3.18.75 \(\int \frac {(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx\) [1775]

Optimal. Leaf size=172 \[ \frac {2 (b c-a d)^2 (d e-c f) \sqrt {e+f x}}{d^4}+\frac {2 (b c-a d)^2 (e+f x)^{3/2}}{3 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}-\frac {2 (b c-a d)^2 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}} \]

[Out]

2/3*(-a*d+b*c)^2*(f*x+e)^(3/2)/d^3-2/5*b*(-2*a*d*f+b*c*f+b*d*e)*(f*x+e)^(5/2)/d^2/f^2+2/7*b^2*(f*x+e)^(7/2)/d/
f^2-2*(-a*d+b*c)^2*(-c*f+d*e)^(3/2)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(9/2)+2*(-a*d+b*c)^2*(-c
*f+d*e)*(f*x+e)^(1/2)/d^4

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Rubi [A]
time = 0.09, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 52, 65, 214} \begin {gather*} -\frac {2 (b c-a d)^2 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}}+\frac {2 \sqrt {e+f x} (b c-a d)^2 (d e-c f)}{d^4}+\frac {2 (e+f x)^{3/2} (b c-a d)^2}{3 d^3}-\frac {2 b (e+f x)^{5/2} (-2 a d f+b c f+b d e)}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*c - a*d)^2*(d*e - c*f)*Sqrt[e + f*x])/d^4 + (2*(b*c - a*d)^2*(e + f*x)^(3/2))/(3*d^3) - (2*b*(b*d*e + b*
c*f - 2*a*d*f)*(e + f*x)^(5/2))/(5*d^2*f^2) + (2*b^2*(e + f*x)^(7/2))/(7*d*f^2) - (2*(b*c - a*d)^2*(d*e - c*f)
^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(9/2)

Rule 52

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

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 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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 {(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx &=\int \left (-\frac {b (b d e+b c f-2 a d f) (e+f x)^{3/2}}{d^2 f}+\frac {(-b c+a d)^2 (e+f x)^{3/2}}{d^2 (c+d x)}+\frac {b^2 (e+f x)^{5/2}}{d f}\right ) \, dx\\ &=-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}+\frac {(b c-a d)^2 \int \frac {(e+f x)^{3/2}}{c+d x} \, dx}{d^2}\\ &=\frac {2 (b c-a d)^2 (e+f x)^{3/2}}{3 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}+\frac {\left ((b c-a d)^2 (d e-c f)\right ) \int \frac {\sqrt {e+f x}}{c+d x} \, dx}{d^3}\\ &=\frac {2 (b c-a d)^2 (d e-c f) \sqrt {e+f x}}{d^4}+\frac {2 (b c-a d)^2 (e+f x)^{3/2}}{3 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}+\frac {\left ((b c-a d)^2 (d e-c f)^2\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d^4}\\ &=\frac {2 (b c-a d)^2 (d e-c f) \sqrt {e+f x}}{d^4}+\frac {2 (b c-a d)^2 (e+f x)^{3/2}}{3 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}+\frac {\left (2 (b c-a d)^2 (d e-c f)^2\right ) \text {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{d^4 f}\\ &=\frac {2 (b c-a d)^2 (d e-c f) \sqrt {e+f x}}{d^4}+\frac {2 (b c-a d)^2 (e+f x)^{3/2}}{3 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}-\frac {2 (b c-a d)^2 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 204, normalized size = 1.19 \begin {gather*} \frac {2 \sqrt {e+f x} \left (35 a^2 d^2 f^2 (4 d e-3 c f+d f x)+14 a b d f \left (15 c^2 f^2+3 d^2 (e+f x)^2-5 c d f (4 e+f x)\right )+b^2 \left (-105 c^3 f^3-21 c d^2 f (e+f x)^2-3 d^3 (2 e-5 f x) (e+f x)^2+35 c^2 d f^2 (4 e+f x)\right )\right )}{105 d^4 f^2}+\frac {2 (b c-a d)^2 (-d e+c f)^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e + f*x]*(35*a^2*d^2*f^2*(4*d*e - 3*c*f + d*f*x) + 14*a*b*d*f*(15*c^2*f^2 + 3*d^2*(e + f*x)^2 - 5*c*d*
f*(4*e + f*x)) + b^2*(-105*c^3*f^3 - 21*c*d^2*f*(e + f*x)^2 - 3*d^3*(2*e - 5*f*x)*(e + f*x)^2 + 35*c^2*d*f^2*(
4*e + f*x))))/(105*d^4*f^2) + (2*(b*c - a*d)^2*(-(d*e) + c*f)^(3/2)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e)
+ c*f]])/d^(9/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(332\) vs. \(2(148)=296\).
time = 0.09, size = 333, normalized size = 1.94

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-\frac {b^{2} \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {\left (-\left (a d f -b c f \right ) b \,d^{2}+b d \left (-a \,d^{2} f +b \,d^{2} e \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {\left (\left (a d f -b c f \right ) \left (-a \,d^{2} f +b \,d^{2} e \right )+b d \left (a c d \,f^{2}-a \,d^{2} e f -b \,c^{2} f^{2}+b c d e f \right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+\left (a d f -b c f \right ) \left (a c d \,f^{2}-a \,d^{2} e f -b \,c^{2} f^{2}+b c d e f \right ) \sqrt {f x +e}\right )}{d^{4}}+\frac {2 f^{2} \left (a^{2} c^{2} d^{2} f^{2}-2 a^{2} c \,d^{3} e f +a^{2} d^{4} e^{2}-2 a b \,c^{3} d \,f^{2}+4 a b \,c^{2} d^{2} e f -2 a b c \,d^{3} e^{2}+b^{2} c^{4} f^{2}-2 b^{2} c^{3} d e f +b^{2} c^{2} d^{2} e^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\)	\(333\)
default	\(\frac {-\frac {2 \left (-\frac {b^{2} \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {\left (-\left (a d f -b c f \right ) b \,d^{2}+b d \left (-a \,d^{2} f +b \,d^{2} e \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {\left (\left (a d f -b c f \right ) \left (-a \,d^{2} f +b \,d^{2} e \right )+b d \left (a c d \,f^{2}-a \,d^{2} e f -b \,c^{2} f^{2}+b c d e f \right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+\left (a d f -b c f \right ) \left (a c d \,f^{2}-a \,d^{2} e f -b \,c^{2} f^{2}+b c d e f \right ) \sqrt {f x +e}\right )}{d^{4}}+\frac {2 f^{2} \left (a^{2} c^{2} d^{2} f^{2}-2 a^{2} c \,d^{3} e f +a^{2} d^{4} e^{2}-2 a b \,c^{3} d \,f^{2}+4 a b \,c^{2} d^{2} e f -2 a b c \,d^{3} e^{2}+b^{2} c^{4} f^{2}-2 b^{2} c^{3} d e f +b^{2} c^{2} d^{2} e^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\)	\(333\)
risch	\(-\frac {2 \left (-15 b^{2} f^{3} d^{3} x^{3}-42 a b \,d^{3} f^{3} x^{2}+21 b^{2} c \,d^{2} f^{3} x^{2}-24 b^{2} d^{3} e \,f^{2} x^{2}-35 a^{2} d^{3} f^{3} x +70 a b c \,d^{2} f^{3} x -84 a b \,d^{3} e \,f^{2} x -35 b^{2} c^{2} d \,f^{3} x +42 b^{2} c \,d^{2} e \,f^{2} x -3 b^{2} d^{3} e^{2} f x +105 a^{2} c \,d^{2} f^{3}-140 a^{2} d^{3} e \,f^{2}-210 a b \,c^{2} d \,f^{3}+280 a b c \,d^{2} e \,f^{2}-42 a b \,d^{3} e^{2} f +105 b^{2} c^{3} f^{3}-140 b^{2} c^{2} d e \,f^{2}+21 b^{2} c \,d^{2} e^{2} f +6 b^{2} d^{3} e^{3}\right ) \sqrt {f x +e}}{105 f^{2} d^{4}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} c^{2} f^{2}}{d^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {4 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} c e f}{d \sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} e^{2}}{\sqrt {\left (c f -d e \right ) d}}-\frac {4 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a b \,c^{3} f^{2}}{d^{3} \sqrt {\left (c f -d e \right ) d}}+\frac {8 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a b \,c^{2} e f}{d^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {4 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a b c \,e^{2}}{d \sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{2} c^{4} f^{2}}{d^{4} \sqrt {\left (c f -d e \right ) d}}-\frac {4 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{2} c^{3} e f}{d^{3} \sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{2} c^{2} e^{2}}{d^{2} \sqrt {\left (c f -d e \right ) d}}\)	\(673\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f^2*(-1/d^4*(-1/7*b^2*(f*x+e)^(7/2)*d^3+1/5*(-(a*d*f-b*c*f)*b*d^2+b*d*(-a*d^2*f+b*d^2*e))*(f*x+e)^(5/2)+1/3*
((a*d*f-b*c*f)*(-a*d^2*f+b*d^2*e)+b*d*(a*c*d*f^2-a*d^2*e*f-b*c^2*f^2+b*c*d*e*f))*(f*x+e)^(3/2)+(a*d*f-b*c*f)*(
a*c*d*f^2-a*d^2*e*f-b*c^2*f^2+b*c*d*e*f)*(f*x+e)^(1/2))+f^2*(a^2*c^2*d^2*f^2-2*a^2*c*d^3*e*f+a^2*d^4*e^2-2*a*b
*c^3*d*f^2+4*a*b*c^2*d^2*e*f-2*a*b*c*d^3*e^2+b^2*c^4*f^2-2*b^2*c^3*d*e*f+b^2*c^2*d^2*e^2)/d^4/((c*f-d*e)*d)^(1
/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(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((b*x+a)^2*(f*x+e)^(3/2)/(d*x+c),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 [B] Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (157) = 314\).
time = 1.44, size = 704, normalized size = 4.09 \begin {gather*} \left [-\frac {105 \, {\left ({\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3} - {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{2} e\right )} \sqrt {-\frac {c f - d e}{d}} \log \left (\frac {d f x - c f - 2 \, \sqrt {f x + e} d \sqrt {-\frac {c f - d e}{d}} + 2 \, d e}{d x + c}\right ) - 2 \, {\left (15 \, b^{2} d^{3} f^{3} x^{3} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f^{3} x^{2} - 6 \, b^{2} d^{3} e^{3} + 35 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3} x - 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3} + 3 \, {\left (b^{2} d^{3} f x - 7 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f\right )} e^{2} + 2 \, {\left (12 \, b^{2} d^{3} f^{2} x^{2} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f^{2} x + 70 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{2}\right )} e\right )} \sqrt {f x + e}}{105 \, d^{4} f^{2}}, -\frac {2 \, {\left (105 \, {\left ({\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3} - {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{2} e\right )} \sqrt {\frac {c f - d e}{d}} \arctan \left (-\frac {\sqrt {f x + e} d \sqrt {\frac {c f - d e}{d}}}{c f - d e}\right ) - {\left (15 \, b^{2} d^{3} f^{3} x^{3} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f^{3} x^{2} - 6 \, b^{2} d^{3} e^{3} + 35 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3} x - 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3} + 3 \, {\left (b^{2} d^{3} f x - 7 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f\right )} e^{2} + 2 \, {\left (12 \, b^{2} d^{3} f^{2} x^{2} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f^{2} x + 70 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{2}\right )} e\right )} \sqrt {f x + e}\right )}}{105 \, d^{4} f^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/105*(105*((b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*f^3 - (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^2*e)*sqrt(-(c*f
- d*e)/d)*log((d*f*x - c*f - 2*sqrt(f*x + e)*d*sqrt(-(c*f - d*e)/d) + 2*d*e)/(d*x + c)) - 2*(15*b^2*d^3*f^3*x^
3 - 21*(b^2*c*d^2 - 2*a*b*d^3)*f^3*x^2 - 6*b^2*d^3*e^3 + 35*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^3*x - 105*(b
^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*f^3 + 3*(b^2*d^3*f*x - 7*(b^2*c*d^2 - 2*a*b*d^3)*f)*e^2 + 2*(12*b^2*d^3*f^2*
x^2 - 21*(b^2*c*d^2 - 2*a*b*d^3)*f^2*x + 70*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^2)*e)*sqrt(f*x + e))/(d^4*f^
2), -2/105*(105*((b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*f^3 - (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^2*e)*sqrt((c*
f - d*e)/d)*arctan(-sqrt(f*x + e)*d*sqrt((c*f - d*e)/d)/(c*f - d*e)) - (15*b^2*d^3*f^3*x^3 - 21*(b^2*c*d^2 - 2
*a*b*d^3)*f^3*x^2 - 6*b^2*d^3*e^3 + 35*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^3*x - 105*(b^2*c^3 - 2*a*b*c^2*d
+ a^2*c*d^2)*f^3 + 3*(b^2*d^3*f*x - 7*(b^2*c*d^2 - 2*a*b*d^3)*f)*e^2 + 2*(12*b^2*d^3*f^2*x^2 - 21*(b^2*c*d^2 -
2*a*b*d^3)*f^2*x + 70*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^2)*e)*sqrt(f*x + e))/(d^4*f^2)]

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Sympy [A]
time = 53.36, size = 236, normalized size = 1.37 \begin {gather*} \frac {2 b^{2} \left (e + f x\right )^{\frac {7}{2}}}{7 d f^{2}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \cdot \left (4 a b d f - 2 b^{2} c f - 2 b^{2} d e\right )}{5 d^{2} f^{2}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \cdot \left (2 a^{2} d^{2} - 4 a b c d + 2 b^{2} c^{2}\right )}{3 d^{3}} + \frac {\sqrt {e + f x} \left (- 2 a^{2} c d^{2} f + 2 a^{2} d^{3} e + 4 a b c^{2} d f - 4 a b c d^{2} e - 2 b^{2} c^{3} f + 2 b^{2} c^{2} d e\right )}{d^{4}} + \frac {2 \left (a d - b c\right )^{2} \left (c f - d e\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{5} \sqrt {\frac {c f - d e}{d}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**2*(e + f*x)**(7/2)/(7*d*f**2) + (e + f*x)**(5/2)*(4*a*b*d*f - 2*b**2*c*f - 2*b**2*d*e)/(5*d**2*f**2) + (e
+ f*x)**(3/2)*(2*a**2*d**2 - 4*a*b*c*d + 2*b**2*c**2)/(3*d**3) + sqrt(e + f*x)*(-2*a**2*c*d**2*f + 2*a**2*d**
3*e + 4*a*b*c**2*d*f - 4*a*b*c*d**2*e - 2*b**2*c**3*f + 2*b**2*c**2*d*e)/d**4 + 2*(a*d - b*c)**2*(c*f - d*e)**
2*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d**5*sqrt((c*f - d*e)/d))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (157) = 314\).
time = 0.69, size = 424, normalized size = 2.47 \begin {gather*} \frac {2 \, {\left (b^{2} c^{4} f^{2} - 2 \, a b c^{3} d f^{2} + a^{2} c^{2} d^{2} f^{2} - 2 \, b^{2} c^{3} d f e + 4 \, a b c^{2} d^{2} f e - 2 \, a^{2} c d^{3} f e + b^{2} c^{2} d^{2} e^{2} - 2 \, a b c d^{3} e^{2} + a^{2} d^{4} e^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{\sqrt {c d f - d^{2} e} d^{4}} + \frac {2 \, {\left (15 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{2} d^{6} f^{12} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{2} c d^{5} f^{13} + 42 \, {\left (f x + e\right )}^{\frac {5}{2}} a b d^{6} f^{13} + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{2} c^{2} d^{4} f^{14} - 70 \, {\left (f x + e\right )}^{\frac {3}{2}} a b c d^{5} f^{14} + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} d^{6} f^{14} - 105 \, \sqrt {f x + e} b^{2} c^{3} d^{3} f^{15} + 210 \, \sqrt {f x + e} a b c^{2} d^{4} f^{15} - 105 \, \sqrt {f x + e} a^{2} c d^{5} f^{15} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{2} d^{6} f^{12} e + 105 \, \sqrt {f x + e} b^{2} c^{2} d^{4} f^{14} e - 210 \, \sqrt {f x + e} a b c d^{5} f^{14} e + 105 \, \sqrt {f x + e} a^{2} d^{6} f^{14} e\right )}}{105 \, d^{7} f^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b^2*c^4*f^2 - 2*a*b*c^3*d*f^2 + a^2*c^2*d^2*f^2 - 2*b^2*c^3*d*f*e + 4*a*b*c^2*d^2*f*e - 2*a^2*c*d^3*f*e + b
^2*c^2*d^2*e^2 - 2*a*b*c*d^3*e^2 + a^2*d^4*e^2)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*
e)*d^4) + 2/105*(15*(f*x + e)^(7/2)*b^2*d^6*f^12 - 21*(f*x + e)^(5/2)*b^2*c*d^5*f^13 + 42*(f*x + e)^(5/2)*a*b*
d^6*f^13 + 35*(f*x + e)^(3/2)*b^2*c^2*d^4*f^14 - 70*(f*x + e)^(3/2)*a*b*c*d^5*f^14 + 35*(f*x + e)^(3/2)*a^2*d^
6*f^14 - 105*sqrt(f*x + e)*b^2*c^3*d^3*f^15 + 210*sqrt(f*x + e)*a*b*c^2*d^4*f^15 - 105*sqrt(f*x + e)*a^2*c*d^5
*f^15 - 21*(f*x + e)^(5/2)*b^2*d^6*f^12*e + 105*sqrt(f*x + e)*b^2*c^2*d^4*f^14*e - 210*sqrt(f*x + e)*a*b*c*d^5
*f^14*e + 105*sqrt(f*x + e)*a^2*d^6*f^14*e)/(d^7*f^14)

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Mupad [B]
time = 1.26, size = 438, normalized size = 2.55 \begin {gather*} {\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{3\,d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{3\,d\,f^2}\right )-{\left (e+f\,x\right )}^{5/2}\,\left (\frac {4\,b^2\,e-4\,a\,b\,f}{5\,d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{5\,d^2\,f^4}\right )+\frac {2\,b^2\,{\left (e+f\,x\right )}^{7/2}}{7\,d\,f^2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^2\,{\left (c\,f-d\,e\right )}^{3/2}}{a^2\,c^2\,d^2\,f^2-2\,a^2\,c\,d^3\,e\,f+a^2\,d^4\,e^2-2\,a\,b\,c^3\,d\,f^2+4\,a\,b\,c^2\,d^2\,e\,f-2\,a\,b\,c\,d^3\,e^2+b^2\,c^4\,f^2-2\,b^2\,c^3\,d\,e\,f+b^2\,c^2\,d^2\,e^2}\right )\,{\left (a\,d-b\,c\right )}^2\,{\left (c\,f-d\,e\right )}^{3/2}}{d^{9/2}}-\frac {\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e + f*x)^(3/2)*((2*(a*f - b*e)^2)/(3*d*f^2) + (((4*b^2*e - 4*a*b*f)/(d*f^2) + (2*b^2*(c*f^3 - d*e*f^2))/(d^2*
f^4))*(c*f^3 - d*e*f^2))/(3*d*f^2)) - (e + f*x)^(5/2)*((4*b^2*e - 4*a*b*f)/(5*d*f^2) + (2*b^2*(c*f^3 - d*e*f^2
))/(5*d^2*f^4)) + (2*b^2*(e + f*x)^(7/2))/(7*d*f^2) + (2*atan((d^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)^2*(c*f - d*
e)^(3/2))/(a^2*d^4*e^2 + b^2*c^4*f^2 + a^2*c^2*d^2*f^2 + b^2*c^2*d^2*e^2 - 2*a*b*c*d^3*e^2 - 2*a*b*c^3*d*f^2 -
2*a^2*c*d^3*e*f - 2*b^2*c^3*d*e*f + 4*a*b*c^2*d^2*e*f))*(a*d - b*c)^2*(c*f - d*e)^(3/2))/d^(9/2) - ((e + f*x)
^(1/2)*((2*(a*f - b*e)^2)/(d*f^2) + (((4*b^2*e - 4*a*b*f)/(d*f^2) + (2*b^2*(c*f^3 - d*e*f^2))/(d^2*f^4))*(c*f^
3 - d*e*f^2))/(d*f^2))*(c*f^3 - d*e*f^2))/(d*f^2)

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