3.7.1 \(\int \frac {x^2 (a+b x)^{3/2}}{\sqrt {c+d x}} \, dx\)

Optimal. Leaf size=254 \[ -\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{64 b^2 d^4}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{96 b^2 d^3}+\frac {(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{9/2}}-\frac {(a+b x)^{5/2} \sqrt {c+d x} (3 a d+7 b c)}{24 b^2 d^2}+\frac {x (a+b x)^{5/2} \sqrt {c+d x}}{4 b d} \]

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {90, 80, 50, 63, 217, 206} \begin {gather*} \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{96 b^2 d^3}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{64 b^2 d^4}+\frac {(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{9/2}}-\frac {(a+b x)^{5/2} \sqrt {c+d x} (3 a d+7 b c)}{24 b^2 d^2}+\frac {x (a+b x)^{5/2} \sqrt {c+d x}}{4 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c - a*d)*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^2*d^4) + ((35*b^2*c^2 +
10*a*b*c*d + 3*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(96*b^2*d^3) - ((7*b*c + 3*a*d)*(a + b*x)^(5/2)*Sqrt[c
+ d*x])/(24*b^2*d^2) + (x*(a + b*x)^(5/2)*Sqrt[c + d*x])/(4*b*d) + ((b*c - a*d)^2*(35*b^2*c^2 + 10*a*b*c*d +
3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(5/2)*d^(9/2))

Rule 50

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 63

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 80

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

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {x^2 (a+b x)^{3/2}}{\sqrt {c+d x}} \, dx &=\frac {x (a+b x)^{5/2} \sqrt {c+d x}}{4 b d}+\frac {\int \frac {(a+b x)^{3/2} \left (-a c-\frac {1}{2} (7 b c+3 a d) x\right )}{\sqrt {c+d x}} \, dx}{4 b d}\\ &=-\frac {(7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b^2 d^2}+\frac {x (a+b x)^{5/2} \sqrt {c+d x}}{4 b d}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{48 b^2 d^2}\\ &=\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b^2 d^3}-\frac {(7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b^2 d^2}+\frac {x (a+b x)^{5/2} \sqrt {c+d x}}{4 b d}-\frac {\left ((b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{64 b^2 d^3}\\ &=-\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^4}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b^2 d^3}-\frac {(7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b^2 d^2}+\frac {x (a+b x)^{5/2} \sqrt {c+d x}}{4 b d}+\frac {\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^2 d^4}\\ &=-\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^4}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b^2 d^3}-\frac {(7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b^2 d^2}+\frac {x (a+b x)^{5/2} \sqrt {c+d x}}{4 b d}+\frac {\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^3 d^4}\\ &=-\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^4}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b^2 d^3}-\frac {(7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b^2 d^2}+\frac {x (a+b x)^{5/2} \sqrt {c+d x}}{4 b d}+\frac {\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 b^3 d^4}\\ &=-\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^4}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b^2 d^3}-\frac {(7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b^2 d^2}+\frac {x (a+b x)^{5/2} \sqrt {c+d x}}{4 b d}+\frac {(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.64, size = 215, normalized size = 0.85 \begin {gather*} \frac {3 (b c-a d)^{5/2} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )-b \sqrt {d} \sqrt {a+b x} (c+d x) \left (9 a^3 d^3+3 a^2 b d^2 (5 c-2 d x)+a b^2 d \left (-145 c^2+92 c d x-72 d^2 x^2\right )+b^3 \left (105 c^3-70 c^2 d x+56 c d^2 x^2-48 d^3 x^3\right )\right )}{192 b^3 d^{9/2} \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(b*Sqrt[d]*Sqrt[a + b*x]*(c + d*x)*(9*a^3*d^3 + 3*a^2*b*d^2*(5*c - 2*d*x) + a*b^2*d*(-145*c^2 + 92*c*d*x - 7
2*d^2*x^2) + b^3*(105*c^3 - 70*c^2*d*x + 56*c*d^2*x^2 - 48*d^3*x^3))) + 3*(b*c - a*d)^(5/2)*(35*b^2*c^2 + 10*a
*b*c*d + 3*a^2*d^2)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(192*b^3
*d^(9/2)*Sqrt[c + d*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.58, size = 371, normalized size = 1.46 \begin {gather*} \frac {(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{5/2} d^{9/2}}-\frac {\sqrt {c+d x} (b c-a d)^2 \left (\frac {9 a^2 b^3 d^2 (c+d x)^3}{(a+b x)^3}-\frac {33 a^2 b^2 d^3 (c+d x)^2}{(a+b x)^2}-\frac {33 a^2 b d^4 (c+d x)}{a+b x}+9 a^2 d^5+\frac {105 b^5 c^2 (c+d x)^3}{(a+b x)^3}-\frac {385 b^4 c^2 d (c+d x)^2}{(a+b x)^2}+\frac {30 a b^4 c d (c+d x)^3}{(a+b x)^3}+\frac {511 b^3 c^2 d^2 (c+d x)}{a+b x}-\frac {110 a b^3 c d^2 (c+d x)^2}{(a+b x)^2}+\frac {146 a b^2 c d^3 (c+d x)}{a+b x}+30 a b c d^4-279 b^2 c^2 d^3\right )}{192 b^2 d^4 \sqrt {a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*(a + b*x)^(3/2))/Sqrt[c + d*x],x]

[Out]

-1/192*((b*c - a*d)^2*Sqrt[c + d*x]*(-279*b^2*c^2*d^3 + 30*a*b*c*d^4 + 9*a^2*d^5 + (511*b^3*c^2*d^2*(c + d*x))
/(a + b*x) + (146*a*b^2*c*d^3*(c + d*x))/(a + b*x) - (33*a^2*b*d^4*(c + d*x))/(a + b*x) - (385*b^4*c^2*d*(c +
d*x)^2)/(a + b*x)^2 - (110*a*b^3*c*d^2*(c + d*x)^2)/(a + b*x)^2 - (33*a^2*b^2*d^3*(c + d*x)^2)/(a + b*x)^2 + (
105*b^5*c^2*(c + d*x)^3)/(a + b*x)^3 + (30*a*b^4*c*d*(c + d*x)^3)/(a + b*x)^3 + (9*a^2*b^3*d^2*(c + d*x)^3)/(a
+ b*x)^3))/(b^2*d^4*Sqrt[a + b*x]*(-d + (b*(c + d*x))/(a + b*x))^4) + ((b*c - a*d)^2*(35*b^2*c^2 + 10*a*b*c*d
+ 3*a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(64*b^(5/2)*d^(9/2))

________________________________________________________________________________________

fricas [A]  time = 1.66, size = 546, normalized size = 2.15 \begin {gather*} \left [\frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 145 \, a b^{3} c^{2} d^{2} - 15 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{3} d^{5}}, -\frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 145 \, a b^{3} c^{2} d^{2} - 15 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{3} d^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(3*(35*b^4*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^4)*sqrt(b*d)*log(8*b^2*d
^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^
2*c*d + a*b*d^2)*x) + 4*(48*b^4*d^4*x^3 - 105*b^4*c^3*d + 145*a*b^3*c^2*d^2 - 15*a^2*b^2*c*d^3 - 9*a^3*b*d^4 -
8*(7*b^4*c*d^3 - 9*a*b^3*d^4)*x^2 + 2*(35*b^4*c^2*d^2 - 46*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt
(d*x + c))/(b^3*d^5), -1/384*(3*(35*b^4*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^4)
*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (
b^2*c*d + a*b*d^2)*x)) - 2*(48*b^4*d^4*x^3 - 105*b^4*c^3*d + 145*a*b^3*c^2*d^2 - 15*a^2*b^2*c*d^3 - 9*a^3*b*d^
4 - 8*(7*b^4*c*d^3 - 9*a*b^3*d^4)*x^2 + 2*(35*b^4*c^2*d^2 - 46*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*x)*sqrt(b*x + a)*s
qrt(d*x + c))/(b^3*d^5)]

________________________________________________________________________________________

giac [A]  time = 1.47, size = 291, normalized size = 1.15 \begin {gather*} \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3} d} - \frac {7 \, b^{7} c d^{5} + 9 \, a b^{6} d^{6}}{b^{9} d^{7}}\right )} + \frac {35 \, b^{8} c^{2} d^{4} + 10 \, a b^{7} c d^{5} + 3 \, a^{2} b^{6} d^{6}}{b^{9} d^{7}}\right )} - \frac {3 \, {\left (35 \, b^{9} c^{3} d^{3} - 25 \, a b^{8} c^{2} d^{4} - 7 \, a^{2} b^{7} c d^{5} - 3 \, a^{3} b^{6} d^{6}\right )}}{b^{9} d^{7}}\right )} \sqrt {b x + a} - \frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{4}}\right )} b}{192 \, {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/(b^3*d) - (7*b^7*c*d^5 + 9*a
*b^6*d^6)/(b^9*d^7)) + (35*b^8*c^2*d^4 + 10*a*b^7*c*d^5 + 3*a^2*b^6*d^6)/(b^9*d^7)) - 3*(35*b^9*c^3*d^3 - 25*a
*b^8*c^2*d^4 - 7*a^2*b^7*c*d^5 - 3*a^3*b^6*d^6)/(b^9*d^7))*sqrt(b*x + a) - 3*(35*b^4*c^4 - 60*a*b^3*c^3*d + 18
*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d -
a*b*d)))/(sqrt(b*d)*b^2*d^4))*b/abs(b)

________________________________________________________________________________________

maple [B]  time = 0.03, size = 574, normalized size = 2.26 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (9 a^{4} d^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+12 a^{3} b c \,d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+54 a^{2} b^{2} c^{2} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-180 a \,b^{3} c^{3} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+105 b^{4} c^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} d^{3} x^{3}+144 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} d^{3} x^{2}-112 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c \,d^{2} x^{2}+12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b \,d^{3} x -184 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c \,d^{2} x +140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{2} d x -18 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} d^{3}-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b c \,d^{2}+290 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c^{2} d -210 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3}\right )}{384 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(96*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^3*d^3*x^3+144*((b*x+a)*(d*x+c))^(1
/2)*(b*d)^(1/2)*a*b^2*d^3*x^2-112*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^3*c*d^2*x^2+9*a^4*d^4*ln(1/2*(2*b*d*x+
a*d+b*c+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+12*a^3*b*c*d^3*ln(1/2*(2*b*d*x+a*d+b*c+2*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+54*a^2*b^2*c^2*d^2*ln(1/2*(2*b*d*x+a*d+b*c+2*((b*x+a)*(d*x+c))^(1/2)*
(b*d)^(1/2))/(b*d)^(1/2))-180*a*b^3*c^3*d*ln(1/2*(2*b*d*x+a*d+b*c+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(b*d)
^(1/2))+105*b^4*c^4*ln(1/2*(2*b*d*x+a*d+b*c+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+12*((b*x+a)*(d
*x+c))^(1/2)*(b*d)^(1/2)*a^2*b*d^3*x-184*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^2*c*d^2*x+140*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)*b^3*c^2*d*x-18*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*d^3-30*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2)*a^2*b*c*d^2+290*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^2*c^2*d-210*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)*b^3*c^3)/b^2/d^4/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)

________________________________________________________________________________________

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(x^2*(b*x+a)^(3/2)/(d*x+c)^(1/2),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 details)Is a*d-b*c zero or nonzero?

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\left (a+b\,x\right )}^{3/2}}{\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  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(x**2*(b*x+a)**(3/2)/(d*x+c)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________