3.1.15 \(\int \frac {\sqrt {a+b x} (c+d x)^3 (e+f x)}{x} \, dx\)

Optimal. Leaf size=226 \[ -\frac {2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (3 c f+d e)+3 a b^2 c d (16 c f+21 d e)-5 b^3 c^2 (4 c f+27 d e)\right )-3 b d x \left (4 (b c-a d) (-2 a d f+2 b c f+3 b d e)+21 b^2 c d e\right )\right )}{315 b^4}+\frac {2 (a+b x)^{3/2} (c+d x)^2 (-2 a d f+2 b c f+3 b d e)}{21 b^2}+2 c^3 e \sqrt {a+b x}-2 \sqrt {a} c^3 e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b} \]

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Rubi [A]  time = 0.25, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {153, 147, 50, 63, 208} \begin {gather*} -\frac {2 (a+b x)^{3/2} \left (2 \left (-12 a^2 b d^2 (3 c f+d e)+8 a^3 d^3 f+3 a b^2 c d (16 c f+21 d e)-5 b^3 c^2 (4 c f+27 d e)\right )-3 b d x \left (4 (b c-a d) (-2 a d f+2 b c f+3 b d e)+21 b^2 c d e\right )\right )}{315 b^4}+\frac {2 (a+b x)^{3/2} (c+d x)^2 (-2 a d f+2 b c f+3 b d e)}{21 b^2}+2 c^3 e \sqrt {a+b x}-2 \sqrt {a} c^3 e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*c^3*e*Sqrt[a + b*x] + (2*(3*b*d*e + 2*b*c*f - 2*a*d*f)*(a + b*x)^(3/2)*(c + d*x)^2)/(21*b^2) + (2*f*(a + b*x
)^(3/2)*(c + d*x)^3)/(9*b) - (2*(a + b*x)^(3/2)*(2*(8*a^3*d^3*f - 12*a^2*b*d^2*(d*e + 3*c*f) - 5*b^3*c^2*(27*d
*e + 4*c*f) + 3*a*b^2*c*d*(21*d*e + 16*c*f)) - 3*b*d*(21*b^2*c*d*e + 4*(b*c - a*d)*(3*b*d*e + 2*b*c*f - 2*a*d*
f))*x))/(315*b^4) - 2*Sqrt[a]*c^3*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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 147

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

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && 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 {\sqrt {a+b x} (c+d x)^3 (e+f x)}{x} \, dx &=\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b}+\frac {2 \int \frac {\sqrt {a+b x} (c+d x)^2 \left (\frac {9 b c e}{2}+\frac {3}{2} (3 b d e+2 b c f-2 a d f) x\right )}{x} \, dx}{9 b}\\ &=\frac {2 (3 b d e+2 b c f-2 a d f) (a+b x)^{3/2} (c+d x)^2}{21 b^2}+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b}+\frac {4 \int \frac {\sqrt {a+b x} (c+d x) \left (\frac {63}{4} b^2 c^2 e+\frac {3}{4} \left (21 b^2 c d e+4 (b c-a d) (3 b d e+2 b c f-2 a d f)\right ) x\right )}{x} \, dx}{63 b^2}\\ &=\frac {2 (3 b d e+2 b c f-2 a d f) (a+b x)^{3/2} (c+d x)^2}{21 b^2}+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b}-\frac {2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (d e+3 c f)-5 b^3 c^2 (27 d e+4 c f)+3 a b^2 c d (21 d e+16 c f)\right )-3 b d \left (21 b^2 c d e+4 (b c-a d) (3 b d e+2 b c f-2 a d f)\right ) x\right )}{315 b^4}+\left (c^3 e\right ) \int \frac {\sqrt {a+b x}}{x} \, dx\\ &=2 c^3 e \sqrt {a+b x}+\frac {2 (3 b d e+2 b c f-2 a d f) (a+b x)^{3/2} (c+d x)^2}{21 b^2}+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b}-\frac {2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (d e+3 c f)-5 b^3 c^2 (27 d e+4 c f)+3 a b^2 c d (21 d e+16 c f)\right )-3 b d \left (21 b^2 c d e+4 (b c-a d) (3 b d e+2 b c f-2 a d f)\right ) x\right )}{315 b^4}+\left (a c^3 e\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=2 c^3 e \sqrt {a+b x}+\frac {2 (3 b d e+2 b c f-2 a d f) (a+b x)^{3/2} (c+d x)^2}{21 b^2}+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b}-\frac {2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (d e+3 c f)-5 b^3 c^2 (27 d e+4 c f)+3 a b^2 c d (21 d e+16 c f)\right )-3 b d \left (21 b^2 c d e+4 (b c-a d) (3 b d e+2 b c f-2 a d f)\right ) x\right )}{315 b^4}+\frac {\left (2 a c^3 e\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=2 c^3 e \sqrt {a+b x}+\frac {2 (3 b d e+2 b c f-2 a d f) (a+b x)^{3/2} (c+d x)^2}{21 b^2}+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b}-\frac {2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (d e+3 c f)-5 b^3 c^2 (27 d e+4 c f)+3 a b^2 c d (21 d e+16 c f)\right )-3 b d \left (21 b^2 c d e+4 (b c-a d) (3 b d e+2 b c f-2 a d f)\right ) x\right )}{315 b^4}-2 \sqrt {a} c^3 e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 204, normalized size = 0.90 \begin {gather*} \frac {2 \left (3 b e \left (35 d (a+b x)^{3/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+105 b^3 c^3 \sqrt {a+b x}-105 \sqrt {a} b^3 c^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+21 d^2 (a+b x)^{5/2} (3 b c-2 a d)+15 d^3 (a+b x)^{7/2}\right )+f (a+b x)^{3/2} \left (135 d^2 (a+b x)^2 (b c-a d)+189 d (a+b x) (b c-a d)^2+105 (b c-a d)^3+35 d^3 (a+b x)^3\right )\right )}{315 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(f*(a + b*x)^(3/2)*(105*(b*c - a*d)^3 + 189*d*(b*c - a*d)^2*(a + b*x) + 135*d^2*(b*c - a*d)*(a + b*x)^2 + 3
5*d^3*(a + b*x)^3) + 3*b*e*(105*b^3*c^3*Sqrt[a + b*x] + 35*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*(a + b*x)^(3/2)
+ 21*d^2*(3*b*c - 2*a*d)*(a + b*x)^(5/2) + 15*d^3*(a + b*x)^(7/2) - 105*Sqrt[a]*b^3*c^3*ArcTanh[Sqrt[a + b*x]
/Sqrt[a]])))/(315*b^4)

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IntegrateAlgebraic [A]  time = 0.18, size = 343, normalized size = 1.52 \begin {gather*} \frac {2 \left (-105 a^3 d^3 f (a+b x)^{3/2}+315 a^2 b c d^2 f (a+b x)^{3/2}+105 a^2 b d^3 e (a+b x)^{3/2}+189 a^2 d^3 f (a+b x)^{5/2}+315 b^4 c^3 e \sqrt {a+b x}+105 b^3 c^3 f (a+b x)^{3/2}+315 b^3 c^2 d e (a+b x)^{3/2}+189 b^2 c^2 d f (a+b x)^{5/2}-315 a b^2 c^2 d f (a+b x)^{3/2}+189 b^2 c d^2 e (a+b x)^{5/2}-315 a b^2 c d^2 e (a+b x)^{3/2}+135 b c d^2 f (a+b x)^{7/2}-378 a b c d^2 f (a+b x)^{5/2}+45 b d^3 e (a+b x)^{7/2}-126 a b d^3 e (a+b x)^{5/2}+35 d^3 f (a+b x)^{9/2}-135 a d^3 f (a+b x)^{7/2}\right )}{315 b^4}-2 \sqrt {a} c^3 e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(315*b^4*c^3*e*Sqrt[a + b*x] + 315*b^3*c^2*d*e*(a + b*x)^(3/2) - 315*a*b^2*c*d^2*e*(a + b*x)^(3/2) + 105*a^
2*b*d^3*e*(a + b*x)^(3/2) + 105*b^3*c^3*f*(a + b*x)^(3/2) - 315*a*b^2*c^2*d*f*(a + b*x)^(3/2) + 315*a^2*b*c*d^
2*f*(a + b*x)^(3/2) - 105*a^3*d^3*f*(a + b*x)^(3/2) + 189*b^2*c*d^2*e*(a + b*x)^(5/2) - 126*a*b*d^3*e*(a + b*x
)^(5/2) + 189*b^2*c^2*d*f*(a + b*x)^(5/2) - 378*a*b*c*d^2*f*(a + b*x)^(5/2) + 189*a^2*d^3*f*(a + b*x)^(5/2) +
45*b*d^3*e*(a + b*x)^(7/2) + 135*b*c*d^2*f*(a + b*x)^(7/2) - 135*a*d^3*f*(a + b*x)^(7/2) + 35*d^3*f*(a + b*x)^
(9/2)))/(315*b^4) - 2*Sqrt[a]*c^3*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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fricas [A]  time = 1.40, size = 641, normalized size = 2.84 \begin {gather*} \left [\frac {315 \, \sqrt {a} b^{4} c^{3} e \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (35 \, b^{4} d^{3} f x^{4} + 5 \, {\left (9 \, b^{4} d^{3} e + {\left (27 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e + {\left (63 \, b^{4} c^{2} d + 9 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} f\right )} x^{2} + 3 \, {\left (105 \, b^{4} c^{3} + 105 \, a b^{3} c^{2} d - 42 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e + {\left (105 \, a b^{3} c^{3} - 126 \, a^{2} b^{2} c^{2} d + 72 \, a^{3} b c d^{2} - 16 \, a^{4} d^{3}\right )} f + {\left (3 \, {\left (105 \, b^{4} c^{2} d + 21 \, a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3}\right )} e + {\left (105 \, b^{4} c^{3} + 63 \, a b^{3} c^{2} d - 36 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} f\right )} x\right )} \sqrt {b x + a}}{315 \, b^{4}}, \frac {2 \, {\left (315 \, \sqrt {-a} b^{4} c^{3} e \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (35 \, b^{4} d^{3} f x^{4} + 5 \, {\left (9 \, b^{4} d^{3} e + {\left (27 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e + {\left (63 \, b^{4} c^{2} d + 9 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} f\right )} x^{2} + 3 \, {\left (105 \, b^{4} c^{3} + 105 \, a b^{3} c^{2} d - 42 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e + {\left (105 \, a b^{3} c^{3} - 126 \, a^{2} b^{2} c^{2} d + 72 \, a^{3} b c d^{2} - 16 \, a^{4} d^{3}\right )} f + {\left (3 \, {\left (105 \, b^{4} c^{2} d + 21 \, a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3}\right )} e + {\left (105 \, b^{4} c^{3} + 63 \, a b^{3} c^{2} d - 36 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} f\right )} x\right )} \sqrt {b x + a}\right )}}{315 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/315*(315*sqrt(a)*b^4*c^3*e*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(35*b^4*d^3*f*x^4 + 5*(9*b^4*d^
3*e + (27*b^4*c*d^2 + a*b^3*d^3)*f)*x^3 + 3*(3*(21*b^4*c*d^2 + a*b^3*d^3)*e + (63*b^4*c^2*d + 9*a*b^3*c*d^2 -
2*a^2*b^2*d^3)*f)*x^2 + 3*(105*b^4*c^3 + 105*a*b^3*c^2*d - 42*a^2*b^2*c*d^2 + 8*a^3*b*d^3)*e + (105*a*b^3*c^3
- 126*a^2*b^2*c^2*d + 72*a^3*b*c*d^2 - 16*a^4*d^3)*f + (3*(105*b^4*c^2*d + 21*a*b^3*c*d^2 - 4*a^2*b^2*d^3)*e +
(105*b^4*c^3 + 63*a*b^3*c^2*d - 36*a^2*b^2*c*d^2 + 8*a^3*b*d^3)*f)*x)*sqrt(b*x + a))/b^4, 2/315*(315*sqrt(-a)
*b^4*c^3*e*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (35*b^4*d^3*f*x^4 + 5*(9*b^4*d^3*e + (27*b^4*c*d^2 + a*b^3*d^3)*
f)*x^3 + 3*(3*(21*b^4*c*d^2 + a*b^3*d^3)*e + (63*b^4*c^2*d + 9*a*b^3*c*d^2 - 2*a^2*b^2*d^3)*f)*x^2 + 3*(105*b^
4*c^3 + 105*a*b^3*c^2*d - 42*a^2*b^2*c*d^2 + 8*a^3*b*d^3)*e + (105*a*b^3*c^3 - 126*a^2*b^2*c^2*d + 72*a^3*b*c*
d^2 - 16*a^4*d^3)*f + (3*(105*b^4*c^2*d + 21*a*b^3*c*d^2 - 4*a^2*b^2*d^3)*e + (105*b^4*c^3 + 63*a*b^3*c^2*d -
36*a^2*b^2*c*d^2 + 8*a^3*b*d^3)*f)*x)*sqrt(b*x + a))/b^4]

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giac [A]  time = 1.39, size = 338, normalized size = 1.50 \begin {gather*} \frac {2 \, a c^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) e}{\sqrt {-a}} + \frac {2 \, {\left (105 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{35} c^{3} f + 189 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{34} c^{2} d f - 315 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{34} c^{2} d f + 135 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{33} c d^{2} f - 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{33} c d^{2} f + 315 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{33} c d^{2} f + 35 \, {\left (b x + a\right )}^{\frac {9}{2}} b^{32} d^{3} f - 135 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{32} d^{3} f + 189 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{32} d^{3} f - 105 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{32} d^{3} f + 315 \, \sqrt {b x + a} b^{36} c^{3} e + 315 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{35} c^{2} d e + 189 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{34} c d^{2} e - 315 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{34} c d^{2} e + 45 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{33} d^{3} e - 126 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{33} d^{3} e + 105 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{33} d^{3} e\right )}}{315 \, b^{36}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*c^3*arctan(sqrt(b*x + a)/sqrt(-a))*e/sqrt(-a) + 2/315*(105*(b*x + a)^(3/2)*b^35*c^3*f + 189*(b*x + a)^(5/2
)*b^34*c^2*d*f - 315*(b*x + a)^(3/2)*a*b^34*c^2*d*f + 135*(b*x + a)^(7/2)*b^33*c*d^2*f - 378*(b*x + a)^(5/2)*a
*b^33*c*d^2*f + 315*(b*x + a)^(3/2)*a^2*b^33*c*d^2*f + 35*(b*x + a)^(9/2)*b^32*d^3*f - 135*(b*x + a)^(7/2)*a*b
^32*d^3*f + 189*(b*x + a)^(5/2)*a^2*b^32*d^3*f - 105*(b*x + a)^(3/2)*a^3*b^32*d^3*f + 315*sqrt(b*x + a)*b^36*c
^3*e + 315*(b*x + a)^(3/2)*b^35*c^2*d*e + 189*(b*x + a)^(5/2)*b^34*c*d^2*e - 315*(b*x + a)^(3/2)*a*b^34*c*d^2*
e + 45*(b*x + a)^(7/2)*b^33*d^3*e - 126*(b*x + a)^(5/2)*a*b^33*d^3*e + 105*(b*x + a)^(3/2)*a^2*b^33*d^3*e)/b^3
6

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maple [A]  time = 0.01, size = 301, normalized size = 1.33 \begin {gather*} \frac {-2 \sqrt {a}\, b^{4} c^{3} e \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 \sqrt {b x +a}\, b^{4} c^{3} e -\frac {2 \left (b x +a \right )^{\frac {3}{2}} a^{3} d^{3} f}{3}+2 \left (b x +a \right )^{\frac {3}{2}} a^{2} b c \,d^{2} f +\frac {2 \left (b x +a \right )^{\frac {3}{2}} a^{2} b \,d^{3} e}{3}-2 \left (b x +a \right )^{\frac {3}{2}} a \,b^{2} c^{2} d f -2 \left (b x +a \right )^{\frac {3}{2}} a \,b^{2} c \,d^{2} e +\frac {2 \left (b x +a \right )^{\frac {3}{2}} b^{3} c^{3} f}{3}+2 \left (b x +a \right )^{\frac {3}{2}} b^{3} c^{2} d e +\frac {6 \left (b x +a \right )^{\frac {5}{2}} a^{2} d^{3} f}{5}-\frac {12 \left (b x +a \right )^{\frac {5}{2}} a b c \,d^{2} f}{5}-\frac {4 \left (b x +a \right )^{\frac {5}{2}} a b \,d^{3} e}{5}+\frac {6 \left (b x +a \right )^{\frac {5}{2}} b^{2} c^{2} d f}{5}+\frac {6 \left (b x +a \right )^{\frac {5}{2}} b^{2} c \,d^{2} e}{5}-\frac {6 \left (b x +a \right )^{\frac {7}{2}} a \,d^{3} f}{7}+\frac {6 \left (b x +a \right )^{\frac {7}{2}} b c \,d^{2} f}{7}+\frac {2 \left (b x +a \right )^{\frac {7}{2}} b \,d^{3} e}{7}+\frac {2 \left (b x +a \right )^{\frac {9}{2}} d^{3} f}{9}}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/b^4*(1/9*f*d^3*(b*x+a)^(9/2)-3/7*(b*x+a)^(7/2)*a*d^3*f+3/7*(b*x+a)^(7/2)*b*c*d^2*f+1/7*(b*x+a)^(7/2)*b*d^3*e
+3/5*(b*x+a)^(5/2)*a^2*d^3*f-6/5*(b*x+a)^(5/2)*a*b*c*d^2*f-2/5*(b*x+a)^(5/2)*a*b*d^3*e+3/5*(b*x+a)^(5/2)*b^2*c
^2*d*f+3/5*(b*x+a)^(5/2)*b^2*c*d^2*e-1/3*(b*x+a)^(3/2)*a^3*d^3*f+(b*x+a)^(3/2)*a^2*b*c*d^2*f+1/3*(b*x+a)^(3/2)
*a^2*b*d^3*e-(b*x+a)^(3/2)*a*b^2*c^2*d*f-(b*x+a)^(3/2)*a*b^2*c*d^2*e+1/3*(b*x+a)^(3/2)*b^3*c^3*f+(b*x+a)^(3/2)
*b^3*c^2*d*e+b^4*c^3*e*(b*x+a)^(1/2)-a^(1/2)*b^4*c^3*e*arctanh((b*x+a)^(1/2)/a^(1/2)))

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maxima [A]  time = 0.97, size = 238, normalized size = 1.05 \begin {gather*} \sqrt {a} c^{3} e \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2 \, {\left (315 \, \sqrt {b x + a} b^{4} c^{3} e + 35 \, {\left (b x + a\right )}^{\frac {9}{2}} d^{3} f + 45 \, {\left (b d^{3} e + 3 \, {\left (b c d^{2} - a d^{3}\right )} f\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 63 \, {\left ({\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} e + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 105 \, {\left ({\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f\right )} {\left (b x + a\right )}^{\frac {3}{2}}\right )}}{315 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a)*c^3*e*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a))) + 2/315*(315*sqrt(b*x + a)*b^4*c^3*e +
35*(b*x + a)^(9/2)*d^3*f + 45*(b*d^3*e + 3*(b*c*d^2 - a*d^3)*f)*(b*x + a)^(7/2) + 63*((3*b^2*c*d^2 - 2*a*b*d^3
)*e + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f)*(b*x + a)^(5/2) + 105*((3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*d^3
)*e + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f)*(b*x + a)^(3/2))/b^4

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mupad [B]  time = 2.53, size = 413, normalized size = 1.83 \begin {gather*} \left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{7\,b^4}+\frac {2\,a\,d^3\,f}{7\,b^4}\right )\,{\left (a+b\,x\right )}^{7/2}+\left (\frac {a\,\left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{b^4}+\frac {2\,a\,d^3\,f}{b^4}\right )}{5}-\frac {6\,d\,\left (a\,d-b\,c\right )\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e\right )}{5\,b^4}\right )\,{\left (a+b\,x\right )}^{5/2}+\left (a\,\left (a\,\left (a\,\left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{b^4}+\frac {2\,a\,d^3\,f}{b^4}\right )-\frac {6\,d\,\left (a\,d-b\,c\right )\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e\right )}{b^4}\right )+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (b\,c\,f-4\,a\,d\,f+3\,b\,d\,e\right )}{b^4}\right )+\frac {2\,{\left (a\,d-b\,c\right )}^3\,\left (a\,f-b\,e\right )}{b^4}\right )\,\sqrt {a+b\,x}+\left (\frac {a\,\left (a\,\left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{b^4}+\frac {2\,a\,d^3\,f}{b^4}\right )-\frac {6\,d\,\left (a\,d-b\,c\right )\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e\right )}{b^4}\right )}{3}+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (b\,c\,f-4\,a\,d\,f+3\,b\,d\,e\right )}{3\,b^4}\right )\,{\left (a+b\,x\right )}^{3/2}+\frac {2\,d^3\,f\,{\left (a+b\,x\right )}^{9/2}}{9\,b^4}+\sqrt {a}\,c^3\,e\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*b*d^3*e - 8*a*d^3*f + 6*b*c*d^2*f)/(7*b^4) + (2*a*d^3*f)/(7*b^4))*(a + b*x)^(7/2) + ((a*((2*b*d^3*e - 8*a*
d^3*f + 6*b*c*d^2*f)/b^4 + (2*a*d^3*f)/b^4))/5 - (6*d*(a*d - b*c)*(b*c*f - 2*a*d*f + b*d*e))/(5*b^4))*(a + b*x
)^(5/2) + (a*(a*(a*((2*b*d^3*e - 8*a*d^3*f + 6*b*c*d^2*f)/b^4 + (2*a*d^3*f)/b^4) - (6*d*(a*d - b*c)*(b*c*f - 2
*a*d*f + b*d*e))/b^4) + (2*(a*d - b*c)^2*(b*c*f - 4*a*d*f + 3*b*d*e))/b^4) + (2*(a*d - b*c)^3*(a*f - b*e))/b^4
)*(a + b*x)^(1/2) + ((a*(a*((2*b*d^3*e - 8*a*d^3*f + 6*b*c*d^2*f)/b^4 + (2*a*d^3*f)/b^4) - (6*d*(a*d - b*c)*(b
*c*f - 2*a*d*f + b*d*e))/b^4))/3 + (2*(a*d - b*c)^2*(b*c*f - 4*a*d*f + 3*b*d*e))/(3*b^4))*(a + b*x)^(3/2) + a^
(1/2)*c^3*e*atan(((a + b*x)^(1/2)*1i)/a^(1/2))*2i + (2*d^3*f*(a + b*x)^(9/2))/(9*b^4)

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sympy [A]  time = 37.95, size = 274, normalized size = 1.21 \begin {gather*} \frac {2 a c^{3} e \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 c^{3} e \sqrt {a + b x} + \frac {2 d^{3} f \left (a + b x\right )^{\frac {9}{2}}}{9 b^{4}} + \frac {2 \left (a + b x\right )^{\frac {7}{2}} \left (- 3 a d^{3} f + 3 b c d^{2} f + b d^{3} e\right )}{7 b^{4}} + \frac {2 \left (a + b x\right )^{\frac {5}{2}} \left (3 a^{2} d^{3} f - 6 a b c d^{2} f - 2 a b d^{3} e + 3 b^{2} c^{2} d f + 3 b^{2} c d^{2} e\right )}{5 b^{4}} + \frac {2 \left (a + b x\right )^{\frac {3}{2}} \left (- a^{3} d^{3} f + 3 a^{2} b c d^{2} f + a^{2} b d^{3} e - 3 a b^{2} c^{2} d f - 3 a b^{2} c d^{2} e + b^{3} c^{3} f + 3 b^{3} c^{2} d e\right )}{3 b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*c**3*e*atan(sqrt(a + b*x)/sqrt(-a))/sqrt(-a) + 2*c**3*e*sqrt(a + b*x) + 2*d**3*f*(a + b*x)**(9/2)/(9*b**4)
+ 2*(a + b*x)**(7/2)*(-3*a*d**3*f + 3*b*c*d**2*f + b*d**3*e)/(7*b**4) + 2*(a + b*x)**(5/2)*(3*a**2*d**3*f - 6
*a*b*c*d**2*f - 2*a*b*d**3*e + 3*b**2*c**2*d*f + 3*b**2*c*d**2*e)/(5*b**4) + 2*(a + b*x)**(3/2)*(-a**3*d**3*f
+ 3*a**2*b*c*d**2*f + a**2*b*d**3*e - 3*a*b**2*c**2*d*f - 3*a*b**2*c*d**2*e + b**3*c**3*f + 3*b**3*c**2*d*e)/(
3*b**4)

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