3.633 \(\int \frac {(a+b x)^{3/2}}{x^5 \sqrt {c+d x}} \, dx\)

Optimal. Leaf size=266 \[ -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (35 a^2 d^2-46 a b c d+3 b^2 c^2\right )}{96 a c^3 x^2}-\frac {\left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{9/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 a^3 d^3-145 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{192 a^2 c^4 x}-\frac {\sqrt {a+b x} \sqrt {c+d x} (9 b c-7 a d)}{24 c^2 x^3}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4} \]

[Out]

-1/64*(-a*d+b*c)^2*(35*a^2*d^2+10*a*b*c*d+3*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(5
/2)/c^(9/2)-1/4*a*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c/x^4-1/24*(-7*a*d+9*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c^2/x^3-1/
96*(35*a^2*d^2-46*a*b*c*d+3*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^3/x^2+1/192*(105*a^3*d^3-145*a^2*b*c*d^2+
15*a*b^2*c^2*d+9*b^3*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^4/x

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Rubi [A]  time = 0.21, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {98, 151, 12, 93, 208} \[ -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (35 a^2 d^2-46 a b c d+3 b^2 c^2\right )}{96 a c^3 x^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-145 a^2 b c d^2+105 a^3 d^3+15 a b^2 c^2 d+9 b^3 c^3\right )}{192 a^2 c^4 x}-\frac {\left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (9 b c-7 a d)}{24 c^2 x^3}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(3/2)/(x^5*Sqrt[c + d*x]),x]

[Out]

-(a*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*c*x^4) - ((9*b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*c^2*x^3) - ((3*
b^2*c^2 - 46*a*b*c*d + 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(96*a*c^3*x^2) + ((9*b^3*c^3 + 15*a*b^2*c^2*d
- 145*a^2*b*c*d^2 + 105*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(192*a^2*c^4*x) - ((b*c - a*d)^2*(3*b^2*c^2 + 10
*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(5/2)*c^(9/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 98

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 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 208

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

Rubi steps

\begin {align*} \int \frac {(a+b x)^{3/2}}{x^5 \sqrt {c+d x}} \, dx &=-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {\int \frac {-\frac {1}{2} a (9 b c-7 a d)-b (4 b c-3 a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{4 c}\\ &=-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(9 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 c^2 x^3}+\frac {\int \frac {\frac {1}{4} a \left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right )-a b d (9 b c-7 a d) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{12 a c^2}\\ &=-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(9 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 c^2 x^3}-\frac {\left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a c^3 x^2}-\frac {\int \frac {\frac {1}{8} a \left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right )+\frac {1}{4} a b d \left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a^2 c^3}\\ &=-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(9 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 c^2 x^3}-\frac {\left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a c^3 x^2}+\frac {\left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^2 c^4 x}+\frac {\int \frac {3 a (b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a^3 c^4}\\ &=-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(9 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 c^2 x^3}-\frac {\left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a c^3 x^2}+\frac {\left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^2 c^4 x}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a^2 c^4}\\ &=-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(9 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 c^2 x^3}-\frac {\left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a c^3 x^2}+\frac {\left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^2 c^4 x}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 a^2 c^4}\\ &=-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(9 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 c^2 x^3}-\frac {\left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a c^3 x^2}+\frac {\left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^2 c^4 x}-\frac {(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 193, normalized size = 0.73 \[ -\frac {\frac {x^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c-3 a d x+5 b c x)\right )}{\sqrt {a} c^{5/2}}+48 a c (a+b x)^{5/2} \sqrt {c+d x}-8 x (a+b x)^{5/2} \sqrt {c+d x} (7 a d+3 b c)}{192 a^2 c^2 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(3/2)/(x^5*Sqrt[c + d*x]),x]

[Out]

-1/192*(48*a*c*(a + b*x)^(5/2)*Sqrt[c + d*x] - 8*(3*b*c + 7*a*d)*x*(a + b*x)^(5/2)*Sqrt[c + d*x] + ((3*b^2*c^2
+ 10*a*b*c*d + 35*a^2*d^2)*x^2*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c + 5*b*c*x - 3*a*d*x) + 3*(
b*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(Sqrt[a]*c^(5/2)))/(a^2*c^2*x^4)

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fricas [A]  time = 6.52, size = 574, normalized size = 2.16 \[ \left [\frac {3 \, {\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (48 \, a^{4} c^{4} - {\left (9 \, a b^{3} c^{4} + 15 \, a^{2} b^{2} c^{3} d - 145 \, a^{3} b c^{2} d^{2} + 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (9 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{3} c^{5} x^{4}}, \frac {3 \, {\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (48 \, a^{4} c^{4} - {\left (9 \, a b^{3} c^{4} + 15 \, a^{2} b^{2} c^{3} d - 145 \, a^{3} b c^{2} d^{2} + 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (9 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{3} c^{5} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)/x^5/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 11.94, size = 3690, normalized size = 13.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)/x^5/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

-1/192*b*(3*(3*sqrt(b*d)*b^5*c^4 + 4*sqrt(b*d)*a*b^4*c^3*d + 18*sqrt(b*d)*a^2*b^3*c^2*d^2 - 60*sqrt(b*d)*a^3*b
^2*c*d^3 + 35*sqrt(b*d)*a^4*b*d^4)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^2*b*c^4) - 2*(9*sqrt(b*d)*b^19*c^11 - 57*sqrt(b*d)*
a*b^18*c^10*d - 13*sqrt(b*d)*a^2*b^17*c^9*d^2 + 1181*sqrt(b*d)*a^3*b^16*c^8*d^3 - 5110*sqrt(b*d)*a^4*b^15*c^7*
d^4 + 11606*sqrt(b*d)*a^5*b^14*c^6*d^5 - 16618*sqrt(b*d)*a^6*b^13*c^5*d^6 + 15818*sqrt(b*d)*a^7*b^12*c^4*d^7 -
10051*sqrt(b*d)*a^8*b^11*c^3*d^8 + 4115*sqrt(b*d)*a^9*b^10*c^2*d^9 - 985*sqrt(b*d)*a^10*b^9*c*d^10 + 105*sqrt
(b*d)*a^11*b^8*d^11 - 63*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^17*c^10
+ 198*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^16*c^9*d + 1229*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^15*c^8*d^2 - 7864*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^14*c^7*d^3 + 17490*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^13*c^6*d^4 - 17692*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^12*c^5*d^5 + 3602*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^11*c^4*d^6 + 10056*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^2*a^7*b^10*c^3*d^7 - 10771*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^2*a^8*b^9*c^2*d^8 + 4550*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
2*a^9*b^8*c*d^9 - 735*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^10*b^7*d^1
0 + 189*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^15*c^9 - 123*sqrt(b*d)*(
sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^14*c^8*d - 5068*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^13*c^7*d^2 + 16164*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^12*c^6*d^3 - 17466*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*b^11*c^5*d^4 + 5510*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^4*a^5*b^10*c^4*d^5 - 2268*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^4*a^6*b^9*c^3*d^6 + 9012*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
4*a^7*b^8*c^2*d^7 - 8155*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^8*b^7*c
*d^8 + 2205*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^9*b^6*d^9 - 315*sqrt
(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^13*c^8 - 360*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^12*c^7*d + 7660*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^11*c^6*d^2 - 12056*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^6*a^3*b^10*c^5*d^3 + 4606*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^6*a^4*b^9*c^4*d^4 - 344*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^6*a^5*b^8*c^3*d^5 - 2516*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6
*b^7*c^2*d^6 + 7000*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^7*b^6*c*d^7
- 3675*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^8*b^5*d^8 + 315*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^11*c^7 + 705*sqrt(b*d)*(sqrt(b*d)*sqrt(b*
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^10*c^6*d - 4613*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(
b^2*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^9*c^5*d^2 + 1225*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^8*a^3*b^8*c^4*d^3 + 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^8*a^4*b^7*c^3*d^4 - 485*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a
^5*b^6*c^2*d^5 - 2975*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^6*b^5*c*d^
6 + 3675*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^7*b^4*d^7 - 189*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^9*c^6 - 498*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^8*c^5*d + 397*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b^7*c^4*d^2 + 2052*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^10*a^3*b^6*c^3*d^3 + 1581*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^10*a^4*b^5*c^2*d^4 + 910*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^10*a^5*b^4*c*d^5 - 2205*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^6*b^3
*d^6 + 63*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^7*c^5 + 147*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a*b^6*c^4*d + 462*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^2*b^5*c^3*d^2 - 882*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^3*b^4*c^2*d^3 - 525*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^12*a^4*b^3*c*d^4 + 735*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^12*a^5*b^2*d^5 - 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14
*b^5*c^4 - 12*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a*b^4*c^3*d - 54*sq
rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^2*b^3*c^2*d^2 + 180*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^3*b^2*c*d^3 - 105*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^4*b*d^4)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)^4*a^2*c^4
))/abs(b)

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maple [B]  time = 0.03, size = 593, normalized size = 2.23 \[ -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 a^{4} d^{4} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-180 a^{3} b c \,d^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+54 a^{2} b^{2} c^{2} d^{2} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+12 a \,b^{3} c^{3} d \,x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+9 b^{4} c^{4} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} d^{3} x^{3}+290 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b c \,d^{2} x^{3}-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{3}-18 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{3} c^{3} x^{3}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-184 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}+12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-112 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{2} d x +144 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{3} x +96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{3}\right )}{384 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} c^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)/x^5/(d*x+c)^(1/2),x)

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)/x^5/(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?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,x\right )}^{3/2}}{x^5\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(3/2)/(x^5*(c + d*x)^(1/2)),x)

[Out]

int((a + b*x)^(3/2)/(x^5*(c + d*x)^(1/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)/x**5/(d*x+c)**(1/2),x)

[Out]

Timed out

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