3.6.90 \(\int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=273 \[ \frac {3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{7/2}}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c) (b c-a d)^3}{128 a^3 c^3 x}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c) (b c-a d)^2}{64 a^2 c^3 x^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (a d+b c) (b c-a d)}{16 a c^3 x^3}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5} \]

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Rubi [A]  time = 0.15, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \begin {gather*} \frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c) (b c-a d)^2}{64 a^2 c^3 x^2}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c) (b c-a d)^3}{128 a^3 c^3 x}+\frac {3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{7/2}}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (a d+b c) (b c-a d)}{16 a c^3 x^3}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(c + d*x)^(3/2))/x^6,x]

[Out]

(-3*(b*c - a*d)^3*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^3*c^3*x) + ((b*c - a*d)^2*(b*c + a*d)*Sqrt[a
+ b*x]*(c + d*x)^(3/2))/(64*a^2*c^3*x^2) + ((b*c - a*d)*(b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(16*a*c^3*
x^3) + ((b*c + a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(8*a*c^2*x^4) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(5*a*c*
x^5) + (3*(b*c - a*d)^4*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(7/2)*c^(
7/2))

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 94

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

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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} (c+d x)^{3/2}}{x^6} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac {(b c+a d) \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^5} \, dx}{2 a c}\\ &=\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac {\left (3 \left (b^2-\frac {a^2 d^2}{c^2}\right )\right ) \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^4} \, dx}{16 a}\\ &=\frac {\left (b^2-\frac {a^2 d^2}{c^2}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac {\left ((b c-a d)^2 (b c+a d)\right ) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx}{32 a c^3}\\ &=\frac {(b c-a d)^2 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac {\left (b^2-\frac {a^2 d^2}{c^2}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}+\frac {\left (3 (b c-a d)^3 (b c+a d)\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{128 a^2 c^3}\\ &=-\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^3 c^3 x}+\frac {(b c-a d)^2 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac {\left (b^2-\frac {a^2 d^2}{c^2}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac {\left (3 (b c-a d)^4 (b c+a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^3 c^3}\\ &=-\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^3 c^3 x}+\frac {(b c-a d)^2 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac {\left (b^2-\frac {a^2 d^2}{c^2}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac {\left (3 (b c-a d)^4 (b c+a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 a^3 c^3}\\ &=-\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^3 c^3 x}+\frac {(b c-a d)^2 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac {\left (b^2-\frac {a^2 d^2}{c^2}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}+\frac {3 (b c-a d)^4 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.74, size = 228, normalized size = 0.84 \begin {gather*} \frac {(a d+b c) \left (16 a^{5/2} c^{3/2} (a+b x)^{3/2} (c+d x)^{5/2}+x (b c-a d) \left (8 a^{5/2} \sqrt {c} \sqrt {a+b x} (c+d x)^{5/2}+x (b c-a d) \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+5 a d x-3 b c x)\right )\right )\right )}{128 a^{7/2} c^{7/2} x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(c + d*x)^(3/2))/x^6,x]

[Out]

-1/5*((a + b*x)^(5/2)*(c + d*x)^(5/2))/(a*c*x^5) + ((b*c + a*d)*(16*a^(5/2)*c^(3/2)*(a + b*x)^(3/2)*(c + d*x)^
(5/2) + (b*c - a*d)*x*(8*a^(5/2)*Sqrt[c]*Sqrt[a + b*x]*(c + d*x)^(5/2) + (b*c - a*d)*x*(Sqrt[a]*Sqrt[c]*Sqrt[a
+ b*x]*Sqrt[c + d*x]*(2*a*c - 3*b*c*x + 5*a*d*x) + 3*(b*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[
a]*Sqrt[c + d*x])]))))/(128*a^(7/2)*c^(7/2)*x^4)

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IntegrateAlgebraic [A]  time = 0.53, size = 305, normalized size = 1.12 \begin {gather*} \frac {3 (a d-b c)^4 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{7/2} c^{7/2}}-\frac {\sqrt {c+d x} (a d-b c)^4 \left (\frac {15 a^5 d (c+d x)^4}{(a+b x)^4}+\frac {15 a^4 b c (c+d x)^4}{(a+b x)^4}-\frac {70 a^4 c d (c+d x)^3}{(a+b x)^3}-\frac {70 a^3 b c^2 (c+d x)^3}{(a+b x)^3}+\frac {128 a^3 c^2 d (c+d x)^2}{(a+b x)^2}-\frac {128 a^2 b c^3 (c+d x)^2}{(a+b x)^2}+\frac {70 a^2 c^3 d (c+d x)}{a+b x}+\frac {70 a b c^4 (c+d x)}{a+b x}-15 a c^4 d-15 b c^5\right )}{640 a^3 c^3 \sqrt {a+b x} \left (\frac {a (c+d x)}{a+b x}-c\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^(3/2)*(c + d*x)^(3/2))/x^6,x]

[Out]

-1/640*((-(b*c) + a*d)^4*Sqrt[c + d*x]*(-15*b*c^5 - 15*a*c^4*d + (70*a*b*c^4*(c + d*x))/(a + b*x) + (70*a^2*c^
3*d*(c + d*x))/(a + b*x) - (128*a^2*b*c^3*(c + d*x)^2)/(a + b*x)^2 + (128*a^3*c^2*d*(c + d*x)^2)/(a + b*x)^2 -
(70*a^3*b*c^2*(c + d*x)^3)/(a + b*x)^3 - (70*a^4*c*d*(c + d*x)^3)/(a + b*x)^3 + (15*a^4*b*c*(c + d*x)^4)/(a +
b*x)^4 + (15*a^5*d*(c + d*x)^4)/(a + b*x)^4))/(a^3*c^3*Sqrt[a + b*x]*(-c + (a*(c + d*x))/(a + b*x))^5) + (3*(
-(b*c) + a*d)^4*(b*c + a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(128*a^(7/2)*c^(7/2))

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fricas [A]  time = 18.46, size = 720, normalized size = 2.64 \begin {gather*} \left [\frac {15 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {a c} x^{5} \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 (128 \, a^{5} c^{5} + {\left (15 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d + 18 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 15 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{5} - 13 \, a^{3} b^{2} c^{4} d - 13 \, a^{4} b c^{3} d^{2} + 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (a^{3} b^{2} c^{5} + 34 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 176 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2560 \, a^{4} c^{4} x^{5}}, -\frac {15 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \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 (128 \, a^{5} c^{5} + {\left (15 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d + 18 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 15 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{5} - 13 \, a^{3} b^{2} c^{4} d - 13 \, a^{4} b c^{3} d^{2} + 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (a^{3} b^{2} c^{5} + 34 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 176 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1280 \, a^{4} c^{4} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(d*x+c)^(3/2)/x^6,x, algorithm="fricas")

[Out]

[1/2560*(15*(b^5*c^5 - 3*a*b^4*c^4*d + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - 3*a^4*b*c*d^4 + a^5*d^5)*sqrt(a
*c)*x^5*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*(128*a^5*c^5 + (15*a*b^4*c^5 - 40*a^2*b^3*c^4*d + 18*a^3*
b^2*c^3*d^2 - 40*a^4*b*c^2*d^3 + 15*a^5*c*d^4)*x^4 - 2*(5*a^2*b^3*c^5 - 13*a^3*b^2*c^4*d - 13*a^4*b*c^3*d^2 +
5*a^5*c^2*d^3)*x^3 + 8*(a^3*b^2*c^5 + 34*a^4*b*c^4*d + a^5*c^3*d^2)*x^2 + 176*(a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(
b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^5), -1/1280*(15*(b^5*c^5 - 3*a*b^4*c^4*d + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^
2*d^3 - 3*a^4*b*c*d^4 + a^5*d^5)*sqrt(-a*c)*x^5*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sq
rt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 2*(128*a^5*c^5 + (15*a*b^4*c^5 - 40*a^2*b^3*c^4
*d + 18*a^3*b^2*c^3*d^2 - 40*a^4*b*c^2*d^3 + 15*a^5*c*d^4)*x^4 - 2*(5*a^2*b^3*c^5 - 13*a^3*b^2*c^4*d - 13*a^4*
b*c^3*d^2 + 5*a^5*c^2*d^3)*x^3 + 8*(a^3*b^2*c^5 + 34*a^4*b*c^4*d + a^5*c^3*d^2)*x^2 + 176*(a^4*b*c^5 + a^5*c^4
*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^5)]

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giac [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((b*x+a)^(3/2)*(d*x+c)^(3/2)/x^6,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.02, size = 967, normalized size = 3.54 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 a^{5} d^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-45 a^{4} b c \,d^{4} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+30 a^{3} b^{2} c^{2} d^{3} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+30 a^{2} b^{3} c^{3} d^{2} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-45 a \,b^{4} c^{4} d \,x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+15 b^{5} c^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-30 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4} x^{4}+80 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3} x^{4}-36 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}+80 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d \,x^{4}-30 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4} x^{4}+20 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c \,d^{3} x^{3}-52 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{2} d^{2} x^{3}-52 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{3} d \,x^{3}+20 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{4} x^{3}-16 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{2} d^{2} x^{2}-544 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{3} d \,x^{2}-16 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{4} x^{2}-352 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{3} d x -352 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{4} x -256 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{4}\right )}{1280 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{3} c^{3} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(d*x+c)^(3/2)/x^6,x)

[Out]

1/1280*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3*(15*a^5*d^5*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2))/x)-45*a^4*b*c*d^4*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x
)+30*a^3*b^2*c^2*d^3*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)+30*a^2*b^3*c^
3*d^2*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)-45*a*b^4*c^4*d*x^5*ln((a*d*x
+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)+15*b^5*c^5*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*d^4*x^4+80*(a*c)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c*d^3*x^4-36*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^2*c^2
*d^2*x^4+80*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^3*d*x^4-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*
c)^(1/2)*b^4*c^4*x^4+20*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c*d^3*x^3-52*(a*c)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*a^3*b*c^2*d^2*x^3-52*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^2*c^3*d*x^3+20*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^4*x^3-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c^2*d^
2*x^2-544*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c^3*d*x^2-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)
^(1/2)*a^2*b^2*c^4*x^2-352*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c^3*d*x-352*(a*c)^(1/2)*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)*a^3*b*c^4*x-256*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(a*c)^(1/2)*a^4*c^4)/(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)/x^5/(a*c)^(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)^(3/2)*(d*x+c)^(3/2)/x^6,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 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(3/2)*(c + d*x)^(3/2))/x^6,x)

[Out]

int(((a + b*x)^(3/2)*(c + d*x)^(3/2))/x^6, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(d*x+c)**(3/2)/x**6,x)

[Out]

Integral((a + b*x)**(3/2)*(c + d*x)**(3/2)/x**6, x)

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