3.9.63 \(\int (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2} \, dx\) [863]

Optimal. Leaf size=119 \[ -\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e (d+e x)^{3/2}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{3/2}}{35 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e} \]

[Out]

-64/105*d^2*(-c*e^2*x^2+c*d^2)^(3/2)/c/e/(e*x+d)^(3/2)-16/35*d*(-c*e^2*x^2+c*d^2)^(3/2)/c/e/(e*x+d)^(1/2)-2/7*
(-c*e^2*x^2+c*d^2)^(3/2)*(e*x+d)^(1/2)/c/e

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663} \begin {gather*} -\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e (d+e x)^{3/2}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{3/2}}{35 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

(-64*d^2*(c*d^2 - c*e^2*x^2)^(3/2))/(105*c*e*(d + e*x)^(3/2)) - (16*d*(c*d^2 - c*e^2*x^2)^(3/2))/(35*c*e*Sqrt[
d + e*x]) - (2*Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2))/(7*c*e)

Rule 663

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*(p + 1))), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rule 671

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[2*c*d*(Simplify[m + p]/(c*(m + 2*p + 1))), Int[(d + e*x)^(m - 1)*(a + c*x^
2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p]
, 0]

Rubi steps

\begin {align*} \int (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2} \, dx &=-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e}+\frac {1}{7} (8 d) \int \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2} \, dx\\ &=-\frac {16 d \left (c d^2-c e^2 x^2\right )^{3/2}}{35 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e}+\frac {1}{35} \left (32 d^2\right ) \int \frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}} \, dx\\ &=-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e (d+e x)^{3/2}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{3/2}}{35 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.07, size = 64, normalized size = 0.54 \begin {gather*} \frac {2 \sqrt {c \left (d^2-e^2 x^2\right )} \left (-71 d^3+17 d^2 e x+39 d e^2 x^2+15 e^3 x^3\right )}{105 e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

(2*Sqrt[c*(d^2 - e^2*x^2)]*(-71*d^3 + 17*d^2*e*x + 39*d*e^2*x^2 + 15*e^3*x^3))/(105*e*Sqrt[d + e*x])

________________________________________________________________________________________

Maple [A]
time = 0.54, size = 54, normalized size = 0.45

method	result	size
default	\(-\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (-e x +d \right ) \left (15 e^{2} x^{2}+54 d x e +71 d^{2}\right )}{105 \sqrt {e x +d}\, e}\)	\(54\)
gosper	\(-\frac {2 \left (-e x +d \right ) \left (15 e^{2} x^{2}+54 d x e +71 d^{2}\right ) \sqrt {-x^{2} c \,e^{2}+c \,d^{2}}}{105 \sqrt {e x +d}\, e}\)	\(55\)
risch	\(-\frac {2 \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c \left (-15 e^{3} x^{3}-39 d \,e^{2} x^{2}-17 d^{2} e x +71 d^{3}\right ) \left (-e x +d \right )}{105 \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(-c*e^2*x^2+c*d^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/105/(e*x+d)^(1/2)*(c*(-e^2*x^2+d^2))^(1/2)*(-e*x+d)*(15*e^2*x^2+54*d*e*x+71*d^2)/e

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 69, normalized size = 0.58 \begin {gather*} \frac {2 \, {\left (15 \, \sqrt {c} x^{3} e^{3} + 39 \, \sqrt {c} d x^{2} e^{2} + 17 \, \sqrt {c} d^{2} x e - 71 \, \sqrt {c} d^{3}\right )} {\left (x e + d\right )} \sqrt {-x e + d}}{105 \, {\left (x e^{2} + d e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*sqrt(c)*x^3*e^3 + 39*sqrt(c)*d*x^2*e^2 + 17*sqrt(c)*d^2*x*e - 71*sqrt(c)*d^3)*(x*e + d)*sqrt(-x*e +
d)/(x*e^2 + d*e)

________________________________________________________________________________________

Fricas [A]
time = 3.74, size = 66, normalized size = 0.55 \begin {gather*} \frac {2 \, {\left (15 \, x^{3} e^{3} + 39 \, d x^{2} e^{2} + 17 \, d^{2} x e - 71 \, d^{3}\right )} \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d}}{105 \, {\left (x e^{2} + d e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*x^3*e^3 + 39*d*x^2*e^2 + 17*d^2*x*e - 71*d^3)*sqrt(-c*x^2*e^2 + c*d^2)*sqrt(x*e + d)/(x*e^2 + d*e)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- c \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(-c*e**2*x**2+c*d**2)**(1/2),x)

[Out]

Integral(sqrt(-c*(-d + e*x)*(d + e*x))*(d + e*x)**(3/2), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (98) = 196\).
time = 1.17, size = 237, normalized size = 1.99 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left (2 \, \sqrt {2} \sqrt {c d} d - \frac {{\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}}{c}\right )} d^{2} - 14 \, {\left (2 \, \sqrt {2} \sqrt {c d} d^{2} + \frac {5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c d - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{c^{2}}\right )} d + {\left (22 \, \sqrt {2} \sqrt {c d} d^{3} e^{\left (-2\right )} - \frac {{\left (35 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{2} d^{2} - 42 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c d - 15 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}\right )} e^{\left (-2\right )}}{c^{3}}\right )} e^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(35*(2*sqrt(2)*sqrt(c*d)*d - (-(x*e + d)*c + 2*c*d)^(3/2)/c)*d^2 - 14*(2*sqrt(2)*sqrt(c*d)*d^2 + (5*(-(x
*e + d)*c + 2*c*d)^(3/2)*c*d - 3*((x*e + d)*c - 2*c*d)^2*sqrt(-(x*e + d)*c + 2*c*d))/c^2)*d + (22*sqrt(2)*sqrt
(c*d)*d^3*e^(-2) - (35*(-(x*e + d)*c + 2*c*d)^(3/2)*c^2*d^2 - 42*((x*e + d)*c - 2*c*d)^2*sqrt(-(x*e + d)*c + 2
*c*d)*c*d - 15*((x*e + d)*c - 2*c*d)^3*sqrt(-(x*e + d)*c + 2*c*d))*e^(-2)/c^3)*e^2)*e^(-1)

________________________________________________________________________________________

Mupad [B]
time = 0.55, size = 85, normalized size = 0.71 \begin {gather*} \frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {26\,d\,x^2\,\sqrt {d+e\,x}}{35}-\frac {142\,d^3\,\sqrt {d+e\,x}}{105\,e^2}+\frac {2\,e\,x^3\,\sqrt {d+e\,x}}{7}+\frac {34\,d^2\,x\,\sqrt {d+e\,x}}{105\,e}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d^2 - c*e^2*x^2)^(1/2)*(d + e*x)^(3/2),x)

[Out]

((c*d^2 - c*e^2*x^2)^(1/2)*((26*d*x^2*(d + e*x)^(1/2))/35 - (142*d^3*(d + e*x)^(1/2))/(105*e^2) + (2*e*x^3*(d
+ e*x)^(1/2))/7 + (34*d^2*x*(d + e*x)^(1/2))/(105*e)))/(x + d/e)

________________________________________________________________________________________