b*f^2 + (b^4*x + a*b^3)*e^2 - 2*(a*b^3*f*
x + a^2*b^2*f)*e), -1/2*(4*(b^2*c*f*x + a*b*c*f - (b^2*d*x + a*b*d)*e)*sqrt(-(c*f - d*e)/f)*arctan(-sqrt(d*x +
c)*f*sqrt(-(c*f - d*e)/f)/(c*f - d*e)) + ((2*b^2*c + a*b*d)*f*x + (2*a*b*c + a^2*d)*f - 3*(b^2*d*x + a*b*d)*e
)*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d - 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) - 2*((a*b*c
- a^2*d)*f - (b^2*c - a*b*d)*e)*sqrt(d*x + c))/(a^2*b^2*f^2*x + a^3*b*f^2 + (b^4*x + a*b^3)*e^2 - 2*(a*b^3*f*
x + a^2*b^2*f)*e), (((2*b^2*c + a*b*d)*f*x + (2*a*b*c + a^2*d)*f - 3*(b^2*d*x + a*b*d)*e)*sqrt(-(b*c - a*d)/b)
*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) - (b^2*c*f*x + a*b*c*f - (b^2*d*x + a*b*d)*e)*sqrt(
(c*f - d*e)/f)*log((d*f*x + 2*c*f + 2*sqrt(d*x + c)*f*sqrt((c*f - d*e)/f) - d*e)/(f*x + e)) + ((a*b*c - a^2*d)
*f - (b^2*c - a*b*d)*e)*sqrt(d*x + c))/(a^2*b^2*f^2*x + a^3*b*f^2 + (b^4*x + a*b^3)*e^2 - 2*(a*b^3*f*x + a^2*b
^2*f)*e), (((2*b^2*c + a*b*d)*f*x + (2*a*b*c + a^2*d)*f - 3*(b^2*d*x + a*b*d)*e)*sqrt(-(b*c - a*d)/b)*arctan(-
sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) - 2*(b^2*c*f*x + a*b*c*f - (b^2*d*x + a*b*d)*e)*sqrt(-(c*f -
d*e)/f)*arctan(-sqrt(d*x + c)*f*sqrt(-(c*f - d*e)/f)/(c*f - d*e)) + ((a*b*c - a^2*d)*f - (b^2*c - a*b*d)*e)*s
qrt(d*x + c))/(a^2*b^2*f^2*x + a^3*b*f^2 + (b^4*x + a*b^3)*e^2 - 2*(a*b^3*f*x + a^2*b^2*f)*e)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1766 vs. \(2 (146) = 292\).
time = 143.95, size = 1766, normalized size = 10.39 \begin {gather*} \frac {a^{2} d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \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 )}}{2 a b f - 2 b^{2} e} - \frac {a^{2} d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \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 )}}{2 a b f - 2 b^{2} e} - \frac {2 a^{2} d^{3} \sqrt {c + d x}}{2 a^{3} b d^{2} f - 2 a^{2} b^{2} c d f - 2 a^{2} b^{2} d^{2} e + 2 a^{2} b^{2} d^{2} f x + 2 a b^{3} c d e - 2 a b^{3} c d f x - 2 a b^{3} d^{2} e x + 2 b^{4} c d e x} + \frac {2 a^{2} d^{2} f \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b^{2} \left (a f - b e\right )^{2} \sqrt {\frac {a d}{b} - c}} - \frac {a b c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \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 )}}{2 a b f - 2 b^{2} e} + \frac {a b c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \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 )}}{2 a b f - 2 b^{2} e} + \frac {2 a b c d^{2} \sqrt {c + d x}}{2 a^{3} b d^{2} f - 2 a^{2} b^{2} c d f - 2 a^{2} b^{2} d^{2} e + 2 a^{2} b^{2} d^{2} f x + 2 a b^{3} c d e - 2 a b^{3} c d f x - 2 a b^{3} d^{2} e x + 2 b^{4} c d e x} - \frac {a c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \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 )}}{2 a f - 2 b e} + \frac {a c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \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 )}}{2 a f - 2 b e} + \frac {2 a c d^{2} \sqrt {c + d x}}{2 a^{3} d^{2} f - 2 a^{2} b c d f - 2 a^{2} b d^{2} e + 2 a^{2} b d^{2} f x + 2 a b^{2} c d e - 2 a b^{2} c d f x - 2 a b^{2} d^{2} e x + 2 b^{3} c d e x} - \frac {4 a d^{2} e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b \left (a f - b e\right )^{2} \sqrt {\frac {a d}{b} - c}} + \frac {b c^{2} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \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 )}}{2 a f - 2 b e} - \frac {b c^{2} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \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 )}}{2 a f - 2 b e} - \frac {2 b c^{2} d \sqrt {c + d x}}{2 a^{3} d^{2} f - 2 a^{2} b c d f - 2 a^{2} b d^{2} e + 2 a^{2} b d^{2} f x + 2 a b^{2} c d e - 2 a b^{2} c d f x - 2 a b^{2} d^{2} e x + 2 b^{3} c d e x} - \frac {2 c^{2} f \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{\left (a f - b e\right )^{2} \sqrt {\frac {a d}{b} - c}} + \frac {2 c^{2} f \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c + \frac {d e}{f}}} \right )}}{\sqrt {- c + \frac {d e}{f}} \left (a f - b e\right )^{2}} + \frac {4 c d e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{\left (a f - b e\right )^{2} \sqrt {\frac {a d}{b} - c}} - \frac {4 c d e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c + \frac {d e}{f}}} \right )}}{\sqrt {- c + \frac {d e}{f}} \left (a f - b e\right )^{2}} + \frac {2 d^{2} e^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c + \frac {d e}{f}}} \right )}}{f \sqrt {- c + \frac {d e}{f}} \left (a f - b e\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d**3*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*a*b*f - 2*b**2*e) - a**2*d**3*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*a*b*f - 2*b**2*e) - 2*a**2*d**3*sqrt(c + d*x)/(2*a**3*b*d**
2*f - 2*a**2*b**2*c*d*f - 2*a**2*b**2*d**2*e + 2*a**2*b**2*d**2*f*x + 2*a*b**3*c*d*e - 2*a*b**3*c*d*f*x - 2*a*
b**3*d**2*e*x + 2*b**4*c*d*e*x) + 2*a**2*d**2*f*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b**2*(a*f - b*e)**2*sqrt(
a*d/b - c)) - a*b*c*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*sq
rt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*a*b*f - 2*b**2*e) + a*b*
c*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*a*b*f - 2*b**2*e) + 2*a*b*c*d**2*sqrt(c + d
*x)/(2*a**3*b*d**2*f - 2*a**2*b**2*c*d*f - 2*a**2*b**2*d**2*e + 2*a**2*b**2*d**2*f*x + 2*a*b**3*c*d*e - 2*a*b*
*3*c*d*f*x - 2*a*b**3*d**2*e*x + 2*b**4*c*d*e*x) - a*c*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*a*f - 2*b*e) + a*c*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*a*f - 2*b*e) +
2*a*c*d**2*sqrt(c + d*x)/(2*a**3*d**2*f - 2*a**2*b*c*d*f - 2*a**2*b*d**2*e + 2*a**2*b*d**2*f*x + 2*a*b**2*c*d*
e - 2*a*b**2*c*d*f*x - 2*a*b**2*d**2*e*x + 2*b**3*c*d*e*x) - 4*a*d**2*e*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b
*(a*f - b*e)**2*sqrt(a*d/b - c)) + b*c**2*d*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*a*f -
2*b*e) - b*c**2*d*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*a*f - 2*b*e) - 2*b*c**2*d*sqrt(
c + d*x)/(2*a**3*d**2*f - 2*a**2*b*c*d*f - 2*a**2*b*d**2*e + 2*a**2*b*d**2*f*x + 2*a*b**2*c*d*e - 2*a*b**2*c*d
*f*x - 2*a*b**2*d**2*e*x + 2*b**3*c*d*e*x) - 2*c**2*f*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/((a*f - b*e)**2*sqrt
(a*d/b - c)) + 2*c**2*f*atan(sqrt(c + d*x)/sqrt(-c + d*e/f))/(sqrt(-c + d*e/f)*(a*f - b*e)**2) + 4*c*d*e*atan(
sqrt(c + d*x)/sqrt(a*d/b - c))/((a*f - b*e)**2*sqrt(a*d/b - c)) - 4*c*d*e*atan(sqrt(c + d*x)/sqrt(-c + d*e/f))
/(sqrt(-c + d*e/f)*(a*f - b*e)**2) + 2*d**2*e**2*atan(sqrt(c + d*x)/sqrt(-c + d*e/f))/(f*sqrt(-c + d*e/f)*(a*f
- b*e)**2)

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Giac [A]
time = 0.58, size = 250, normalized size = 1.47 \begin {gather*} -\frac {{\left (2 \, b^{2} c^{2} f - a b c d f - a^{2} d^{2} f - 3 \, b^{2} c d e + 3 \, a b d^{2} e\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{2} b f^{2} - 2 \, a b^{2} f e + b^{3} e^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (c^{2} f^{2} - 2 \, c d f e + d^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} f}{\sqrt {-c f^{2} + d f e}}\right )}{{\left (a^{2} f^{2} - 2 \, a b f e + b^{2} e^{2}\right )} \sqrt {-c f^{2} + d f e}} + \frac {\sqrt {d x + c} b c d - \sqrt {d x + c} a d^{2}}{{\left (a b f - b^{2} e\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^2/(f*x+e),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 3.82, size = 2500, normalized size = 14.71 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(3/2)/((e + f*x)*(a + b*x)^2),x)

[Out]

- (atan(((((2*(c + d*x)^(1/2)*(a^4*d^6*f^5 + 4*b^4*d^6*e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*b^2*c^2*d^4*f^5 + 9*a
^2*b^2*d^6*e^2*f^3 + 33*b^4*c^2*d^4*e^2*f^3 + 2*a^3*b*c*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a*b^3*c^3*d^3*f^5 - 16
*b^4*c*d^5*e^3*f^2 - 28*b^4*c^3*d^3*e*f^4 - 18*a*b^3*c*d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f^4))/(b^3*e^2 + a^2*b
*f^2 - 2*a*b^2*e*f) + ((-b^3*(a*d - b*c))^(1/2)*((2*(2*a^4*b^3*c^2*d^3*f^7 - 8*a^2*b^5*d^5*e^4*f^3 + 12*a^3*b^
4*d^5*e^3*f^4 - 8*a^4*b^3*d^5*e^2*f^5 + 2*b^7*c^2*d^3*e^4*f^3 - 2*a^5*b^2*c*d^4*f^7 + 2*a*b^6*d^5*e^5*f^2 + 2*
a^5*b^2*d^5*e*f^6 - 2*b^7*c*d^4*e^5*f^2 + 6*a*b^6*c*d^4*e^4*f^3 + 6*a^4*b^3*c*d^4*e*f^6 - 8*a*b^6*c^2*d^3*e^3*
f^4 - 4*a^2*b^5*c*d^4*e^3*f^4 - 4*a^3*b^4*c*d^4*e^2*f^5 - 8*a^3*b^4*c^2*d^3*e*f^6 + 12*a^2*b^5*c^2*d^3*e^2*f^5
))/(b^4*e^3 - a^3*b*f^3 + 3*a^2*b^2*e*f^2 - 3*a*b^3*e^2*f) + ((-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2)*(a*d*f
+ 2*b*c*f - 3*b*d*e)*(4*a^5*b^3*d^3*f^7 + 4*b^8*d^3*e^5*f^2 + 8*a^2*b^6*d^3*e^3*f^4 + 8*a^3*b^5*d^3*e^2*f^5 -
8*a^4*b^4*c*d^2*f^7 - 12*a*b^7*d^3*e^4*f^3 - 12*a^4*b^4*d^3*e*f^6 - 8*b^8*c*d^2*e^4*f^3 + 32*a*b^7*c*d^2*e^3*f
^4 + 32*a^3*b^5*c*d^2*e*f^6 - 48*a^2*b^6*c*d^2*e^2*f^5))/((b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)*(b^3*e^2 + a^2
*b*f^2 - 2*a*b^2*e*f)))*(a*d*f + 2*b*c*f - 3*b*d*e))/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)))*(-b^3*(a*d - b
*c))^(1/2)*(a*d*f + 2*b*c*f - 3*b*d*e)*1i)/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)) + (((2*(c + d*x)^(1/2)*(a
^4*d^6*f^5 + 4*b^4*d^6*e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*b^2*c^2*d^4*f^5 + 9*a^2*b^2*d^6*e^2*f^3 + 33*b^4*c^2*
d^4*e^2*f^3 + 2*a^3*b*c*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a*b^3*c^3*d^3*f^5 - 16*b^4*c*d^5*e^3*f^2 - 28*b^4*c^3*
d^3*e*f^4 - 18*a*b^3*c*d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f^4))/(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f) - ((-b^3*(a*
d - b*c))^(1/2)*((2*(2*a^4*b^3*c^2*d^3*f^7 - 8*a^2*b^5*d^5*e^4*f^3 + 12*a^3*b^4*d^5*e^3*f^4 - 8*a^4*b^3*d^5*e^
2*f^5 + 2*b^7*c^2*d^3*e^4*f^3 - 2*a^5*b^2*c*d^4*f^7 + 2*a*b^6*d^5*e^5*f^2 + 2*a^5*b^2*d^5*e*f^6 - 2*b^7*c*d^4*
e^5*f^2 + 6*a*b^6*c*d^4*e^4*f^3 + 6*a^4*b^3*c*d^4*e*f^6 - 8*a*b^6*c^2*d^3*e^3*f^4 - 4*a^2*b^5*c*d^4*e^3*f^4 -
4*a^3*b^4*c*d^4*e^2*f^5 - 8*a^3*b^4*c^2*d^3*e*f^6 + 12*a^2*b^5*c^2*d^3*e^2*f^5))/(b^4*e^3 - a^3*b*f^3 + 3*a^2*
b^2*e*f^2 - 3*a*b^3*e^2*f) - ((-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2)*(a*d*f + 2*b*c*f - 3*b*d*e)*(4*a^5*b^3*
d^3*f^7 + 4*b^8*d^3*e^5*f^2 + 8*a^2*b^6*d^3*e^3*f^4 + 8*a^3*b^5*d^3*e^2*f^5 - 8*a^4*b^4*c*d^2*f^7 - 12*a*b^7*d
^3*e^4*f^3 - 12*a^4*b^4*d^3*e*f^6 - 8*b^8*c*d^2*e^4*f^3 + 32*a*b^7*c*d^2*e^3*f^4 + 32*a^3*b^5*c*d^2*e*f^6 - 48
*a^2*b^6*c*d^2*e^2*f^5))/((b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)*(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f)))*(a*d*f +
2*b*c*f - 3*b*d*e))/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)))*(-b^3*(a*d - b*c))^(1/2)*(a*d*f + 2*b*c*f - 3*
b*d*e)*1i)/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)))/((4*(a^3*c^2*d^6*f^5 - 2*b^3*c^5*d^3*f^5 + a^3*d^8*e^2*f
^3 + 19*b^3*c^2*d^6*e^3*f^2 - 22*b^3*c^3*d^5*e^2*f^3 + 6*a*b^2*d^8*e^4*f - 2*a^3*c*d^7*e*f^4 - 6*b^3*c*d^7*e^4
*f - a*b^2*c^4*d^4*f^5 + 2*a^2*b*c^3*d^5*f^5 - 5*a^2*b*d^8*e^3*f^2 + 11*b^3*c^4*d^4*e*f^4 - 14*a*b^2*c*d^7*e^3
*f^2 + 12*a^2*b*c*d^7*e^2*f^3 - 9*a^2*b*c^2*d^6*e*f^4 + 9*a*b^2*c^2*d^6*e^2*f^3))/(b^4*e^3 - a^3*b*f^3 + 3*a^2
*b^2*e*f^2 - 3*a*b^3*e^2*f) + (((2*(c + d*x)^(1/2)*(a^4*d^6*f^5 + 4*b^4*d^6*e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*
b^2*c^2*d^4*f^5 + 9*a^2*b^2*d^6*e^2*f^3 + 33*b^4*c^2*d^4*e^2*f^3 + 2*a^3*b*c*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a
*b^3*c^3*d^3*f^5 - 16*b^4*c*d^5*e^3*f^2 - 28*b^4*c^3*d^3*e*f^4 - 18*a*b^3*c*d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f
^4))/(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f) + ((-b^3*(a*d - b*c))^(1/2)*((2*(2*a^4*b^3*c^2*d^3*f^7 - 8*a^2*b^5*d^
5*e^4*f^3 + 12*a^3*b^4*d^5*e^3*f^4 - 8*a^4*b^3*d^5*e^2*f^5 + 2*b^7*c^2*d^3*e^4*f^3 - 2*a^5*b^2*c*d^4*f^7 + 2*a
*b^6*d^5*e^5*f^2 + 2*a^5*b^2*d^5*e*f^6 - 2*b^7*c*d^4*e^5*f^2 + 6*a*b^6*c*d^4*e^4*f^3 + 6*a^4*b^3*c*d^4*e*f^6 -
8*a*b^6*c^2*d^3*e^3*f^4 - 4*a^2*b^5*c*d^4*e^3*f^4 - 4*a^3*b^4*c*d^4*e^2*f^5 - 8*a^3*b^4*c^2*d^3*e*f^6 + 12*a^
2*b^5*c^2*d^3*e^2*f^5))/(b^4*e^3 - a^3*b*f^3 + 3*a^2*b^2*e*f^2 - 3*a*b^3*e^2*f) + ((-b^3*(a*d - b*c))^(1/2)*(c
+ d*x)^(1/2)*(a*d*f + 2*b*c*f - 3*b*d*e)*(4*a^5*b^3*d^3*f^7 + 4*b^8*d^3*e^5*f^2 + 8*a^2*b^6*d^3*e^3*f^4 + 8*a
^3*b^5*d^3*e^2*f^5 - 8*a^4*b^4*c*d^2*f^7 - 12*a*b^7*d^3*e^4*f^3 - 12*a^4*b^4*d^3*e*f^6 - 8*b^8*c*d^2*e^4*f^3 +
32*a*b^7*c*d^2*e^3*f^4 + 32*a^3*b^5*c*d^2*e*f^6 - 48*a^2*b^6*c*d^2*e^2*f^5))/((b^5*e^2 + a^2*b^3*f^2 - 2*a*b^
4*e*f)*(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f)))*(a*d*f + 2*b*c*f - 3*b*d*e))/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*
e*f)))*(-b^3*(a*d - b*c))^(1/2)*(a*d*f + 2*b*c*f - 3*b*d*e))/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)) - (((2*
(c + d*x)^(1/2)*(a^4*d^6*f^5 + 4*b^4*d^6*e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*b^2*c^2*d^4*f^5 + 9*a^2*b^2*d^6*e^2
*f^3 + 33*b^4*c^2*d^4*e^2*f^3 + 2*a^3*b*c*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a*b^3*c^3*d^3*f^5 - 16*b^4*c*d^5*e^3
*f^2 - 28*b^4*c^3*d^3*e*f^4 - 18*a*b^3*c*d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f^4))/(b^3*e^2 + a^2*b*f^2 - 2*a*b^2
*e*f) - ((-b^3*(a*d - b*c))^(1/2)*((2*(2*a^4*b^...

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