3.5.55 \(\int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx\) [455]

Optimal. Leaf size=209 \[ -\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}+\frac {2 a^3 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}} \]

[Out]

-2/3*a^3*(-a*d+b*c)*(d*x+c)^(3/2)/b^5-2/5*a^3*(d*x+c)^(5/2)/b^4+2/7*(a^2*d^2+a*b*c*d+b^2*c^2)*(d*x+c)^(7/2)/b^
3/d^3-2/9*(a*d+2*b*c)*(d*x+c)^(9/2)/b^2/d^3+2/11*(d*x+c)^(11/2)/b/d^3+2*a^3*(-a*d+b*c)^(5/2)*arctanh(b^(1/2)*(
d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(13/2)-2*a^3*(-a*d+b*c)^2*(d*x+c)^(1/2)/b^6

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Rubi [A]
time = 0.12, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {90, 52, 65, 214} \begin {gather*} \frac {2 a^3 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}-\frac {2 a^3 \sqrt {c+d x} (b c-a d)^2}{b^6}-\frac {2 a^3 (c+d x)^{3/2} (b c-a d)}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 (c+d x)^{7/2} \left (a^2 d^2+a b c d+b^2 c^2\right )}{7 b^3 d^3}-\frac {2 (c+d x)^{9/2} (a d+2 b c)}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*(b*c - a*d)^2*Sqrt[c + d*x])/b^6 - (2*a^3*(b*c - a*d)*(c + d*x)^(3/2))/(3*b^5) - (2*a^3*(c + d*x)^(5/2
))/(5*b^4) + (2*(b^2*c^2 + a*b*c*d + a^2*d^2)*(c + d*x)^(7/2))/(7*b^3*d^3) - (2*(2*b*c + a*d)*(c + d*x)^(9/2))
/(9*b^2*d^3) + (2*(c + d*x)^(11/2))/(11*b*d^3) + (2*a^3*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt
[b*c - a*d]])/b^(13/2)

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 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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 {x^3 (c+d x)^{5/2}}{a+b x} \, dx &=\int \left (\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{5/2}}{b^3 d^2}-\frac {a^3 (c+d x)^{5/2}}{b^3 (a+b x)}+\frac {(-2 b c-a d) (c+d x)^{7/2}}{b^2 d^2}+\frac {(c+d x)^{9/2}}{b d^2}\right ) \, dx\\ &=\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {a^3 \int \frac {(c+d x)^{5/2}}{a+b x} \, dx}{b^3}\\ &=-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (a^3 (b c-a d)\right ) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{b^4}\\ &=-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (a^3 (b c-a d)^2\right ) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{b^5}\\ &=-\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (a^3 (b c-a d)^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{b^6}\\ &=-\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (2 a^3 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b^6 d}\\ &=-\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}+\frac {2 a^3 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 196, normalized size = 0.94 \begin {gather*} \frac {2 \sqrt {c+d x} \left (-3465 a^5 d^5+495 a^2 b^3 d^2 (c+d x)^3+55 a b^4 d (2 c-7 d x) (c+d x)^3+1155 a^4 b d^4 (7 c+d x)-231 a^3 b^2 d^3 \left (23 c^2+11 c d x+3 d^2 x^2\right )+5 b^5 (c+d x)^3 \left (8 c^2-28 c d x+63 d^2 x^2\right )\right )}{3465 b^6 d^3}+\frac {2 a^3 (-b c+a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(-3465*a^5*d^5 + 495*a^2*b^3*d^2*(c + d*x)^3 + 55*a*b^4*d*(2*c - 7*d*x)*(c + d*x)^3 + 1155*a^
4*b*d^4*(7*c + d*x) - 231*a^3*b^2*d^3*(23*c^2 + 11*c*d*x + 3*d^2*x^2) + 5*b^5*(c + d*x)^3*(8*c^2 - 28*c*d*x +
63*d^2*x^2)))/(3465*b^6*d^3) + (2*a^3*(-(b*c) + a*d)^(5/2)*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])
/b^(13/2)

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Maple [A]
time = 0.10, size = 348, normalized size = 1.67

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{\frac {11}{2}} b^{5}}{11}+\frac {\left (a \,b^{4} d +2 b^{5} c \right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {\left (-2 a d \,b^{4} c -b \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {\left (a d \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )-b \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {\left (a d \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )-b \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}+a d \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right ) \sqrt {d x +c}\right )}{b^{6}}+\frac {2 a^{3} d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{6} \sqrt {\left (a d -b c \right ) b}}}{d^{3}}\)	\(348\)
default	\(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{\frac {11}{2}} b^{5}}{11}+\frac {\left (a \,b^{4} d +2 b^{5} c \right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {\left (-2 a d \,b^{4} c -b \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {\left (a d \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )-b \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {\left (a d \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )-b \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}+a d \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right ) \sqrt {d x +c}\right )}{b^{6}}+\frac {2 a^{3} d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{6} \sqrt {\left (a d -b c \right ) b}}}{d^{3}}\)	\(348\)
risch	\(-\frac {2 \left (-315 d^{5} b^{5} x^{5}+385 a \,b^{4} d^{5} x^{4}-805 b^{5} c \,d^{4} x^{4}-495 a^{2} b^{3} d^{5} x^{3}+1045 a \,b^{4} c \,d^{4} x^{3}-565 b^{5} c^{2} d^{3} x^{3}+693 a^{3} b^{2} d^{5} x^{2}-1485 a^{2} b^{3} c \,d^{4} x^{2}+825 a \,b^{4} c^{2} d^{3} x^{2}-15 b^{5} c^{3} d^{2} x^{2}-1155 a^{4} b \,d^{5} x +2541 a^{3} b^{2} c \,d^{4} x -1485 a^{2} b^{3} c^{2} d^{3} x +55 a \,b^{4} c^{3} d^{2} x +20 b^{5} c^{4} d x +3465 a^{5} d^{5}-8085 a^{4} b c \,d^{4}+5313 a^{3} b^{2} c^{2} d^{3}-495 a^{2} b^{3} c^{3} d^{2}-110 a \,b^{4} c^{4} d -40 b^{5} c^{5}\right ) \sqrt {d x +c}}{3465 d^{3} b^{6}}+\frac {2 a^{6} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) d^{3}}{b^{6} \sqrt {\left (a d -b c \right ) b}}-\frac {6 a^{5} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c \,d^{2}}{b^{5} \sqrt {\left (a d -b c \right ) b}}+\frac {6 a^{4} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c^{2} d}{b^{4} \sqrt {\left (a d -b c \right ) b}}-\frac {2 a^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c^{3}}{b^{3} \sqrt {\left (a d -b c \right ) b}}\)	\(459\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d^3*(-1/b^6*(-1/11*(d*x+c)^(11/2)*b^5+1/9*(a*b^4*d+2*b^5*c)*(d*x+c)^(9/2)+1/7*(-2*a*d*b^4*c-b*(a^2*b^2*d^2-a
*b^3*c*d+b^4*c^2))*(d*x+c)^(7/2)+1/5*(a*d*(a^2*b^2*d^2-a*b^3*c*d+b^4*c^2)-b*(-a^2*b^2*c*d^2+a*b^3*c^2*d))*(d*x
+c)^(5/2)+1/3*(a*d*(-a^2*b^2*c*d^2+a*b^3*c^2*d)-b*(a^4*d^4-2*a^3*b*c*d^3+a^2*b^2*c^2*d^2))*(d*x+c)^(3/2)+a*d*(
a^4*d^4-2*a^3*b*c*d^3+a^2*b^2*c^2*d^2)*(d*x+c)^(1/2))+a^3*d^3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^
6/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*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(x^3*(d*x+c)^(5/2)/(b*x+a),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 detail

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Fricas [A]
time = 0.86, size = 713, normalized size = 3.41 \begin {gather*} \left [\frac {3465 \, {\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (315 \, b^{5} d^{5} x^{5} + 40 \, b^{5} c^{5} + 110 \, a b^{4} c^{4} d + 495 \, a^{2} b^{3} c^{3} d^{2} - 5313 \, a^{3} b^{2} c^{2} d^{3} + 8085 \, a^{4} b c d^{4} - 3465 \, a^{5} d^{5} + 35 \, {\left (23 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 5 \, {\left (113 \, b^{5} c^{2} d^{3} - 209 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (5 \, b^{5} c^{3} d^{2} - 275 \, a b^{4} c^{2} d^{3} + 495 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} - {\left (20 \, b^{5} c^{4} d + 55 \, a b^{4} c^{3} d^{2} - 1485 \, a^{2} b^{3} c^{2} d^{3} + 2541 \, a^{3} b^{2} c d^{4} - 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}}{3465 \, b^{6} d^{3}}, \frac {2 \, {\left (3465 \, {\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (315 \, b^{5} d^{5} x^{5} + 40 \, b^{5} c^{5} + 110 \, a b^{4} c^{4} d + 495 \, a^{2} b^{3} c^{3} d^{2} - 5313 \, a^{3} b^{2} c^{2} d^{3} + 8085 \, a^{4} b c d^{4} - 3465 \, a^{5} d^{5} + 35 \, {\left (23 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 5 \, {\left (113 \, b^{5} c^{2} d^{3} - 209 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (5 \, b^{5} c^{3} d^{2} - 275 \, a b^{4} c^{2} d^{3} + 495 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} - {\left (20 \, b^{5} c^{4} d + 55 \, a b^{4} c^{3} d^{2} - 1485 \, a^{2} b^{3} c^{2} d^{3} + 2541 \, a^{3} b^{2} c d^{4} - 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}\right )}}{3465 \, b^{6} d^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/3465*(3465*(a^3*b^2*c^2*d^3 - 2*a^4*b*c*d^4 + a^5*d^5)*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d + 2*sqr
t(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*(315*b^5*d^5*x^5 + 40*b^5*c^5 + 110*a*b^4*c^4*d + 495*a^2*b^3
*c^3*d^2 - 5313*a^3*b^2*c^2*d^3 + 8085*a^4*b*c*d^4 - 3465*a^5*d^5 + 35*(23*b^5*c*d^4 - 11*a*b^4*d^5)*x^4 + 5*(
113*b^5*c^2*d^3 - 209*a*b^4*c*d^4 + 99*a^2*b^3*d^5)*x^3 + 3*(5*b^5*c^3*d^2 - 275*a*b^4*c^2*d^3 + 495*a^2*b^3*c
*d^4 - 231*a^3*b^2*d^5)*x^2 - (20*b^5*c^4*d + 55*a*b^4*c^3*d^2 - 1485*a^2*b^3*c^2*d^3 + 2541*a^3*b^2*c*d^4 - 1
155*a^4*b*d^5)*x)*sqrt(d*x + c))/(b^6*d^3), 2/3465*(3465*(a^3*b^2*c^2*d^3 - 2*a^4*b*c*d^4 + a^5*d^5)*sqrt(-(b*
c - a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + (315*b^5*d^5*x^5 + 40*b^5*c^5 + 110*a*
b^4*c^4*d + 495*a^2*b^3*c^3*d^2 - 5313*a^3*b^2*c^2*d^3 + 8085*a^4*b*c*d^4 - 3465*a^5*d^5 + 35*(23*b^5*c*d^4 -
11*a*b^4*d^5)*x^4 + 5*(113*b^5*c^2*d^3 - 209*a*b^4*c*d^4 + 99*a^2*b^3*d^5)*x^3 + 3*(5*b^5*c^3*d^2 - 275*a*b^4*
c^2*d^3 + 495*a^2*b^3*c*d^4 - 231*a^3*b^2*d^5)*x^2 - (20*b^5*c^4*d + 55*a*b^4*c^3*d^2 - 1485*a^2*b^3*c^2*d^3 +
2541*a^3*b^2*c*d^4 - 1155*a^4*b*d^5)*x)*sqrt(d*x + c))/(b^6*d^3)]

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Sympy [A]
time = 37.77, size = 228, normalized size = 1.09 \begin {gather*} - \frac {2 a^{3} \left (c + d x\right )^{\frac {5}{2}}}{5 b^{4}} + \frac {2 a^{3} \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{7} \sqrt {\frac {a d - b c}{b}}} + \frac {2 \left (c + d x\right )^{\frac {11}{2}}}{11 b d^{3}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (- 2 a d - 4 b c\right )}{9 b^{2} d^{3}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (2 a^{2} d^{2} + 2 a b c d + 2 b^{2} c^{2}\right )}{7 b^{3} d^{3}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (2 a^{4} d - 2 a^{3} b c\right )}{3 b^{5}} + \frac {\sqrt {c + d x} \left (- 2 a^{5} d^{2} + 4 a^{4} b c d - 2 a^{3} b^{2} c^{2}\right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**3*(c + d*x)**(5/2)/(5*b**4) + 2*a**3*(a*d - b*c)**3*atan(sqrt(c + d*x)/sqrt((a*d - b*c)/b))/(b**7*sqrt((
a*d - b*c)/b)) + 2*(c + d*x)**(11/2)/(11*b*d**3) + (c + d*x)**(9/2)*(-2*a*d - 4*b*c)/(9*b**2*d**3) + (c + d*x)
**(7/2)*(2*a**2*d**2 + 2*a*b*c*d + 2*b**2*c**2)/(7*b**3*d**3) + (c + d*x)**(3/2)*(2*a**4*d - 2*a**3*b*c)/(3*b*
*5) + sqrt(c + d*x)*(-2*a**5*d**2 + 4*a**4*b*c*d - 2*a**3*b**2*c**2)/b**6

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Giac [A]
time = 0.91, size = 305, normalized size = 1.46 \begin {gather*} -\frac {2 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{6}} + \frac {2 \, {\left (315 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{10} d^{30} - 770 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{10} c d^{30} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{10} c^{2} d^{30} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} a b^{9} d^{31} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{9} c d^{31} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} a^{2} b^{8} d^{32} - 693 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{3} b^{7} d^{33} - 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{7} c d^{33} - 3465 \, \sqrt {d x + c} a^{3} b^{7} c^{2} d^{33} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{4} b^{6} d^{34} + 6930 \, \sqrt {d x + c} a^{4} b^{6} c d^{34} - 3465 \, \sqrt {d x + c} a^{5} b^{5} d^{35}\right )}}{3465 \, b^{11} d^{33}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqr
t(-b^2*c + a*b*d)*b^6) + 2/3465*(315*(d*x + c)^(11/2)*b^10*d^30 - 770*(d*x + c)^(9/2)*b^10*c*d^30 + 495*(d*x +
c)^(7/2)*b^10*c^2*d^30 - 385*(d*x + c)^(9/2)*a*b^9*d^31 + 495*(d*x + c)^(7/2)*a*b^9*c*d^31 + 495*(d*x + c)^(7
/2)*a^2*b^8*d^32 - 693*(d*x + c)^(5/2)*a^3*b^7*d^33 - 1155*(d*x + c)^(3/2)*a^3*b^7*c*d^33 - 3465*sqrt(d*x + c)
*a^3*b^7*c^2*d^33 + 1155*(d*x + c)^(3/2)*a^4*b^6*d^34 + 6930*sqrt(d*x + c)*a^4*b^6*c*d^34 - 3465*sqrt(d*x + c)
*a^5*b^5*d^35)/(b^11*d^33)

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Mupad [B]
time = 0.10, size = 567, normalized size = 2.71 \begin {gather*} \left (\frac {6\,c^2}{7\,b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{7\,b\,d^3}\right )\,{\left (c+d\,x\right )}^{7/2}-\left (\frac {2\,c^3}{5\,b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{5\,b\,d^3}\right )\,{\left (c+d\,x\right )}^{5/2}-\left (\frac {2\,c}{3\,b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{9\,b^2\,d^6}\right )\,{\left (c+d\,x\right )}^{9/2}+\frac {2\,{\left (c+d\,x\right )}^{11/2}}{11\,b\,d^3}+\frac {2\,a^3\,\mathrm {atan}\left (\frac {a^3\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}\,\sqrt {c+d\,x}}{a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{5/2}}{b^{13/2}}-\frac {\left (\frac {2\,c^3}{b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,{\left (a\,d^4-b\,c\,d^3\right )}^2\,\sqrt {c+d\,x}}{b^2\,d^6}+\frac {\left (\frac {2\,c^3}{b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,b\,d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((6*c^2)/(7*b*d^3) + (((6*c)/(b*d^3) + (2*(a*d^4 - b*c*d^3))/(b^2*d^6))*(a*d^4 - b*c*d^3))/(7*b*d^3))*(c + d*x
)^(7/2) - ((2*c^3)/(5*b*d^3) + (((6*c^2)/(b*d^3) + (((6*c)/(b*d^3) + (2*(a*d^4 - b*c*d^3))/(b^2*d^6))*(a*d^4 -
b*c*d^3))/(b*d^3))*(a*d^4 - b*c*d^3))/(5*b*d^3))*(c + d*x)^(5/2) - ((2*c)/(3*b*d^3) + (2*(a*d^4 - b*c*d^3))/(
9*b^2*d^6))*(c + d*x)^(9/2) + (2*(c + d*x)^(11/2))/(11*b*d^3) + (2*a^3*atan((a^3*b^(1/2)*(a*d - b*c)^(5/2)*(c
+ d*x)^(1/2))/(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2))*(a*d - b*c)^(5/2))/b^(13/2) - (((2*c^
3)/(b*d^3) + (((6*c^2)/(b*d^3) + (((6*c)/(b*d^3) + (2*(a*d^4 - b*c*d^3))/(b^2*d^6))*(a*d^4 - b*c*d^3))/(b*d^3)
)*(a*d^4 - b*c*d^3))/(b*d^3))*(a*d^4 - b*c*d^3)^2*(c + d*x)^(1/2))/(b^2*d^6) + (((2*c^3)/(b*d^3) + (((6*c^2)/(
b*d^3) + (((6*c)/(b*d^3) + (2*(a*d^4 - b*c*d^3))/(b^2*d^6))*(a*d^4 - b*c*d^3))/(b*d^3))*(a*d^4 - b*c*d^3))/(b*
d^3))*(a*d^4 - b*c*d^3)*(c + d*x)^(3/2))/(3*b*d^3)

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