3.18.59 \(\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^3} \, dx\) [1759]

Optimal. Leaf size=233 \[ \frac {5 e (4 b B d+3 A b e-7 a B e) \sqrt {d+e x}}{4 b^4}+\frac {5 e (4 b B d+3 A b e-7 a B e) (d+e x)^{3/2}}{12 b^3 (b d-a e)}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}-\frac {5 e \sqrt {b d-a e} (4 b B d+3 A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2}} \]

[Out]

5/12*e*(3*A*b*e-7*B*a*e+4*B*b*d)*(e*x+d)^(3/2)/b^3/(-a*e+b*d)-1/4*(3*A*b*e-7*B*a*e+4*B*b*d)*(e*x+d)^(5/2)/b^2/
(-a*e+b*d)/(b*x+a)-1/2*(A*b-B*a)*(e*x+d)^(7/2)/b/(-a*e+b*d)/(b*x+a)^2-5/4*e*(3*A*b*e-7*B*a*e+4*B*b*d)*arctanh(
b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*(-a*e+b*d)^(1/2)/b^(9/2)+5/4*e*(3*A*b*e-7*B*a*e+4*B*b*d)*(e*x+d)^(1/2)
/b^4

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Rubi [A]
time = 0.12, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 52, 65, 214} \begin {gather*} -\frac {5 e \sqrt {b d-a e} (-7 a B e+3 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2}}+\frac {5 e \sqrt {d+e x} (-7 a B e+3 A b e+4 b B d)}{4 b^4}+\frac {5 e (d+e x)^{3/2} (-7 a B e+3 A b e+4 b B d)}{12 b^3 (b d-a e)}-\frac {(d+e x)^{5/2} (-7 a B e+3 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac {(d+e x)^{7/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^3,x]

[Out]

(5*e*(4*b*B*d + 3*A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(4*b^4) + (5*e*(4*b*B*d + 3*A*b*e - 7*a*B*e)*(d + e*x)^(3/2)
)/(12*b^3*(b*d - a*e)) - ((4*b*B*d + 3*A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(4*b^2*(b*d - a*e)*(a + b*x)) - ((A*b
- a*B)*(d + e*x)^(7/2))/(2*b*(b*d - a*e)*(a + b*x)^2) - (5*e*Sqrt[b*d - a*e]*(4*b*B*d + 3*A*b*e - 7*a*B*e)*Ar
cTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(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 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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

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) (d+e x)^{5/2}}{(a+b x)^3} \, dx &=-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(4 b B d+3 A b e-7 a B e) \int \frac {(d+e x)^{5/2}}{(a+b x)^2} \, dx}{4 b (b d-a e)}\\ &=-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(5 e (4 b B d+3 A b e-7 a B e)) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{8 b^2 (b d-a e)}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) (d+e x)^{3/2}}{12 b^3 (b d-a e)}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(5 e (4 b B d+3 A b e-7 a B e)) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{8 b^3}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) \sqrt {d+e x}}{4 b^4}+\frac {5 e (4 b B d+3 A b e-7 a B e) (d+e x)^{3/2}}{12 b^3 (b d-a e)}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(5 e (b d-a e) (4 b B d+3 A b e-7 a B e)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 b^4}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) \sqrt {d+e x}}{4 b^4}+\frac {5 e (4 b B d+3 A b e-7 a B e) (d+e x)^{3/2}}{12 b^3 (b d-a e)}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(5 (b d-a e) (4 b B d+3 A b e-7 a B e)) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 b^4}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) \sqrt {d+e x}}{4 b^4}+\frac {5 e (4 b B d+3 A b e-7 a B e) (d+e x)^{3/2}}{12 b^3 (b d-a e)}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}-\frac {5 e \sqrt {b d-a e} (4 b B d+3 A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.77, size = 213, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {d+e x} \left (3 A b \left (-15 a^2 e^2+5 a b e (d-5 e x)+b^2 \left (2 d^2+9 d e x-8 e^2 x^2\right )\right )+B \left (105 a^3 e^2+5 a^2 b e (-19 d+35 e x)-4 b^3 x \left (-3 d^2+14 d e x+2 e^2 x^2\right )+a b^2 \left (6 d^2-163 d e x+56 e^2 x^2\right )\right )\right )}{12 b^4 (a+b x)^2}-\frac {5 e \sqrt {-b d+a e} (4 b B d+3 A b e-7 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{4 b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^3,x]

[Out]

-1/12*(Sqrt[d + e*x]*(3*A*b*(-15*a^2*e^2 + 5*a*b*e*(d - 5*e*x) + b^2*(2*d^2 + 9*d*e*x - 8*e^2*x^2)) + B*(105*a
^3*e^2 + 5*a^2*b*e*(-19*d + 35*e*x) - 4*b^3*x*(-3*d^2 + 14*d*e*x + 2*e^2*x^2) + a*b^2*(6*d^2 - 163*d*e*x + 56*
e^2*x^2))))/(b^4*(a + b*x)^2) - (5*e*Sqrt[-(b*d) + a*e]*(4*b*B*d + 3*A*b*e - 7*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d +
e*x])/Sqrt[-(b*d) + a*e]])/(4*b^(9/2))

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Maple [A]
time = 0.13, size = 294, normalized size = 1.26

method	result	size
derivativedivides	\(2 e \left (\frac {\frac {B b \left (e x +d \right )^{\frac {3}{2}}}{3}+A b e \sqrt {e x +d}-3 a e B \sqrt {e x +d}+2 B b d \sqrt {e x +d}}{b^{4}}-\frac {\frac {\left (-\frac {9}{8} A a \,b^{2} e^{2}+\frac {9}{8} A \,b^{3} d e +\frac {13}{8} B \,a^{2} b \,e^{2}-\frac {17}{8} B a \,b^{2} d e +\frac {1}{2} b^{3} B \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {7}{8} A \,a^{2} b \,e^{3}+\frac {7}{4} A a \,b^{2} d \,e^{2}-\frac {7}{8} A \,b^{3} d^{2} e +\frac {11}{8} B \,a^{3} e^{3}-\frac {13}{4} B \,a^{2} b d \,e^{2}+\frac {19}{8} B a \,b^{2} d^{2} e -\frac {1}{2} b^{3} B \,d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {5 \left (3 A a b \,e^{2}-3 A \,b^{2} d e -7 B \,a^{2} e^{2}+11 B a b d e -4 b^{2} B \,d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}}{b^{4}}\right )\)	\(294\)
default	\(2 e \left (\frac {\frac {B b \left (e x +d \right )^{\frac {3}{2}}}{3}+A b e \sqrt {e x +d}-3 a e B \sqrt {e x +d}+2 B b d \sqrt {e x +d}}{b^{4}}-\frac {\frac {\left (-\frac {9}{8} A a \,b^{2} e^{2}+\frac {9}{8} A \,b^{3} d e +\frac {13}{8} B \,a^{2} b \,e^{2}-\frac {17}{8} B a \,b^{2} d e +\frac {1}{2} b^{3} B \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {7}{8} A \,a^{2} b \,e^{3}+\frac {7}{4} A a \,b^{2} d \,e^{2}-\frac {7}{8} A \,b^{3} d^{2} e +\frac {11}{8} B \,a^{3} e^{3}-\frac {13}{4} B \,a^{2} b d \,e^{2}+\frac {19}{8} B a \,b^{2} d^{2} e -\frac {1}{2} b^{3} B \,d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {5 \left (3 A a b \,e^{2}-3 A \,b^{2} d e -7 B \,a^{2} e^{2}+11 B a b d e -4 b^{2} B \,d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}}{b^{4}}\right )\)	\(294\)
risch	\(\frac {2 e \left (b B x e +3 A b e -9 B a e +7 B b d \right ) \sqrt {e x +d}}{3 b^{4}}+\frac {9 e^{3} \left (e x +d \right )^{\frac {3}{2}} A a}{4 b^{2} \left (b e x +a e \right )^{2}}-\frac {9 e^{2} \left (e x +d \right )^{\frac {3}{2}} A d}{4 b \left (b e x +a e \right )^{2}}-\frac {13 e^{3} \left (e x +d \right )^{\frac {3}{2}} B \,a^{2}}{4 b^{3} \left (b e x +a e \right )^{2}}+\frac {17 e^{2} \left (e x +d \right )^{\frac {3}{2}} B a d}{4 b^{2} \left (b e x +a e \right )^{2}}-\frac {e \left (e x +d \right )^{\frac {3}{2}} B \,d^{2}}{b \left (b e x +a e \right )^{2}}+\frac {7 e^{4} \sqrt {e x +d}\, A \,a^{2}}{4 b^{3} \left (b e x +a e \right )^{2}}-\frac {7 e^{3} \sqrt {e x +d}\, A a d}{2 b^{2} \left (b e x +a e \right )^{2}}+\frac {7 e^{2} \sqrt {e x +d}\, A \,d^{2}}{4 b \left (b e x +a e \right )^{2}}-\frac {11 e^{4} \sqrt {e x +d}\, B \,a^{3}}{4 b^{4} \left (b e x +a e \right )^{2}}+\frac {13 e^{3} \sqrt {e x +d}\, B \,a^{2} d}{2 b^{3} \left (b e x +a e \right )^{2}}-\frac {19 e^{2} \sqrt {e x +d}\, B a \,d^{2}}{4 b^{2} \left (b e x +a e \right )^{2}}+\frac {e \sqrt {e x +d}\, B \,d^{3}}{b \left (b e x +a e \right )^{2}}-\frac {15 e^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A a}{4 b^{3} \sqrt {\left (a e -b d \right ) b}}+\frac {15 e^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A d}{4 b^{2} \sqrt {\left (a e -b d \right ) b}}+\frac {35 e^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,a^{2}}{4 b^{4} \sqrt {\left (a e -b d \right ) b}}-\frac {55 e^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B a d}{4 b^{3} \sqrt {\left (a e -b d \right ) b}}+\frac {5 e \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,d^{2}}{b^{2} \sqrt {\left (a e -b d \right ) b}}\)	\(598\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

2*e*(1/b^4*(1/3*B*b*(e*x+d)^(3/2)+A*b*e*(e*x+d)^(1/2)-3*a*e*B*(e*x+d)^(1/2)+2*B*b*d*(e*x+d)^(1/2))-1/b^4*(((-9
/8*A*a*b^2*e^2+9/8*A*b^3*d*e+13/8*B*a^2*b*e^2-17/8*B*a*b^2*d*e+1/2*b^3*B*d^2)*(e*x+d)^(3/2)+(-7/8*A*a^2*b*e^3+
7/4*A*a*b^2*d*e^2-7/8*A*b^3*d^2*e+11/8*B*a^3*e^3-13/4*B*a^2*b*d*e^2+19/8*B*a*b^2*d^2*e-1/2*b^3*B*d^3)*(e*x+d)^
(1/2))/(b*(e*x+d)+a*e-b*d)^2+5/8*(3*A*a*b*e^2-3*A*b^2*d*e-7*B*a^2*e^2+11*B*a*b*d*e-4*B*b^2*d^2)/((a*e-b*d)*b)^
(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*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((B*x+A)*(e*x+d)^(5/2)/(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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 0.89, size = 651, normalized size = 2.79 \begin {gather*} \left [\frac {15 \, {\left ({\left (7 \, B a^{3} - 3 \, A a^{2} b + {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} e^{2} - 4 \, {\left (B b^{3} d x^{2} + 2 \, B a b^{2} d x + B a^{2} b d\right )} e\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d + 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (12 \, B b^{3} d^{2} x + 6 \, {\left (B a b^{2} + A b^{3}\right )} d^{2} - {\left (8 \, B b^{3} x^{3} - 105 \, B a^{3} + 45 \, A a^{2} b - 8 \, {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} - 25 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} e^{2} - {\left (56 \, B b^{3} d x^{2} + {\left (163 \, B a b^{2} - 27 \, A b^{3}\right )} d x + 5 \, {\left (19 \, B a^{2} b - 3 \, A a b^{2}\right )} d\right )} e\right )} \sqrt {x e + d}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {15 \, {\left ({\left (7 \, B a^{3} - 3 \, A a^{2} b + {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} e^{2} - 4 \, {\left (B b^{3} d x^{2} + 2 \, B a b^{2} d x + B a^{2} b d\right )} e\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (12 \, B b^{3} d^{2} x + 6 \, {\left (B a b^{2} + A b^{3}\right )} d^{2} - {\left (8 \, B b^{3} x^{3} - 105 \, B a^{3} + 45 \, A a^{2} b - 8 \, {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} - 25 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} e^{2} - {\left (56 \, B b^{3} d x^{2} + {\left (163 \, B a b^{2} - 27 \, A b^{3}\right )} d x + 5 \, {\left (19 \, B a^{2} b - 3 \, A a b^{2}\right )} d\right )} e\right )} \sqrt {x e + d}}{12 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^3,x, algorithm="fricas")

[Out]

[1/24*(15*((7*B*a^3 - 3*A*a^2*b + (7*B*a*b^2 - 3*A*b^3)*x^2 + 2*(7*B*a^2*b - 3*A*a*b^2)*x)*e^2 - 4*(B*b^3*d*x^
2 + 2*B*a*b^2*d*x + B*a^2*b*d)*e)*sqrt((b*d - a*e)/b)*log((2*b*d + 2*sqrt(x*e + d)*b*sqrt((b*d - a*e)/b) + (b*
x - a)*e)/(b*x + a)) - 2*(12*B*b^3*d^2*x + 6*(B*a*b^2 + A*b^3)*d^2 - (8*B*b^3*x^3 - 105*B*a^3 + 45*A*a^2*b - 8
*(7*B*a*b^2 - 3*A*b^3)*x^2 - 25*(7*B*a^2*b - 3*A*a*b^2)*x)*e^2 - (56*B*b^3*d*x^2 + (163*B*a*b^2 - 27*A*b^3)*d*
x + 5*(19*B*a^2*b - 3*A*a*b^2)*d)*e)*sqrt(x*e + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), 1/12*(15*((7*B*a^3 - 3*A*
a^2*b + (7*B*a*b^2 - 3*A*b^3)*x^2 + 2*(7*B*a^2*b - 3*A*a*b^2)*x)*e^2 - 4*(B*b^3*d*x^2 + 2*B*a*b^2*d*x + B*a^2*
b*d)*e)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (12*B*b^3*d^2*x + 6*(
B*a*b^2 + A*b^3)*d^2 - (8*B*b^3*x^3 - 105*B*a^3 + 45*A*a^2*b - 8*(7*B*a*b^2 - 3*A*b^3)*x^2 - 25*(7*B*a^2*b - 3
*A*a*b^2)*x)*e^2 - (56*B*b^3*d*x^2 + (163*B*a*b^2 - 27*A*b^3)*d*x + 5*(19*B*a^2*b - 3*A*a*b^2)*d)*e)*sqrt(x*e
+ d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)]

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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((B*x+A)*(e*x+d)**(5/2)/(b*x+a)**3,x)

[Out]

Timed out

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Giac [A]
time = 1.87, size = 400, normalized size = 1.72 \begin {gather*} \frac {5 \, {\left (4 \, B b^{2} d^{2} e - 11 \, B a b d e^{2} + 3 \, A b^{2} d e^{2} + 7 \, B a^{2} e^{3} - 3 \, A a b e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, \sqrt {-b^{2} d + a b e} b^{4}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d^{2} e - 4 \, \sqrt {x e + d} B b^{3} d^{3} e - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} d e^{2} + 9 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} d e^{2} + 19 \, \sqrt {x e + d} B a b^{2} d^{2} e^{2} - 7 \, \sqrt {x e + d} A b^{3} d^{2} e^{2} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b e^{3} - 9 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{2} e^{3} - 26 \, \sqrt {x e + d} B a^{2} b d e^{3} + 14 \, \sqrt {x e + d} A a b^{2} d e^{3} + 11 \, \sqrt {x e + d} B a^{3} e^{4} - 7 \, \sqrt {x e + d} A a^{2} b e^{4}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{4}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{6} e + 6 \, \sqrt {x e + d} B b^{6} d e - 9 \, \sqrt {x e + d} B a b^{5} e^{2} + 3 \, \sqrt {x e + d} A b^{6} e^{2}\right )}}{3 \, b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^3,x, algorithm="giac")

[Out]

5/4*(4*B*b^2*d^2*e - 11*B*a*b*d*e^2 + 3*A*b^2*d*e^2 + 7*B*a^2*e^3 - 3*A*a*b*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-
b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4) - 1/4*(4*(x*e + d)^(3/2)*B*b^3*d^2*e - 4*sqrt(x*e + d)*B*b^3*d^3*e
- 17*(x*e + d)^(3/2)*B*a*b^2*d*e^2 + 9*(x*e + d)^(3/2)*A*b^3*d*e^2 + 19*sqrt(x*e + d)*B*a*b^2*d^2*e^2 - 7*sqrt
(x*e + d)*A*b^3*d^2*e^2 + 13*(x*e + d)^(3/2)*B*a^2*b*e^3 - 9*(x*e + d)^(3/2)*A*a*b^2*e^3 - 26*sqrt(x*e + d)*B*
a^2*b*d*e^3 + 14*sqrt(x*e + d)*A*a*b^2*d*e^3 + 11*sqrt(x*e + d)*B*a^3*e^4 - 7*sqrt(x*e + d)*A*a^2*b*e^4)/(((x*
e + d)*b - b*d + a*e)^2*b^4) + 2/3*((x*e + d)^(3/2)*B*b^6*e + 6*sqrt(x*e + d)*B*b^6*d*e - 9*sqrt(x*e + d)*B*a*
b^5*e^2 + 3*sqrt(x*e + d)*A*b^6*e^2)/b^9

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Mupad [B]
time = 1.35, size = 325, normalized size = 1.39 \begin {gather*} \left (\frac {2\,A\,e^2-2\,B\,d\,e}{b^3}+\frac {2\,B\,e\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{b^6}\right )\,\sqrt {d+e\,x}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (\frac {13\,B\,a^2\,b\,e^3}{4}-\frac {17\,B\,a\,b^2\,d\,e^2}{4}-\frac {9\,A\,a\,b^2\,e^3}{4}+B\,b^3\,d^2\,e+\frac {9\,A\,b^3\,d\,e^2}{4}\right )-\sqrt {d+e\,x}\,\left (-\frac {11\,B\,a^3\,e^4}{4}+\frac {13\,B\,a^2\,b\,d\,e^3}{2}+\frac {7\,A\,a^2\,b\,e^4}{4}-\frac {19\,B\,a\,b^2\,d^2\,e^2}{4}-\frac {7\,A\,a\,b^2\,d\,e^3}{2}+B\,b^3\,d^3\,e+\frac {7\,A\,b^3\,d^2\,e^2}{4}\right )}{b^6\,{\left (d+e\,x\right )}^2-\left (2\,b^6\,d-2\,a\,b^5\,e\right )\,\left (d+e\,x\right )+b^6\,d^2+a^2\,b^4\,e^2-2\,a\,b^5\,d\,e}+\frac {2\,B\,e\,{\left (d+e\,x\right )}^{3/2}}{3\,b^3}+\frac {e\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,1{}\mathrm {i}}{\sqrt {b\,d-a\,e}}\right )\,\sqrt {b\,d-a\,e}\,\left (3\,A\,b\,e-7\,B\,a\,e+4\,B\,b\,d\right )\,5{}\mathrm {i}}{4\,b^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^3,x)

[Out]

((2*A*e^2 - 2*B*d*e)/b^3 + (2*B*e*(3*b^3*d - 3*a*b^2*e))/b^6)*(d + e*x)^(1/2) - ((d + e*x)^(3/2)*((13*B*a^2*b*
e^3)/4 - (9*A*a*b^2*e^3)/4 + (9*A*b^3*d*e^2)/4 + B*b^3*d^2*e - (17*B*a*b^2*d*e^2)/4) - (d + e*x)^(1/2)*((7*A*a
^2*b*e^4)/4 - (11*B*a^3*e^4)/4 + B*b^3*d^3*e + (7*A*b^3*d^2*e^2)/4 - (19*B*a*b^2*d^2*e^2)/4 - (7*A*a*b^2*d*e^3
)/2 + (13*B*a^2*b*d*e^3)/2))/(b^6*(d + e*x)^2 - (2*b^6*d - 2*a*b^5*e)*(d + e*x) + b^6*d^2 + a^2*b^4*e^2 - 2*a*
b^5*d*e) + (2*B*e*(d + e*x)^(3/2))/(3*b^3) + (e*atan((b^(1/2)*(d + e*x)^(1/2)*1i)/(b*d - a*e)^(1/2))*(b*d - a*
e)^(1/2)*(3*A*b*e - 7*B*a*e + 4*B*b*d)*5i)/(4*b^(9/2))

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