3.6.47 \(\int (d+e x)^2 (a+c x^2)^{5/2} \, dx\) [547]

Optimal. Leaf size=189 \[ \frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}} \]

[Out]

5/192*a*(-a*e^2+8*c*d^2)*x*(c*x^2+a)^(3/2)/c+1/48*(-a*e^2+8*c*d^2)*x*(c*x^2+a)^(5/2)/c+9/56*d*e*(c*x^2+a)^(7/2
)/c+1/8*e*(e*x+d)*(c*x^2+a)^(7/2)/c+5/128*a^3*(-a*e^2+8*c*d^2)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(3/2)+5/12
8*a^2*(-a*e^2+8*c*d^2)*x*(c*x^2+a)^(1/2)/c

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Rubi [A]
time = 0.06, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {757, 655, 201, 223, 212} \begin {gather*} \frac {5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}}+\frac {5 a^2 x \sqrt {a+c x^2} \left (8 c d^2-a e^2\right )}{128 c}+\frac {x \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right )}{48 c}+\frac {5 a x \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right )}{192 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + c*x^2)^(5/2),x]

[Out]

(5*a^2*(8*c*d^2 - a*e^2)*x*Sqrt[a + c*x^2])/(128*c) + (5*a*(8*c*d^2 - a*e^2)*x*(a + c*x^2)^(3/2))/(192*c) + ((
8*c*d^2 - a*e^2)*x*(a + c*x^2)^(5/2))/(48*c) + (9*d*e*(a + c*x^2)^(7/2))/(56*c) + (e*(d + e*x)*(a + c*x^2)^(7/
2))/(8*c) + (5*a^3*(8*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(128*c^(3/2))

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 212

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

Rule 223

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]

Rule 655

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

Rule 757

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2,
0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rubi steps

\begin {align*} \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx &=\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\int \left (8 c d^2-a e^2+9 c d e x\right ) \left (a+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (8 c d^2-a e^2\right ) \int \left (a+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a \left (8 c d^2-a e^2\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{48 c}\\ &=\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a^2 \left (8 c d^2-a e^2\right )\right ) \int \sqrt {a+c x^2} \, dx}{64 c}\\ &=\frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a^3 \left (8 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{128 c}\\ &=\frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a^3 \left (8 c d^2-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{128 c}\\ &=\frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.49, size = 161, normalized size = 0.85 \begin {gather*} \frac {\sqrt {c} \sqrt {a+c x^2} \left (3 a^3 e (256 d+35 e x)+16 c^3 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+8 a c^2 x^3 \left (182 d^2+288 d e x+119 e^2 x^2\right )+2 a^2 c x \left (924 d^2+1152 d e x+413 e^2 x^2\right )\right )+105 a^3 \left (-8 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{2688 c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*Sqrt[a + c*x^2]*(3*a^3*e*(256*d + 35*e*x) + 16*c^3*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 8*a*c^2*x^3
*(182*d^2 + 288*d*e*x + 119*e^2*x^2) + 2*a^2*c*x*(924*d^2 + 1152*d*e*x + 413*e^2*x^2)) + 105*a^3*(-8*c*d^2 + a
*e^2)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/(2688*c^(3/2))

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Maple [A]
time = 0.43, size = 182, normalized size = 0.96

method	result	size
default	\(e^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{8 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )}{8 c}\right )+\frac {2 d e \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{7 c}+d^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )\)	\(182\)
risch	\(\frac {\left (336 e^{2} c^{3} x^{7}+768 d e \,c^{3} x^{6}+952 e^{2} c^{2} a \,x^{5}+448 c^{3} d^{2} x^{5}+2304 a \,c^{2} d e \,x^{4}+826 a^{2} c \,e^{2} x^{3}+1456 a \,c^{2} d^{2} x^{3}+2304 x^{2} a^{2} c d e +105 a^{3} e^{2} x +1848 a^{2} c \,d^{2} x +768 d e \,a^{3}\right ) \sqrt {c \,x^{2}+a}}{2688 c}-\frac {5 a^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right ) e^{2}}{128 c^{\frac {3}{2}}}+\frac {5 a^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right ) d^{2}}{16 \sqrt {c}}\)	\(187\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^2*(1/8*x*(c*x^2+a)^(7/2)/c-1/8*a/c*(1/6*x*(c*x^2+a)^(5/2)+5/6*a*(1/4*x*(c*x^2+a)^(3/2)+3/4*a*(1/2*x*(c*x^2+a
)^(1/2)+1/2*a/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))))))+2/7*d*e*(c*x^2+a)^(7/2)/c+d^2*(1/6*x*(c*x^2+a)^(5/2)+5
/6*a*(1/4*x*(c*x^2+a)^(3/2)+3/4*a*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2)))))

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Maxima [A]
time = 0.30, size = 181, normalized size = 0.96 \begin {gather*} \frac {1}{6} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d^{2} x + \frac {5}{24} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d^{2} x + \frac {5}{16} \, \sqrt {c x^{2} + a} a^{2} d^{2} x + \frac {5 \, a^{3} d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} x e^{2}}{8 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a x e^{2}}{48 \, c} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} x e^{2}}{192 \, c} - \frac {5 \, \sqrt {c x^{2} + a} a^{3} x e^{2}}{128 \, c} - \frac {5 \, a^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{2}}{128 \, c^{\frac {3}{2}}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d e}{7 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

1/6*(c*x^2 + a)^(5/2)*d^2*x + 5/24*(c*x^2 + a)^(3/2)*a*d^2*x + 5/16*sqrt(c*x^2 + a)*a^2*d^2*x + 5/16*a^3*d^2*a
rcsinh(c*x/sqrt(a*c))/sqrt(c) + 1/8*(c*x^2 + a)^(7/2)*x*e^2/c - 1/48*(c*x^2 + a)^(5/2)*a*x*e^2/c - 5/192*(c*x^
2 + a)^(3/2)*a^2*x*e^2/c - 5/128*sqrt(c*x^2 + a)*a^3*x*e^2/c - 5/128*a^4*arcsinh(c*x/sqrt(a*c))*e^2/c^(3/2) +
2/7*(c*x^2 + a)^(7/2)*d*e/c

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Fricas [A]
time = 1.67, size = 358, normalized size = 1.89 \begin {gather*} \left [\frac {105 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (448 \, c^{4} d^{2} x^{5} + 1456 \, a c^{3} d^{2} x^{3} + 1848 \, a^{2} c^{2} d^{2} x + 7 \, {\left (48 \, c^{4} x^{7} + 136 \, a c^{3} x^{5} + 118 \, a^{2} c^{2} x^{3} + 15 \, a^{3} c x\right )} e^{2} + 768 \, {\left (c^{4} d x^{6} + 3 \, a c^{3} d x^{4} + 3 \, a^{2} c^{2} d x^{2} + a^{3} c d\right )} e\right )} \sqrt {c x^{2} + a}}{5376 \, c^{2}}, -\frac {105 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (448 \, c^{4} d^{2} x^{5} + 1456 \, a c^{3} d^{2} x^{3} + 1848 \, a^{2} c^{2} d^{2} x + 7 \, {\left (48 \, c^{4} x^{7} + 136 \, a c^{3} x^{5} + 118 \, a^{2} c^{2} x^{3} + 15 \, a^{3} c x\right )} e^{2} + 768 \, {\left (c^{4} d x^{6} + 3 \, a c^{3} d x^{4} + 3 \, a^{2} c^{2} d x^{2} + a^{3} c d\right )} e\right )} \sqrt {c x^{2} + a}}{2688 \, c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/5376*(105*(8*a^3*c*d^2 - a^4*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(448*c^4*d^2*
x^5 + 1456*a*c^3*d^2*x^3 + 1848*a^2*c^2*d^2*x + 7*(48*c^4*x^7 + 136*a*c^3*x^5 + 118*a^2*c^2*x^3 + 15*a^3*c*x)*
e^2 + 768*(c^4*d*x^6 + 3*a*c^3*d*x^4 + 3*a^2*c^2*d*x^2 + a^3*c*d)*e)*sqrt(c*x^2 + a))/c^2, -1/2688*(105*(8*a^3
*c*d^2 - a^4*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (448*c^4*d^2*x^5 + 1456*a*c^3*d^2*x^3 + 1848*a
^2*c^2*d^2*x + 7*(48*c^4*x^7 + 136*a*c^3*x^5 + 118*a^2*c^2*x^3 + 15*a^3*c*x)*e^2 + 768*(c^4*d*x^6 + 3*a*c^3*d*
x^4 + 3*a^2*c^2*d*x^2 + a^3*c*d)*e)*sqrt(c*x^2 + a))/c^2]

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Sympy [A]
time = 34.53, size = 539, normalized size = 2.85 \begin {gather*} \frac {5 a^{\frac {7}{2}} e^{2} x}{128 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {5}{2}} d^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {3 a^{\frac {5}{2}} d^{2} x}{16 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {133 a^{\frac {5}{2}} e^{2} x^{3}}{384 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {35 a^{\frac {3}{2}} c d^{2} x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {127 a^{\frac {3}{2}} c e^{2} x^{5}}{192 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 \sqrt {a} c^{2} d^{2} x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {23 \sqrt {a} c^{2} e^{2} x^{7}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {5 a^{4} e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{128 c^{\frac {3}{2}}} + \frac {5 a^{3} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 \sqrt {c}} + 2 a^{2} d e \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + 4 a c d e \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 c^{2} d e \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + \frac {c^{3} d^{2} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{3} e^{2} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*a**(7/2)*e**2*x/(128*c*sqrt(1 + c*x**2/a)) + a**(5/2)*d**2*x*sqrt(1 + c*x**2/a)/2 + 3*a**(5/2)*d**2*x/(16*sq
rt(1 + c*x**2/a)) + 133*a**(5/2)*e**2*x**3/(384*sqrt(1 + c*x**2/a)) + 35*a**(3/2)*c*d**2*x**3/(48*sqrt(1 + c*x
**2/a)) + 127*a**(3/2)*c*e**2*x**5/(192*sqrt(1 + c*x**2/a)) + 17*sqrt(a)*c**2*d**2*x**5/(24*sqrt(1 + c*x**2/a)
) + 23*sqrt(a)*c**2*e**2*x**7/(48*sqrt(1 + c*x**2/a)) - 5*a**4*e**2*asinh(sqrt(c)*x/sqrt(a))/(128*c**(3/2)) +
5*a**3*d**2*asinh(sqrt(c)*x/sqrt(a))/(16*sqrt(c)) + 2*a**2*d*e*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x
**2)**(3/2)/(3*c), True)) + 4*a*c*d*e*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/
(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + 2*c**2*d*e*Piecewise((8*a**3*sqrt(a + c
*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c
*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6, True)) + c**3*d**2*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + c**3*e**2*x**9/
(8*sqrt(a)*sqrt(1 + c*x**2/a))

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Giac [A]
time = 1.37, size = 190, normalized size = 1.01 \begin {gather*} \frac {1}{2688} \, {\left (\frac {768 \, a^{3} d e}{c} + {\left (2 \, {\left (1152 \, a^{2} d e + {\left (4 \, {\left (288 \, a c d e + {\left (6 \, {\left (7 \, c^{2} x e^{2} + 16 \, c^{2} d e\right )} x + \frac {7 \, {\left (8 \, c^{8} d^{2} + 17 \, a c^{7} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac {7 \, {\left (104 \, a c^{7} d^{2} + 59 \, a^{2} c^{6} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac {21 \, {\left (88 \, a^{2} c^{6} d^{2} + 5 \, a^{3} c^{5} e^{2}\right )}}{c^{6}}\right )} x\right )} \sqrt {c x^{2} + a} - \frac {5 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2688*(768*a^3*d*e/c + (2*(1152*a^2*d*e + (4*(288*a*c*d*e + (6*(7*c^2*x*e^2 + 16*c^2*d*e)*x + 7*(8*c^8*d^2 +
17*a*c^7*e^2)/c^6)*x)*x + 7*(104*a*c^7*d^2 + 59*a^2*c^6*e^2)/c^6)*x)*x + 21*(88*a^2*c^6*d^2 + 5*a^3*c^5*e^2)/c
^6)*x)*sqrt(c*x^2 + a) - 5/128*(8*a^3*c*d^2 - a^4*e^2)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^(5/2)*(d + e*x)^2,x)

[Out]

int((a + c*x^2)^(5/2)*(d + e*x)^2, x)

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